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6 Week 5 Introduction to Hypothesis Testing Reading
An introduction to hypothesis testing.
What are you interested in learning about? Perhaps you’d like to know if there is a difference in average final grade between two different versions of a college class? Does the Fort Lewis women’s soccer team score more goals than the national Division II women’s average? Which outdoor sport do Fort Lewis students prefer the most? Do the pine trees on campus differ in mean height from the aspen trees? For all of these questions, we can collect a sample, analyze the data, then make a statistical inference based on the analysis. This means determining whether we have enough evidence to reject our null hypothesis (what was originally assumed to be true, until we prove otherwise). The process is called hypothesis testing .
A really good Khan Academy video to introduce the hypothesis test process: Khan Academy Hypothesis Testing . As you watch, please don’t get caught up in the calculations, as we will use SPSS to do these calculations. We will also use SPSS pvalues, instead of the referenced Ztable, to make statistical decisions.
The SixStep Process
Hypothesis testing requires very specific, detailed steps. Think of it as a mathematical lab report where you have to write out your work in a particular way. There are six steps that we will follow for ALL of the hypothesis tests that we learn this semester.
1. Research Question
All hypothesis tests start with a research question. This is literally a question that includes what you are trying to prove, like the examples earlier: Which outdoor sport do Fort Lewis students prefer the most? Is there sufficient evidence to show that the Fort Lewis women’s soccer team scores more goals than the national Division 2 women’s average?
In this step, besides literally being a question, you’ll want to include:
 mention of your variable(s)
 wording specific to the type of test that you’ll be conducting (mean, mean difference, relationship, pattern)
 specific wording that indicates directionality (are you looking for a ‘difference’, are you looking for something to be ‘more than’ or ‘less than’ something else, or are you comparing one pattern to another?)
Consider this research question: Do the pine trees on campus differ in mean height from the aspen trees?
 The wording of this research question clearly mentions the variables being studied. The independent variable is the type of tree (pine or aspen), and these trees are having their heights compared, so the dependent variable is height.
 ‘Mean’ is mentioned, so this indicates a test with a quantitative dependent variable.
 The question also asks if the tree heights ‘differ’. This specific word indicates that the test being performed is a twotailed (i.e. nondirectional) test. More about the meaning of one/twotailed will come later.
2. Statistical Hypotheses
A statistical hypothesis test has a null hypothesis, the status quo, what we assume to be true. Notation is H 0, read as “H naught”. The alternative hypothesis is what you are trying to prove (mentioned in your research question), H 1 or H A . All hypothesis tests must include a null and an alternative hypothesis. We also note which hypothesis test is being done in this step.
The notation for your statistical hypotheses will vary depending on the type of test that you’re doing. Writing statistical hypotheses is NOT the same as most scientific hypotheses. You are not writing sentences explaining what you think will happen in the study. Here is an example of what statistical hypotheses look like using the research question: Do the pine trees on campus differ in mean height from the aspen trees?
3. Decision Rule
In this step, you state which alpha value you will use, and when appropriate, the directionality, or tail, of the test. You also write a statement: “I will reject the null hypothesis if p < alpha” (insert actual alpha value here). In this introductory class, alpha is the level of significance, how willing we are to make the wrong statistical decision, and it will be set to 0.05 or 0.01.
Example of a Decision Rule:
Let alpha=0.01, twotailed. I will reject the null hypothesis if p<0.01.
4. Assumptions, Analysis and Calculations
Quite a bit goes on in this step. Assumptions for the particular hypothesis test must be done. SPSS will be used to create appropriate graphs, and test output tables. Where appropriate, calculations of the test’s effect size will also be done in this step.
All hypothesis tests have assumptions that we hope to meet. For example, tests with a quantitative dependent variable consider a histogram(s) to check if the distribution is normal, and whether there are any obvious outliers. Each hypothesis test has different assumptions, so it is important to pay attention to the specific test’s requirements.
Required SPSS output will also depend on the test.
5. Statistical Decision
It is in Step 5 that we determine if we have enough statistical evidence to reject our null hypothesis. We will consult the SPSS pvalue and compare to our chosen alpha (from Step 3: Decision Rule).
Put very simply, the p value is the probability that, if the null hypothesis is true, the results from another randomly selected sample will be as extreme or more extreme as the results obtained from the given sample. The p value can also be thought of as the probability that the results (from the sample) that we are seeing are solely due to chance. This concept will be discussed in much further detail in the class notes.
Based on this numerical comparison between the pvalue and alpha, we’ll either reject or retain our null hypothesis. Note: You may NEVER ‘accept’ the null hypothesis. This is because it is impossible to prove a null hypothesis to be true.
Retaining the null means that you just don’t have enough evidence to prove your alternative hypothesis to be true, so you fall back to your null. (You retain the null when p is greater than or equal to alpha.)
Rejecting the null means that you did find enough evidence to prove your alternative hypothesis as true. (You reject the null when p is less than alpha.)
Example of a Statistical Decision:
Retain the null hypothesis, because p=0.12 > alpha=0.01.
The pvalue will come from SPSS output, and the alpha will have already been determined back in Step 3. You must be very careful when you compare the decimal values of the pvalue and alpha. If, for example, you mistakenly think that p=0.12 < alpha=0.01, then you will make the incorrect statistical decision, which will likely lead to an incorrect interpretation of the study’s findings.
6. Interpretation
The interpretation is where you write up your findings. The specifics will vary depending on the type of hypothesis test you performed, but you will always include a plain English, contextual conclusion of what your study found (i.e. what it means to reject or retain the null hypothesis in that particular study). You’ll have statistics that you quote to support your decision. Some of the statistics will need to be written in APA style citation (the American Psychological Association style of citation). For some hypothesis tests, you’ll also include an interpretation of the effect size.
Some hypothesis tests will also require an additional (nonParametric) test after the completion of your original test, if the test’s assumptions have not been met. These tests are also call “PostHoc tests”.
As previously stated, hypothesis testing is a very detailed process. Do not be concerned if you have read through all of the steps above, and have many questions (and are possibly very confused). It will take time, and a lot of practice to learn and apply these steps!
This Reading is just meant as an overview of hypothesis testing. Much more information is forthcoming in the various sets of Notes about the specifics needed in each of these steps. The Hypothesis Test Checklist will be a critical resource for you to refer to during homeworks and tests.
Student Course Learning Objectives
4. Choose, administer and interpret the correct tests based on the situation, including identification of appropriate sampling and potential errors
c. Choose the appropriate hypothesis test given a situation
d. Describe the meaning and uses of alpha and pvalues
e. Write the appropriate null and alternative hypotheses, including whether the alternative should be onesided or twosided
f. Determine and calculate the appropriate test statistic (e.g. ztest, multiple ttests, ChiSquare, ANOVA)
g. Determine and interpret effect sizes.
h. Interpret results of a hypothesis test
 Use technology in the statistical analysis of data
 Communicate in writing the results of statistical analyses of data
Attributions
Adapted from “Week 5 Introduction to Hypothesis Testing Reading” by Sherri Spriggs and Sandi Dang is licensed under CC BYNCSA 4.0 .
Math 132 Introduction to Statistics Readings Copyright © by Sherri Spriggs is licensed under a Creative Commons AttributionNonCommercial 4.0 International License , except where otherwise noted.
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9.1 Null and Alternative Hypotheses
The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.
H 0 , the — null hypothesis: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.
H a —, the alternative hypothesis: a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .
Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.
After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are reject H 0 if the sample information favors the alternative hypothesis or do not reject H 0 or decline to reject H 0 if the sample information is insufficient to reject the null hypothesis.
Mathematical Symbols Used in H 0 and H a :
equal (=)  not equal (≠) greater than (>) less than (<) 
greater than or equal to (≥)  less than (<) 
less than or equal to (≤)  more than (>) 
H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.
Example 9.1
H 0 : No more than 30 percent of the registered voters in Santa Clara County voted in the primary election. p ≤ 30 H a : More than 30 percent of the registered voters in Santa Clara County voted in the primary election. p > 30
A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25 percent. State the null and alternative hypotheses.
Example 9.2
We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are the following: H 0 : μ = 2.0 H a : μ ≠ 2.0
We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.
 H 0 : μ __ 66
 H a : μ __ 66
Example 9.3
We want to test if college students take fewer than five years to graduate from college, on the average. The null and alternative hypotheses are the following: H 0 : μ ≥ 5 H a : μ < 5
We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.
 H 0 : μ __ 45
 H a : μ __ 45
Example 9.4
An article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third of the students pass. The same article stated that 6.6 percent of U.S. students take advanced placement exams and 4.4 percent pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6 percent. State the null and alternative hypotheses. H 0 : p ≤ 0.066 H a : p > 0.066
On a state driver’s test, about 40 percent pass the test on the first try. We want to test if more than 40 percent pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.
 H 0 : p __ 0.40
 H a : p __ 0.40
Collaborative Exercise
Bring to class a newspaper, some news magazines, and some internet articles. In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.
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 Null and Alternative Hypotheses  Definitions & Examples
Null and Alternative Hypotheses  Definitions & Examples
Published on 5 October 2022 by Shaun Turney . Revised on 6 December 2022.
The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test :
 Null hypothesis (H 0 ): There’s no effect in the population .
 Alternative hypothesis (H A ): There’s an effect in the population.
The effect is usually the effect of the independent variable on the dependent variable .
Table of contents
Answering your research question with hypotheses, what is a null hypothesis, what is an alternative hypothesis, differences between null and alternative hypotheses, how to write null and alternative hypotheses, frequently asked questions about null and alternative hypotheses.
The null and alternative hypotheses offer competing answers to your research question . When the research question asks “Does the independent variable affect the dependent variable?”, the null hypothesis (H 0 ) answers “No, there’s no effect in the population.” On the other hand, the alternative hypothesis (H A ) answers “Yes, there is an effect in the population.”
The null and alternative are always claims about the population. That’s because the goal of hypothesis testing is to make inferences about a population based on a sample . Often, we infer whether there’s an effect in the population by looking at differences between groups or relationships between variables in the sample.
You can use a statistical test to decide whether the evidence favors the null or alternative hypothesis. Each type of statistical test comes with a specific way of phrasing the null and alternative hypothesis. However, the hypotheses can also be phrased in a general way that applies to any test.
The null hypothesis is the claim that there’s no effect in the population.
If the sample provides enough evidence against the claim that there’s no effect in the population ( p ≤ α), then we can reject the null hypothesis . Otherwise, we fail to reject the null hypothesis.
Although “fail to reject” may sound awkward, it’s the only wording that statisticians accept. Be careful not to say you “prove” or “accept” the null hypothesis.
Null hypotheses often include phrases such as “no effect”, “no difference”, or “no relationship”. When written in mathematical terms, they always include an equality (usually =, but sometimes ≥ or ≤).
Examples of null hypotheses
The table below gives examples of research questions and null hypotheses. There’s always more than one way to answer a research question, but these null hypotheses can help you get started.
( )  
Does tooth flossing affect the number of cavities?  Tooth flossing has on the number of cavities.  test: The mean number of cavities per person does not differ between the flossing group (µ ) and the nonflossing group (µ ) in the population; µ = µ . 
Does the amount of text highlighted in the textbook affect exam scores?  The amount of text highlighted in the textbook has on exam scores.  : There is no relationship between the amount of text highlighted and exam scores in the population; β = 0. 
Does daily meditation decrease the incidence of depression?  Daily meditation the incidence of depression.*  test: The proportion of people with depression in the dailymeditation group ( ) is greater than or equal to the nomeditation group ( ) in the population; ≥ . 
*Note that some researchers prefer to always write the null hypothesis in terms of “no effect” and “=”. It would be fine to say that daily meditation has no effect on the incidence of depression and p 1 = p 2 .
The alternative hypothesis (H A ) is the other answer to your research question . It claims that there’s an effect in the population.
Often, your alternative hypothesis is the same as your research hypothesis. In other words, it’s the claim that you expect or hope will be true.
The alternative hypothesis is the complement to the null hypothesis. Null and alternative hypotheses are exhaustive, meaning that together they cover every possible outcome. They are also mutually exclusive, meaning that only one can be true at a time.
Alternative hypotheses often include phrases such as “an effect”, “a difference”, or “a relationship”. When alternative hypotheses are written in mathematical terms, they always include an inequality (usually ≠, but sometimes > or <). As with null hypotheses, there are many acceptable ways to phrase an alternative hypothesis.
Examples of alternative hypotheses
The table below gives examples of research questions and alternative hypotheses to help you get started with formulating your own.
Does tooth flossing affect the number of cavities?  Tooth flossing has an on the number of cavities.  test: The mean number of cavities per person differs between the flossing group (µ ) and the nonflossing group (µ ) in the population; µ ≠ µ . 
Does the amount of text highlighted in a textbook affect exam scores?  The amount of text highlighted in the textbook has an on exam scores.  : There is a relationship between the amount of text highlighted and exam scores in the population; β ≠ 0. 
Does daily meditation decrease the incidence of depression?  Daily meditation the incidence of depression.  test: The proportion of people with depression in the dailymeditation group ( ) is less than the nomeditation group ( ) in the population; < . 
Null and alternative hypotheses are similar in some ways:
 They’re both answers to the research question
 They both make claims about the population
 They’re both evaluated by statistical tests.
However, there are important differences between the two types of hypotheses, summarized in the following table.
A claim that there is in the population.  A claim that there is in the population.  
 
Equality symbol (=, ≥, or ≤)  Inequality symbol (≠, <, or >)  
Rejected  Supported  
Failed to reject  Not supported 
To help you write your hypotheses, you can use the template sentences below. If you know which statistical test you’re going to use, you can use the testspecific template sentences. Otherwise, you can use the general template sentences.
The only thing you need to know to use these general template sentences are your dependent and independent variables. To write your research question, null hypothesis, and alternative hypothesis, fill in the following sentences with your variables:
Does independent variable affect dependent variable ?
 Null hypothesis (H 0 ): Independent variable does not affect dependent variable .
 Alternative hypothesis (H A ): Independent variable affects dependent variable .
Testspecific
Once you know the statistical test you’ll be using, you can write your hypotheses in a more precise and mathematical way specific to the test you chose. The table below provides template sentences for common statistical tests.
( )  
test
with two groups  The mean dependent variable does not differ between group 1 (µ ) and group 2 (µ ) in the population; µ = µ .  The mean dependent variable differs between group 1 (µ ) and group 2 (µ ) in the population; µ ≠ µ . 
with three groups  The mean dependent variable does not differ between group 1 (µ ), group 2 (µ ), and group 3 (µ ) in the population; µ = µ = µ .  The mean dependent variable of group 1 (µ ), group 2 (µ ), and group 3 (µ ) are not all equal in the population. 
There is no correlation between independent variable and dependent variable in the population; ρ = 0.  There is a correlation between independent variable and dependent variable in the population; ρ ≠ 0.  
There is no relationship between independent variable and dependent variable in the population; β = 0.  There is a relationship between independent variable and dependent variable in the population; β ≠ 0.  
Twoproportions test  The dependent variable expressed as a proportion does not differ between group 1 ( ) and group 2 ( ) in the population; = .  The dependent variable expressed as a proportion differs between group 1 ( ) and group 2 ( ) in the population; ≠ . 
Note: The template sentences above assume that you’re performing onetailed tests . Onetailed tests are appropriate for most studies.
The null hypothesis is often abbreviated as H 0 . When the null hypothesis is written using mathematical symbols, it always includes an equality symbol (usually =, but sometimes ≥ or ≤).
The alternative hypothesis is often abbreviated as H a or H 1 . When the alternative hypothesis is written using mathematical symbols, it always includes an inequality symbol (usually ≠, but sometimes < or >).
A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (‘ x affects y because …’).
A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses. In a welldesigned study , the statistical hypotheses correspond logically to the research hypothesis.
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Oneway ANOVA  When and How to Use It (With Examples)
Published on March 6, 2020 by Rebecca Bevans . Revised on May 10, 2024.
ANOVA , which stands for Analysis of Variance, is a statistical test used to analyze the difference between the means of more than two groups.
A oneway ANOVA uses one independent variable , while a twoway ANOVA uses two independent variables.
Table of contents
When to use a oneway anova, how does an anova test work, assumptions of anova, performing a oneway anova, interpreting the results, posthoc testing, reporting the results of anova, other interesting articles, frequently asked questions about oneway anova.
Use a oneway ANOVA when you have collected data about one categorical independent variable and one quantitative dependent variable . The independent variable should have at least three levels (i.e. at least three different groups or categories).
ANOVA tells you if the dependent variable changes according to the level of the independent variable. For example:
 Your independent variable is social media use , and you assign groups to low , medium , and high levels of social media use to find out if there is a difference in hours of sleep per night .
 Your independent variable is brand of soda , and you collect data on Coke , Pepsi , Sprite , and Fanta to find out if there is a difference in the price per 100ml .
 You independent variable is type of fertilizer , and you treat crop fields with mixtures 1 , 2 and 3 to find out if there is a difference in crop yield .
The null hypothesis ( H 0 ) of ANOVA is that there is no difference among group means. The alternative hypothesis ( H a ) is that at least one group differs significantly from the overall mean of the dependent variable.
If you only want to compare two groups, use a t test instead.
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ANOVA determines whether the groups created by the levels of the independent variable are statistically different by calculating whether the means of the treatment levels are different from the overall mean of the dependent variable.
If any of the group means is significantly different from the overall mean, then the null hypothesis is rejected.
ANOVA uses the F test for statistical significance . This allows for comparison of multiple means at once, because the error is calculated for the whole set of comparisons rather than for each individual twoway comparison (which would happen with a t test).
The F test compares the variance in each group mean from the overall group variance. If the variance within groups is smaller than the variance between groups , the F test will find a higher F value, and therefore a higher likelihood that the difference observed is real and not due to chance.
The assumptions of the ANOVA test are the same as the general assumptions for any parametric test:
 Independence of observations : the data were collected using statistically valid sampling methods , and there are no hidden relationships among observations. If your data fail to meet this assumption because you have a confounding variable that you need to control for statistically, use an ANOVA with blocking variables.
 Normallydistributed response variable : The values of the dependent variable follow a normal distribution .
 Homogeneity of variance : The variation within each group being compared is similar for every group. If the variances are different among the groups, then ANOVA probably isn’t the right fit for the data.
While you can perform an ANOVA by hand , it is difficult to do so with more than a few observations. We will perform our analysis in the R statistical program because it is free, powerful, and widely available. For a full walkthrough of this ANOVA example, see our guide to performing ANOVA in R .
The sample dataset from our imaginary crop yield experiment contains data about:
 fertilizer type (type 1, 2, or 3)
 planting density (1 = low density, 2 = high density)
 planting location in the field (blocks 1, 2, 3, or 4)
 final crop yield (in bushels per acre).
This gives us enough information to run various different ANOVA tests and see which model is the best fit for the data.
For the oneway ANOVA, we will only analyze the effect of fertilizer type on crop yield.
Sample dataset for ANOVA
After loading the dataset into our R environment, we can use the command aov() to run an ANOVA. In this example we will model the differences in the mean of the response variable , crop yield, as a function of type of fertilizer.
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To view the summary of a statistical model in R, use the summary() function.
The summary of an ANOVA test (in R) looks like this:
The ANOVA output provides an estimate of how much variation in the dependent variable that can be explained by the independent variable.
 The first column lists the independent variable along with the model residuals (aka the model error).
 The Df column displays the degrees of freedom for the independent variable (calculated by taking the number of levels within the variable and subtracting 1), and the degrees of freedom for the residuals (calculated by taking the total number of observations minus 1, then subtracting the number of levels in each of the independent variables).
 The Sum Sq column displays the sum of squares (a.k.a. the total variation) between the group means and the overall mean explained by that variable. The sum of squares for the fertilizer variable is 6.07, while the sum of squares of the residuals is 35.89.
 The Mean Sq column is the mean of the sum of squares, which is calculated by dividing the sum of squares by the degrees of freedom.
 The F value column is the test statistic from the F test: the mean square of each independent variable divided by the mean square of the residuals. The larger the F value, the more likely it is that the variation associated with the independent variable is real and not due to chance.
 The Pr(>F) column is the p value of the F statistic. This shows how likely it is that the F value calculated from the test would have occurred if the null hypothesis of no difference among group means were true.
Because the p value of the independent variable, fertilizer, is statistically significant ( p < 0.05), it is likely that fertilizer type does have a significant effect on average crop yield.
ANOVA will tell you if there are differences among the levels of the independent variable, but not which differences are significant. To find how the treatment levels differ from one another, perform a TukeyHSD (Tukey’s HonestlySignificant Difference) posthoc test.
The Tukey test runs pairwise comparisons among each of the groups, and uses a conservative error estimate to find the groups which are statistically different from one another.
The output of the TukeyHSD looks like this:
First, the table reports the model being tested (‘Fit’). Next it lists the pairwise differences among groups for the independent variable.
Under the ‘$fertilizer’ section, we see the mean difference between each fertilizer treatment (‘diff’), the lower and upper bounds of the 95% confidence interval (‘lwr’ and ‘upr’), and the p value , adjusted for multiple pairwise comparisons.
The pairwise comparisons show that fertilizer type 3 has a significantly higher mean yield than both fertilizer 2 and fertilizer 1, but the difference between the mean yields of fertilizers 2 and 1 is not statistically significant.
When reporting the results of an ANOVA, include a brief description of the variables you tested, the F value, degrees of freedom, and p values for each independent variable, and explain what the results mean.
If you want to provide more detailed information about the differences found in your test, you can also include a graph of the ANOVA results , with grouping letters above each level of the independent variable to show which groups are statistically different from one another:
If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.
 Chi square test of independence
 Statistical power
 Descriptive statistics
 Degrees of freedom
 Pearson correlation
 Null hypothesis
Methodology
 Doubleblind study
 Casecontrol study
 Research ethics
 Data collection
 Hypothesis testing
 Structured interviews
Research bias
 Hawthorne effect
 Unconscious bias
 Recall bias
 Halo effect
 Selfserving bias
 Information bias
The only difference between oneway and twoway ANOVA is the number of independent variables . A oneway ANOVA has one independent variable, while a twoway ANOVA has two.
 Oneway ANOVA : Testing the relationship between shoe brand (Nike, Adidas, Saucony, Hoka) and race finish times in a marathon.
 Twoway ANOVA : Testing the relationship between shoe brand (Nike, Adidas, Saucony, Hoka), runner age group (junior, senior, master’s), and race finishing times in a marathon.
All ANOVAs are designed to test for differences among three or more groups. If you are only testing for a difference between two groups, use a ttest instead.
A factorial ANOVA is any ANOVA that uses more than one categorical independent variable . A twoway ANOVA is a type of factorial ANOVA.
Some examples of factorial ANOVAs include:
 Testing the combined effects of vaccination (vaccinated or not vaccinated) and health status (healthy or preexisting condition) on the rate of flu infection in a population.
 Testing the effects of marital status (married, single, divorced, widowed), job status (employed, selfemployed, unemployed, retired), and family history (no family history, some family history) on the incidence of depression in a population.
 Testing the effects of feed type (type A, B, or C) and barn crowding (not crowded, somewhat crowded, very crowded) on the final weight of chickens in a commercial farming operation.
In ANOVA, the null hypothesis is that there is no difference among group means. If any group differs significantly from the overall group mean, then the ANOVA will report a statistically significant result.
Significant differences among group means are calculated using the F statistic, which is the ratio of the mean sum of squares (the variance explained by the independent variable) to the mean square error (the variance left over).
If the F statistic is higher than the critical value (the value of F that corresponds with your alpha value, usually 0.05), then the difference among groups is deemed statistically significant.
Quantitative variables are any variables where the data represent amounts (e.g. height, weight, or age).
Categorical variables are any variables where the data represent groups. This includes rankings (e.g. finishing places in a race), classifications (e.g. brands of cereal), and binary outcomes (e.g. coin flips).
You need to know what type of variables you are working with to choose the right statistical test for your data and interpret your results .
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Null Hypothesis
Null Hypothesis , often denoted as H 0, is a foundational concept in statistical hypothesis testing. It represents an assumption that no significant difference, effect, or relationship exists between variables within a population. It serves as a baseline assumption, positing no observed change or effect occurring. The null is t he truth or falsity of an idea in analysis.
In this article, we will discuss the null hypothesis in detail, along with some solved examples and questions on the null hypothesis.
Table of Content
What is Null Hypothesis?
Null hypothesis symbol, formula of null hypothesis, types of null hypothesis, null hypothesis examples, principle of null hypothesis, how do you find null hypothesis, null hypothesis in statistics, null hypothesis and alternative hypothesis, null hypothesis and alternative hypothesis examples, null hypothesis – practice problems.
Null Hypothesis in statistical analysis suggests the absence of statistical significance within a specific set of observed data. Hypothesis testing, using sample data, evaluates the validity of this hypothesis. Commonly denoted as H 0 or simply “null,” it plays an important role in quantitative analysis, examining theories related to markets, investment strategies, or economies to determine their validity.
Null Hypothesis Meaning
Null Hypothesis represents a default position, often suggesting no effect or difference, against which researchers compare their experimental results. The Null Hypothesis, often denoted as H 0 asserts a default assumption in statistical analysis. It posits no significant difference or effect, serving as a baseline for comparison in hypothesis testing.
The null Hypothesis is represented as H 0 , the Null Hypothesis symbolizes the absence of a measurable effect or difference in the variables under examination.
Certainly, a simple example would be asserting that the mean score of a group is equal to a specified value like stating that the average IQ of a population is 100.
The Null Hypothesis is typically formulated as a statement of equality or absence of a specific parameter in the population being studied. It provides a clear and testable prediction for comparison with the alternative hypothesis. The formulation of the Null Hypothesis typically follows a concise structure, stating the equality or absence of a specific parameter in the population.
Mean Comparison (Twosample ttest)
H 0 : μ 1 = μ 2
This asserts that there is no significant difference between the means of two populations or groups.
Proportion Comparison
H 0 : p 1 − p 2 = 0
This suggests no significant difference in proportions between two populations or conditions.
Equality in Variance (Ftest in ANOVA)
H 0 : σ 1 = σ 2
This states that there’s no significant difference in variances between groups or populations.
Independence (Chisquare Test of Independence):
H 0 : Variables are independent
This asserts that there’s no association or relationship between categorical variables.
Null Hypotheses vary including simple and composite forms, each tailored to the complexity of the research question. Understanding these types is pivotal for effective hypothesis testing.
Equality Null Hypothesis (Simple Null Hypothesis)
The Equality Null Hypothesis, also known as the Simple Null Hypothesis, is a fundamental concept in statistical hypothesis testing that assumes no difference, effect or relationship between groups, conditions or populations being compared.
NonInferiority Null Hypothesis
In some studies, the focus might be on demonstrating that a new treatment or method is not significantly worse than the standard or existing one.
Superiority Null Hypothesis
The concept of a superiority null hypothesis comes into play when a study aims to demonstrate that a new treatment, method, or intervention is significantly better than an existing or standard one.
Independence Null Hypothesis
In certain statistical tests, such as chisquare tests for independence, the null hypothesis assumes no association or independence between categorical variables.
Homogeneity Null Hypothesis
In tests like ANOVA (Analysis of Variance), the null hypothesis suggests that there’s no difference in population means across different groups.
 Medicine: Null Hypothesis: “No significant difference exists in blood pressure levels between patients given the experimental drug versus those given a placebo.”
 Education: Null Hypothesis: “There’s no significant variation in test scores between students using a new teaching method and those using traditional teaching.”
 Economics: Null Hypothesis: “There’s no significant change in consumer spending pre and postimplementation of a new taxation policy.”
 Environmental Science: Null Hypothesis: “There’s no substantial difference in pollution levels before and after a water treatment plant’s establishment.”
The principle of the null hypothesis is a fundamental concept in statistical hypothesis testing. It involves making an assumption about the population parameter or the absence of an effect or relationship between variables.
In essence, the null hypothesis (H 0 ) proposes that there is no significant difference, effect, or relationship between variables. It serves as a starting point or a default assumption that there is no real change, no effect or no difference between groups or conditions.
The null hypothesis is usually formulated to be tested against an alternative hypothesis (H 1 or H [Tex]\alpha [/Tex] ) which suggests that there is an effect, difference or relationship present in the population.
Null Hypothesis Rejection
Rejecting the Null Hypothesis occurs when statistical evidence suggests a significant departure from the assumed baseline. It implies that there is enough evidence to support the alternative hypothesis, indicating a meaningful effect or difference. Null Hypothesis rejection occurs when statistical evidence suggests a deviation from the assumed baseline, prompting a reconsideration of the initial hypothesis.
Identifying the Null Hypothesis involves defining the status quotient, asserting no effect and formulating a statement suitable for statistical analysis.
When is Null Hypothesis Rejected?
The Null Hypothesis is rejected when statistical tests indicate a significant departure from the expected outcome, leading to the consideration of alternative hypotheses. It occurs when statistical evidence suggests a deviation from the assumed baseline, prompting a reconsideration of the initial hypothesis.
In statistical hypothesis testing, researchers begin by stating the null hypothesis, often based on theoretical considerations or previous research. The null hypothesis is then tested against an alternative hypothesis (Ha), which represents the researcher’s claim or the hypothesis they seek to support.
The process of hypothesis testing involves collecting sample data and using statistical methods to assess the likelihood of observing the data if the null hypothesis were true. This assessment is typically done by calculating a test statistic, which measures the difference between the observed data and what would be expected under the null hypothesis.
In the realm of hypothesis testing, the null hypothesis (H 0 ) and alternative hypothesis (H₁ or Ha) play critical roles. The null hypothesis generally assumes no difference, effect, or relationship between variables, suggesting that any observed change or effect is due to random chance. Its counterpart, the alternative hypothesis, asserts the presence of a significant difference, effect, or relationship between variables, challenging the null hypothesis. These hypotheses are formulated based on the research question and guide statistical analyses.
Difference Between Null Hypothesis and Alternative Hypothesis
The null hypothesis (H 0 ) serves as the baseline assumption in statistical testing, suggesting no significant effect, relationship, or difference within the data. It often proposes that any observed change or correlation is merely due to chance or random variation. Conversely, the alternative hypothesis (H 1 or Ha) contradicts the null hypothesis, positing the existence of a genuine effect, relationship or difference in the data. It represents the researcher’s intended focus, seeking to provide evidence against the null hypothesis and support for a specific outcome or theory. These hypotheses form the crux of hypothesis testing, guiding the assessment of data to draw conclusions about the population being studied.
Criteria  Null Hypothesis  Alternative Hypothesis 

Definition  Assumes no effect or difference  Asserts a specific effect or difference 
Symbol  H  H (or Ha) 
Formulation  States equality or absence of parameter  States a specific value or relationship 
Testing Outcome  Rejected if evidence of a significant effect  Accepted if evidence supports the hypothesis 
Let’s envision a scenario where a researcher aims to examine the impact of a new medication on reducing blood pressure among patients. In this context:
Null Hypothesis (H 0 ): “The new medication does not produce a significant effect in reducing blood pressure levels among patients.”
Alternative Hypothesis (H 1 or Ha): “The new medication yields a significant effect in reducing blood pressure levels among patients.”
The null hypothesis implies that any observed alterations in blood pressure subsequent to the medication’s administration are a result of random fluctuations rather than a consequence of the medication itself. Conversely, the alternative hypothesis contends that the medication does indeed generate a meaningful alteration in blood pressure levels, distinct from what might naturally occur or by random chance.
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Example 1: A researcher claims that the average time students spend on homework is 2 hours per night.
Null Hypothesis (H 0 ): The average time students spend on homework is equal to 2 hours per night. Data: A random sample of 30 students has an average homework time of 1.8 hours with a standard deviation of 0.5 hours. Test Statistic and Decision: Using a ttest, if the calculated tstatistic falls within the acceptance region, we fail to reject the null hypothesis. If it falls in the rejection region, we reject the null hypothesis. Conclusion: Based on the statistical analysis, we fail to reject the null hypothesis, suggesting that there is not enough evidence to dispute the claim of the average homework time being 2 hours per night.
Example 2: A company asserts that the error rate in its production process is less than 1%.
Null Hypothesis (H 0 ): The error rate in the production process is 1% or higher. Data: A sample of 500 products shows an error rate of 0.8%. Test Statistic and Decision: Using a ztest, if the calculated zstatistic falls within the acceptance region, we fail to reject the null hypothesis. If it falls in the rejection region, we reject the null hypothesis. Conclusion: The statistical analysis supports rejecting the null hypothesis, indicating that there is enough evidence to dispute the company’s claim of an error rate of 1% or higher.
Q1. A researcher claims that the average time spent by students on homework is less than 2 hours per day. Formulate the null hypothesis for this claim?
Q2. A manufacturing company states that their new machine produces widgets with a defect rate of less than 5%. Write the null hypothesis to test this claim?
Q3. An educational institute believes that their online course completion rate is at least 60%. Develop the null hypothesis to validate this assertion?
Q4. A restaurant claims that the waiting time for customers during peak hours is not more than 15 minutes. Formulate the null hypothesis for this claim?
Q5. A study suggests that the mean weight loss after following a specific diet plan for a month is more than 8 pounds. Construct the null hypothesis to evaluate this statement?
Summary – Null Hypothesis and Alternative Hypothesis
The null hypothesis (H 0 ) and alternative hypothesis (H a ) are fundamental concepts in statistical hypothesis testing. The null hypothesis represents the default assumption, stating that there is no significant effect, difference, or relationship between variables. It serves as the baseline against which the alternative hypothesis is tested. In contrast, the alternative hypothesis represents the researcher’s hypothesis or the claim to be tested, suggesting that there is a significant effect, difference, or relationship between variables. The relationship between the null and alternative hypotheses is such that they are complementary, and statistical tests are conducted to determine whether the evidence from the data is strong enough to reject the null hypothesis in favor of the alternative hypothesis. This decision is based on the strength of the evidence and the chosen level of significance. Ultimately, the choice between the null and alternative hypotheses depends on the specific research question and the direction of the effect being investigated.
FAQs on Null Hypothesis
What does null hypothesis stands for.
The null hypothesis, denoted as H 0 , is a fundamental concept in statistics used for hypothesis testing. It represents the statement that there is no effect or no difference, and it is the hypothesis that the researcher typically aims to provide evidence against.
How to Form a Null Hypothesis?
A null hypothesis is formed based on the assumption that there is no significant difference or effect between the groups being compared or no association between variables being tested. It often involves stating that there is no relationship, no change, or no effect in the population being studied.
When Do we reject the Null Hypothesis?
In statistical hypothesis testing, if the pvalue (the probability of obtaining the observed results) is lower than the chosen significance level (commonly 0.05), we reject the null hypothesis. This suggests that the data provides enough evidence to refute the assumption made in the null hypothesis.
What is a Null Hypothesis in Research?
In research, the null hypothesis represents the default assumption or position that there is no significant difference or effect. Researchers often try to test this hypothesis by collecting data and performing statistical analyses to see if the observed results contradict the assumption.
What Are Alternative and Null Hypotheses?
The null hypothesis (H0) is the default assumption that there is no significant difference or effect. The alternative hypothesis (H1 or Ha) is the opposite, suggesting there is a significant difference, effect or relationship.
What Does it Mean to Reject the Null Hypothesis?
Rejecting the null hypothesis implies that there is enough evidence in the data to support the alternative hypothesis. In simpler terms, it suggests that there might be a significant difference, effect or relationship between the groups or variables being studied.
How to Find Null Hypothesis?
Formulating a null hypothesis often involves considering the research question and assuming that no difference or effect exists. It should be a statement that can be tested through data collection and statistical analysis, typically stating no relationship or no change between variables or groups.
How is Null Hypothesis denoted?
The null hypothesis is commonly symbolized as H 0 in statistical notation.
What is the Purpose of the Null hypothesis in Statistical Analysis?
The null hypothesis serves as a starting point for hypothesis testing, enabling researchers to assess if there’s enough evidence to reject it in favor of an alternative hypothesis.
What happens if we Reject the Null hypothesis?
Rejecting the null hypothesis implies that there is sufficient evidence to support an alternative hypothesis, suggesting a significant effect or relationship between variables.
What are Test for Null Hypothesis?
Various statistical tests, such as ttests or chisquare tests, are employed to evaluate the validity of the Null Hypothesis in different scenarios.
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Hypothesis Testing  Analysis of Variance (ANOVA)
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The ANOVA Approach
Test statistic for anova.
All Modules
Table of FStatistic Values
Consider an example with four independent groups and a continuous outcome measure. The independent groups might be defined by a particular characteristic of the participants such as BMI (e.g., underweight, normal weight, overweight, obese) or by the investigator (e.g., randomizing participants to one of four competing treatments, call them A, B, C and D). Suppose that the outcome is systolic blood pressure, and we wish to test whether there is a statistically significant difference in mean systolic blood pressures among the four groups. The sample data are organized as follows:






 n  n  n  n 





 s  s  s  s 
The hypotheses of interest in an ANOVA are as follows:
 H 0 : μ 1 = μ 2 = μ 3 ... = μ k
 H 1 : Means are not all equal.
where k = the number of independent comparison groups.
In this example, the hypotheses are:
 H 0 : μ 1 = μ 2 = μ 3 = μ 4
 H 1 : The means are not all equal.
The null hypothesis in ANOVA is always that there is no difference in means. The research or alternative hypothesis is always that the means are not all equal and is usually written in words rather than in mathematical symbols. The research hypothesis captures any difference in means and includes, for example, the situation where all four means are unequal, where one is different from the other three, where two are different, and so on. The alternative hypothesis, as shown above, capture all possible situations other than equality of all means specified in the null hypothesis.
The test statistic for testing H 0 : μ 1 = μ 2 = ... = μ k is:
and the critical value is found in a table of probability values for the F distribution with (degrees of freedom) df 1 = k1, df 2 =Nk. The table can be found in "Other Resources" on the left side of the pages.
NOTE: The test statistic F assumes equal variability in the k populations (i.e., the population variances are equal, or s 1 2 = s 2 2 = ... = s k 2 ). This means that the outcome is equally variable in each of the comparison populations. This assumption is the same as that assumed for appropriate use of the test statistic to test equality of two independent means. It is possible to assess the likelihood that the assumption of equal variances is true and the test can be conducted in most statistical computing packages. If the variability in the k comparison groups is not similar, then alternative techniques must be used.
The F statistic is computed by taking the ratio of what is called the "between treatment" variability to the "residual or error" variability. This is where the name of the procedure originates. In analysis of variance we are testing for a difference in means (H 0 : means are all equal versus H 1 : means are not all equal) by evaluating variability in the data. The numerator captures between treatment variability (i.e., differences among the sample means) and the denominator contains an estimate of the variability in the outcome. The test statistic is a measure that allows us to assess whether the differences among the sample means (numerator) are more than would be expected by chance if the null hypothesis is true. Recall in the two independent sample test, the test statistic was computed by taking the ratio of the difference in sample means (numerator) to the variability in the outcome (estimated by Sp).
The decision rule for the F test in ANOVA is set up in a similar way to decision rules we established for t tests. The decision rule again depends on the level of significance and the degrees of freedom. The F statistic has two degrees of freedom. These are denoted df 1 and df 2 , and called the numerator and denominator degrees of freedom, respectively. The degrees of freedom are defined as follows:
df 1 = k1 and df 2 =Nk,
where k is the number of comparison groups and N is the total number of observations in the analysis. If the null hypothesis is true, the between treatment variation (numerator) will not exceed the residual or error variation (denominator) and the F statistic will small. If the null hypothesis is false, then the F statistic will be large. The rejection region for the F test is always in the upper (righthand) tail of the distribution as shown below.
Rejection Region for F Test with a =0.05, df 1 =3 and df 2 =36 (k=4, N=40)
For the scenario depicted here, the decision rule is: Reject H 0 if F > 2.87.
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10.2  a statistical test for oneway anova.
Before we go into the details of the test, we need to determine the null and alternative hypotheses. Recall that for a test for two independent means, the null hypothesis was \(\mu_1=\mu_2\). In oneway ANOVA, we want to compare \(t\) population means, where \(t>2\). Therefore, the null hypothesis for analysis of variance for \(t\) population means is:
\(H_0\colon \mu_1=\mu_2=...\mu_t\)
The alternative, however, cannot be set up similarly to the twosample case. If we wanted to see if two population means are different, the alternative would be \(\mu_1\ne\mu_2\). With more than two groups, the research question is “Are some of the means different?." If we set up the alternative to be \(\mu_1\ne\mu_2\ne…\ne\mu_t\), then we would have a test to see if ALL the means are different. This is not what we want. We need to be careful how we set up the alternative. The mathematical version of the alternative is...
\(H_a\colon \mu_i\ne\mu_j\text{ for some }i \text{ and }j \text{ where }i\ne j\)
This means that at least one of the pairs is not equal. The more common presentation of the alternative is:
\(H_a\colon \text{ at least one mean is different}\) or \(H_a\colon \text{ not all the means are equal}\)
Recall that when we compare the means of two populations for independent samples, we use a 2sample t test with pooled variance when the population variances can be assumed equal.
For more than two populations, the test statistic, \(F\), is the ratio of between group sample variance and the withingroupsample variance. That is,
\(F=\dfrac{\text{between group variance}}{\text{within group variance}}\)
Under the null hypothesis (and with certain assumptions), both quantities estimate the variance of the random error, and thus the ratio should be close to 1. If the ratio is large, then we have evidence against the null, and hence, we would reject the null hypothesis.
In the next section, we present the assumptions for this test. In the following section, we present how to find the between group variance, the within group variance, and the Fstatistic in the ANOVA table.
Hypothesis Testing and ANOVA
 First Online: 20 July 2018
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 Marko Sarstedt 3 &
 Erik Mooi 4
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We first describe the essentials of hypothesis testing and how testing helps make critical business decisions of statistical and practical significance. Without using difficult mathematical formulas, we discuss the steps involved in hypothesis testing, the types of errors that may occur, and provide strategies on how to best deal with these errors. We also discuss common types of test statistics and explain how to determine which type you should use in which specific situation. We explain that the test selection depends on the testing situation, the nature of the samples, the choice of test, and the region of rejection. Drawing on a case study, we show how to link hypothesis testing logic to empirics in SPSS. The case study touches upon different test situations and helps you interpret the tables and graphics in a quick and meaningful way.
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In experimental studies, if respondents were paired with others (as in a matched case control sample), each person would be sampled once, but it still would be a paired sample.
The exact calculation of this test is shown on https://www.ibm.com/support/knowledgecenter/en/SSLVMB_20.0.0/com.ibm.spss.statistics.help/alg_npar_tests_mannwhitney.htm
The fundamental difference between the z  and t distributions is that the t distribution is dependent on sample size n (which the z distribution is not). The distributions become more similar with larger values of n .
To obtain the critical value, you can also use the TINV function provided in Microsoft Excel, whose general form is “TINV( α, df ).” Here, α represents the desired Type I error rate and df the degrees of freedom. To carry out this computation, open a new Excel spreadsheet and type in “ = TINV(2*0.025,9).” Note that we have to specify “2*0.025” (or, directly 0.05) under α , because we are applying a twotailed instead of a onetailed test.
Unfortunately, there is some confusion about the difference between the α and p value. See Hubbard and Bayarri ( 2003 ) for a discussion.
Note that this is convention and most textbooks discuss hypothesis testing in this way. Originally, two testing procedures were developed, one by Neyman and Pearson and another by Fisher (for more details, see Lehmann 1993 ). Agresti and Finlay ( 2014 ) explain the differences between the convention and the two original procedures.
Note that this rule doesn't always apply such as for exact tests of probabilities.
We don’t have to conduct manual calculations and tables when working with SPSS. However, we can calculate the p value using the TDIST function in Microsoft Excel. The function has the general form “TDIST( t, df , tails)”, where t describes the test value, df the degrees of freedom, and tails specifies whether it’s a onetailed test (tails = 1) or twotailed test (tails = 2). Just open a new spreadsheet for our example and type in “ = TDIST(2.274,9,1)”. Likewise, there are several webpages with Javabased modules (e.g., https://graphpad.com/quickcalcs/pvalue1.cfm ) that calculate p values and test statistic values.
The number of pairwise comparisons is calculated as follows : k· ( k − 1)/2, with k the number of groups to compare.
In fact, these two assumptions are interrelated, since unequal group sample sizes result in a greater probability that we will violate the homogeneity assumption.
SS is an abbreviation of “sum of squares,” because the variation is calculated using the squared differences between different types of values.
Note that the groupspecific sample size in this example is too small to draw conclusions and is only used to show the calculation of the statistics.
Note that when initiating the analysis by going to ► Analyze ► General Linear Model ► Univariate, we can request these statistics under Options ( Estimates of effect size ).
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Kanji, G. K. (2006). 100 statistical tests (3rd ed.). London: Sage.
Van Belle, G. (2011). Statistical rules of thumb (2nd ed.). Hoboken, N.J.: John Wiley & Sons.
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Hypothesis testing involves the careful construction of two statements: the null hypothesis and the alternative hypothesis. These hypotheses can look very similar but are actually different.
How do we know which hypothesis is the null and which one is the alternative? We will see that there are a few ways to tell the difference.
The Null Hypothesis
The null hypothesis reflects that there will be no observed effect in our experiment. In a mathematical formulation of the null hypothesis, there will typically be an equal sign. This hypothesis is denoted by H 0 .
The null hypothesis is what we attempt to find evidence against in our hypothesis test. We hope to obtain a small enough pvalue that it is lower than our level of significance alpha and we are justified in rejecting the null hypothesis. If our pvalue is greater than alpha, then we fail to reject the null hypothesis.
If the null hypothesis is not rejected, then we must be careful to say what this means. The thinking on this is similar to a legal verdict. Just because a person has been declared "not guilty", it does not mean that he is innocent. In the same way, just because we failed to reject a null hypothesis it does not mean that the statement is true.
For example, we may want to investigate the claim that despite what convention has told us, the mean adult body temperature is not the accepted value of 98.6 degrees Fahrenheit . The null hypothesis for an experiment to investigate this is “The mean adult body temperature for healthy individuals is 98.6 degrees Fahrenheit.” If we fail to reject the null hypothesis, then our working hypothesis remains that the average adult who is healthy has a temperature of 98.6 degrees. We do not prove that this is true.
If we are studying a new treatment, the null hypothesis is that our treatment will not change our subjects in any meaningful way. In other words, the treatment will not produce any effect in our subjects.
The Alternative Hypothesis
The alternative or experimental hypothesis reflects that there will be an observed effect for our experiment. In a mathematical formulation of the alternative hypothesis, there will typically be an inequality, or not equal to symbol. This hypothesis is denoted by either H a or by H 1 .
The alternative hypothesis is what we are attempting to demonstrate in an indirect way by the use of our hypothesis test. If the null hypothesis is rejected, then we accept the alternative hypothesis. If the null hypothesis is not rejected, then we do not accept the alternative hypothesis. Going back to the above example of mean human body temperature, the alternative hypothesis is “The average adult human body temperature is not 98.6 degrees Fahrenheit.”
If we are studying a new treatment, then the alternative hypothesis is that our treatment does, in fact, change our subjects in a meaningful and measurable way.
The following set of negations may help when you are forming your null and alternative hypotheses. Most technical papers rely on just the first formulation, even though you may see some of the others in a statistics textbook.
 Null hypothesis: “ x is equal to y .” Alternative hypothesis “ x is not equal to y .”
 Null hypothesis: “ x is at least y .” Alternative hypothesis “ x is less than y .”
 Null hypothesis: “ x is at most y .” Alternative hypothesis “ x is greater than y .”
 What 'Fail to Reject' Means in a Hypothesis Test
 Type I and Type II Errors in Statistics
 An Example of a Hypothesis Test
 The Runs Test for Random Sequences
 An Example of ChiSquare Test for a Multinomial Experiment
 The Difference Between Type I and Type II Errors in Hypothesis Testing
 What Level of Alpha Determines Statistical Significance?
 What Is the Difference Between Alpha and PValues?
 What Is ANOVA?
 How to Find Critical Values with a ChiSquare Table
 Example of a Permutation Test
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What is the null hypothesis of a MANOVA?
In order to analyze differences in some continuous variable between different groups (given by a categorical variable), one can perform a oneway ANOVA. If there are several explanatory (categorical) variables, one can perform a factorial ANOVA. If one wants to analyze differences between groups in several continuous variables (i.e., several response variables), one has to perform a multivariate ANOVA (MANOVA).
I hardly understand how one can perform an ANOVAlike test on several response variables and more importantly, I don't understand what the null hypothesis could be. Is the null hypothesis:
 "For each response variable, the means of all groups are equal",
 "For at least one response variable, the means of all groups are equal",
or is $H_0$ something else?
 hypothesistesting
 $\begingroup$ I can't tell, are you also asking how an ANOVA works? In the context of discussing what a standard error is, I essentially explain the basic idea behind an ANOVA here: How does the standard error work? $\endgroup$ – gung  Reinstate Monica Commented Jan 13, 2015 at 21:33
 $\begingroup$ Neither of your two statements. H0 of MANOVA is that there is no difference in multivariate space . The multivariate case is considerably more complex than univariate because we have to deal with covariances, not just variances. There exist several ways to formulate the H0H1 hypotheses in MANOVA. Read Wikipedia. $\endgroup$ – ttnphns Commented Jan 13, 2015 at 21:38
 $\begingroup$ @ttnphns: Why neither? The $H_0$ of ANOVA is that the means of all groups are equal. The $H_0$ of MANOVA is that the multivariate means of all groups are equal. This is exactly alternative 1 in the OP. Covariances etc. enter the assumptions and the computations of MANOVA, not the null hypothesis. $\endgroup$ – amoeba Commented Jan 13, 2015 at 21:41
 $\begingroup$ @amoeba, I didn't like For each response variable . To me it sounds like (or I read it as) "testing is done univarietly on each" (and then somehow combined). $\endgroup$ – ttnphns Commented Jan 13, 2015 at 21:47
2 Answers 2
The null hypothesis $H_0$ of a oneway ANOVA is that the means of all groups are equal: $$H_0: \mu_1 = \mu_2 = ... = \mu_k.$$ The null hypothesis $H_0$ of a oneway MANOVA is that the [multivariate] means of all groups are equal: $$H_0: \boldsymbol \mu_1 = \boldsymbol \mu_2 = ... = \boldsymbol \mu_k.$$ This is equivalent to saying that the means are equal for each response variable, i.e. your first option is correct .
In both cases the alternative hypothesis $H_1$ is the negation of the null. In both cases the assumptions are (a) Gaussian withingroup distributions, and (b) equal variances (for ANOVA) / covariance matrices (for MANOVA) across groups.
Difference between MANOVA and ANOVAs
This might appear a bit confusing: the null hypothesis of MANOVA is exactly the same as the combination of null hypotheses for a collection of univariate ANOVAs, but at the same time we know that doing MANOVA is not equivalent to doing univariate ANOVAs and then somehow "combining" the results (one could come up with various ways of combining). Why not?
The answer is that running all univariate ANOVAs, even though would test the same null hypothesis, will have less power. See my answer here for an illustration: How can MANOVA report a significant difference when none of the univariate ANOVAs reaches significance? Naive method of "combining" (reject the global null if at least one ANOVA rejects the null) would also lead to a huge inflation of type I error rate; but even if one chooses some smart way of "combining" to maintain the correct error rate, one would lose in power.
How the testing works
ANOVA decomposes the total sumofsquares $T$ into betweengroup sumofsquares $B$ and withingroup sumofsquares $W$, so that $T=B+W$. It then computes the ratio $B/W$. Under the null hypothesis, this ratio should be small (around $1$); one can work out the exact distribution of this ratio expected under the null hypothesis (it will depend on $n$ and on the number of groups). Comparing the observed value $B/W$ with this distribution yields a pvalue.
MANOVA decomposes the total scatter matrix $\mathbf T$ into betweengroup scatter matrix $\mathbf B$ and withingroup scatter matrix $\mathbf W$, so that $\mathbf T = \mathbf B + \mathbf W$. It then computes the matrix $\mathbf W^{1} \mathbf B$. Under the null hypothesis, this matrix should be "small" (around $\mathbf{I}$); but how to quantify how "small" it is? MANOVA looks at the eigenvalues $\lambda_i$ of this matrix (they are all positive). Again, under the null hypothesis, these eigenvalues should be "small" (all around $1$). But to compute a pvalue, we need one number (called "statistic") in order to be able to compare it with its expected distribution under the null. There are several ways to do it: take the sum of all eigenvalues $\sum \lambda_i$; take maximal eigenvalue $\max\{\lambda_i\}$, etc. In each case, this number is compared with the distribution of this quantity expected under the null, resulting in a pvalue.
Different choices of the test statistic lead to slightly different pvalues, but it is important to realize that in each case the same null hypothesis is being tested.
 $\begingroup$ Also, if you don't correct for multiple testing, the allunivariateANOVAs approach will yield type I error inflation as well. $\endgroup$ – gung  Reinstate Monica Commented Jan 13, 2015 at 22:20
 1 $\begingroup$ @gung: Yes, that is true as well. However, one can be smarter in "combining" than just rejecting the null as soon as at least one of the ANOVAs rejects the null. My point was that however smart one tries to be in "combining", one will still lose in power as compared to MANOVA (even if one manages to maintain the size of the test without inflating the error rate). $\endgroup$ – amoeba Commented Jan 13, 2015 at 22:24
 $\begingroup$ But isn't now that "power" directly related to the notion of the covariance? The moral is that with a (series of) univariate test we test only for marginal effect which is SSdifference/SSerror scalar. In MANOVA the multivariate effect is SSCPerror^(1)SSCPdifference matrix (covariances total and withingroups accounted for). But since there are several eigenvalues in it which could be "combined" not in a single manner in a test statistic, several possible alternative hypotheses exist. More power  more theoretical complexity. $\endgroup$ – ttnphns Commented Jan 14, 2015 at 7:03
 $\begingroup$ @ttnphns, yes, this is all correct, but I think does not change the fact that the null hypothesis is what I wrote it is (and that's what the question was about). Whatever test statistic is used (Wilks/Roy/PillaiBartlett/LawleyHotelling), they are trying to test the same null hypothesis. I might expand my answer later to discuss this in more detail. $\endgroup$ – amoeba Commented Jan 14, 2015 at 9:18
 1 $\begingroup$ @gung asked me to chime in (not sure why... I taught MANOVA some 7 years ago, and never applied it)  I would say that amoeba is right in saying that $H_1$ is a full negation of the null $H_0: \mu_{\mbox{group }1} = \ldots = \mu_{\mbox{group }k}$, which is a $p$dimensional hyperspace in $kp$ dimensional space of parameters (if $p$ is the dimension that nobody bothered defining so far). And it is option 1 given by the OP. Option 2 is significantly more difficult to test. $\endgroup$ – StasK Commented Jan 20, 2015 at 4:28
It is the former.
However, the way it does it isn't literally to compare the means of each of the original variables in turn. Instead the response variables are linearly transformed in a way that is very similar to principal components analysis . (There is an excellent thread on PCA here: Making sense of principal component analysis, eigenvectors & eigenvalues .) The difference is that PCA orients your axes so as to align with the directions of maximal variation, whereas MANOVA rotates your axes in the directions that maximize the separation of your groups.
To be clear though, none of the tests associated with a MANOVA is testing all the means one after another in a direct sense, either with the means in the original space or in the transformed space. There are several different test statistics that each work in a slightly different way, nonetheless they tend to operate over the eigenvalues of the decomposition that transforms the space. But as far as the nature of the null hypothesis goes, it is that all means of all groups are the same on each response variable, not that they can differ on some variables but are the same on at least one.
 $\begingroup$ Ooh...So Manova makes a linear discriminant analysis (to maximize the distance between the mean of the groups) and then, it runs a standard anova using the first axis as response variable? So, $Ho$ is "the means  in term of PC1  of all groups are the same". Is that right? $\endgroup$ – Remi.b Commented Jan 13, 2015 at 21:19
 $\begingroup$ There are several different possible tests. Testing only the the 1st axis is essentially using Roy's largest root as your test. This will often be the most powerful test, but it is also more limited. I gather there is ongoing discussion over which test is 'best'. $\endgroup$ – gung  Reinstate Monica Commented Jan 13, 2015 at 21:23
 $\begingroup$ I guess we use MANOVA rather than several ANOVAs in order to avoid multiple testing issues. But if, by doing an MANOVA we just make an ANOVA on PC1 of a LDR , then we still have a multiple testing issue to consider when looking at the Pvalue. Is this right? (Hope that makes more sense. I deleted my previous unclear comment) $\endgroup$ – Remi.b Commented Jan 13, 2015 at 21:28
 $\begingroup$ That's an insightful point, but there are two issues: 1) the axes are now orthogonal, & that can change the issues w/ multiple testing; 2) the sampling distributions of the MANOVA test statistics take the multiple axes into account. $\endgroup$ – gung  Reinstate Monica Commented Jan 13, 2015 at 21:32
 1 $\begingroup$ @Remi.b: These are good questions, but just to be clear: MANOVA is not equivalent to a ANOVA on the first discriminant axis of LDA! See here for a the relation between MANOVA and LDA: How is MANOVA related to LDA? $\endgroup$ – amoeba Commented Jan 13, 2015 at 21:34
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Rejecting the null means that you did find enough evidence to prove your alternative hypothesis as true. (You reject the null when p is less than alpha.) Example of a Statistical Decision: Retain the null hypothesis, because p=0.12 > alpha=0.01. The pvalue will come from SPSS output, and the alpha will have already been determined back in Step 3.
Appendix: Comparison of Hypothesis Tests While the only statistical hypothesis test that will be used in this class is Welch's ttest, for completeness this section describes a few alternative tests, and when they are applicable. • Student's ttest: Requires normally distributed sample means, equality of variance. Robust to moderate
To decide if we should reject or fail to reject the null hypothesis, we must refer to the pvalue in the output of the ANOVA table. If the pvalue is less than some significance level (e.g. 0.05) then we can reject the null hypothesis and conclude that not all group means are equal.
The null hypothesis (H0) answers "No, there's no effect in the population.". The alternative hypothesis (Ha) answers "Yes, there is an effect in the population.". The null and alternative are always claims about the population. That's because the goal of hypothesis testing is to make inferences about a population based on a sample.
The null hypothesis in ANOVA is always that there is no difference in means. The research or alternative hypothesis is always that the means are not all equal and is usually written in words rather than in mathematical symbols. The research hypothesis captures any difference in means and includes, for example, the situation where all four means ...
The Null and Alternative Hypothesis •States the assumption (numerical) to be tested •Begin with the assumption that the null hypothesis is TRUE •Always contains the '=' sign The null hypothesis, H 0: The alternative hypothesis, H a: •Is the opposite of the null hypothesis •Challenges the status quo •Never contains just the ...
The actual test begins by considering two hypotheses.They are called the null hypothesis and the alternative hypothesis.These hypotheses contain opposing viewpoints. H 0, the —null hypothesis: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.
The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test: Null hypothesis (H0): There's no effect in the population. Alternative hypothesis (HA): There's an effect in the population. The effect is usually the effect of the independent variable on the dependent ...
To conduct the ANOVA: 1. State the hypotheses: Null Hypothesis (H0): There is no difference in mean stress levels between the three types of exercise. Alternative Hypothesis (H1): There is a difference in mean stress levels between at least two of the types of exercise. 2. Calculate the ANOVA statistics:
The null hypothesis (H 0) of ANOVA is that there is no difference among group means. The alternative hypothesis (H a) is that at least one group differs significantly from the overall mean of the dependent variable. If you only want to compare two groups, use a t test instead.
The alternative hypothesis is: the means are not the same. For the twoway ANOVA, the possible null hypotheses are: There is no difference in the means of factor A. There is no difference in means of factor B. There is no interaction between factors A and B. The alternative hypothesis for cases 1 and 2 is: the means are not equal.
OneWay ANOVA: The Process. A oneway ANOVA uses the following null and alternative hypotheses: H0 (null hypothesis): μ1 = μ2 = μ3 = … = μk (all the population means are equal) H1 (alternative hypothesis): at least one population mean is different from the rest. You will typically use some statistical software (such as R, Excel, Stata ...
The actual test begins by considering two hypotheses.They are called the null hypothesis and the alternative hypothesis.These hypotheses contain opposing viewpoints. \(H_0\): The null hypothesis: It is a statement of no difference between the variables—they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.
In tests like ANOVA (Analysis of Variance), the null hypothesis suggests that there's no difference in population means across different groups. ... Difference Between Null Hypothesis and Alternative Hypothesis. The null hypothesis (H 0) serves as the baseline assumption in statistical testing, suggesting no significant effect, relationship ...
The sample data are organized as follows: The hypotheses of interest in an ANOVA are as follows: H 1: Means are not all equal. where k = the number of independent comparison groups. In this example, the hypotheses are: H 1: The means are not all equal. The null hypothesis in ANOVA is always that there is no difference in means.
Null Hypothesis and Alternative Hypothesis  ANOVA  Hypothesis Statistics  Hypothesis Testing  Statistics and Data Analysis  Business Statistics  Lean S...
In ANOVA, we will still adopt the alternative hypothesis as the best explanation of our data if we reject the null hypothesis. However, when we look at the alternative hypothesis, we can see that it does not give us much information. We will know that a difference exists somewhere, but we will not know where that difference is.
In oneway ANOVA, we want to compare t population means, where t > 2. Therefore, the null hypothesis for analysis of variance for t population means is: H 0: μ 1 = μ 2 =... μ t. The alternative, however, cannot be set up similarly to the twosample case. If we wanted to see if two population means are different, the alternative would be μ 1 ...
Hypothesis testing starts with the formulation of a null and alternative hypothesis. A null hypothesis (indicated as H 0) is a statement expecting no difference or effect. ... This means that although the underlying alternative hypothesis for the ANOVA analysis is twosided, all the group differences are assumed to be in the same side of the ...
Most technical papers rely on just the first formulation, even though you may see some of the others in a statistics textbook. Null hypothesis: " x is equal to y.". Alternative hypothesis " x is not equal to y.". Null hypothesis: " x is at least y.". Alternative hypothesis " x is less than y.". Null hypothesis: " x is at most ...
x: The value of the predictor variable. Simple linear regression uses the following null and alternative hypotheses: H0: β1 = 0. HA: β1 ≠ 0. The null hypothesis states that the coefficient β1 is equal to zero. In other words, there is no statistically significant relationship between the predictor variable, x, and the response variable, y.
I hardly understand how one can perform an ANOVAlike test on several response variables and more importantly, I don't understand what the null hypothesis could be. Is the null hypothesis: "For each response variable, the means of all groups are equal", or is it "For at least one response variable, the means of all groups are equal",