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Hypothesis Testing and Statistical Significance

This tutorial is a part of the Zero to Data Analyst Bootcamp by Jovian

Hypothesis testing is a technique used by statisticians, scientists and data analysts for measuring whether the results of an experiment are meaningful and reliable. The statistical significance of the results of an experiment is often quantified using a P-value. This tutorial aims to build intuition for hypothesis testing using some real-world examples.

How to Run the Code

The best way to learn the material is to execute the code and experiment with it yourself. This tutorial is an executable Jupyter notebook . You can run this tutorial and experiment with the code examples in a couple of ways: using free online resources (recommended) or on your computer .

Option 1: Running using free online resources (1-click, recommended)

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Option 2: Running on your computer locally

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Problem Statement

Let's work through a real-world example to understand how statistical tests are performed:

QUESTION : You're an analyst at the investment firm Capital Ventures, and you're evaluating the company Jovian for a potential investment. The founders of Jovian claim that completing a data science bootcamp offered by Jovian helps you land a data science job faster. A 2020 McKinley report suggests that candidates apply for an average of 37 data science job roles before getting hired. You've surveyed 42 Jovian bootcamp graduates who are now working in data science roles, and compiled data for the number of jobs each one applied to before getting hired: 31, 23, 19, 42, 37, 18, 7, 53, 33, 17, 27, 41, 36, 29, 60, 34, 21, 18, 45, 33, 16, 10, 48, 32, 19, 29, 40, 35, 28, 57, 25, 31, 19, 40, 37, 33, 38, 28, 40, 36, 42, 39 Is there a statistically significant decrease in the number of jobs candidates need to apply to before getting hired if they've completed a bootcamp offered by Jovian?

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• Knowledge Base

Hypothesis Testing | A Step-by-Step Guide with Easy Examples

Published on November 8, 2019 by Rebecca Bevans . Revised on June 22, 2023.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics . It is most often used by scientists to test specific predictions, called hypotheses, that arise from theories.

There are 5 main steps in hypothesis testing:

• State your research hypothesis as a null hypothesis and alternate hypothesis (H o ) and (H a  or H 1 ).
• Collect data in a way designed to test the hypothesis.
• Perform an appropriate statistical test .
• Decide whether to reject or fail to reject your null hypothesis.
• Present the findings in your results and discussion section.

Though the specific details might vary, the procedure you will use when testing a hypothesis will always follow some version of these steps.

Table of contents

Step 1: state your null and alternate hypothesis, step 2: collect data, step 3: perform a statistical test, step 4: decide whether to reject or fail to reject your null hypothesis, step 5: present your findings, other interesting articles, frequently asked questions about hypothesis testing.

After developing your initial research hypothesis (the prediction that you want to investigate), it is important to restate it as a null (H o ) and alternate (H a ) hypothesis so that you can test it mathematically.

The alternate hypothesis is usually your initial hypothesis that predicts a relationship between variables. The null hypothesis is a prediction of no relationship between the variables you are interested in.

• H 0 : Men are, on average, not taller than women. H a : Men are, on average, taller than women.

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See an example

For a statistical test to be valid , it is important to perform sampling and collect data in a way that is designed to test your hypothesis. If your data are not representative, then you cannot make statistical inferences about the population you are interested in.

There are a variety of statistical tests available, but they are all based on the comparison of within-group variance (how spread out the data is within a category) versus between-group variance (how different the categories are from one another).

If the between-group variance is large enough that there is little or no overlap between groups, then your statistical test will reflect that by showing a low p -value . This means it is unlikely that the differences between these groups came about by chance.

Alternatively, if there is high within-group variance and low between-group variance, then your statistical test will reflect that with a high p -value. This means it is likely that any difference you measure between groups is due to chance.

Your choice of statistical test will be based on the type of variables and the level of measurement of your collected data .

• an estimate of the difference in average height between the two groups.
• a p -value showing how likely you are to see this difference if the null hypothesis of no difference is true.

Based on the outcome of your statistical test, you will have to decide whether to reject or fail to reject your null hypothesis.

In most cases you will use the p -value generated by your statistical test to guide your decision. And in most cases, your predetermined level of significance for rejecting the null hypothesis will be 0.05 – that is, when there is a less than 5% chance that you would see these results if the null hypothesis were true.

In some cases, researchers choose a more conservative level of significance, such as 0.01 (1%). This minimizes the risk of incorrectly rejecting the null hypothesis ( Type I error ).

The results of hypothesis testing will be presented in the results and discussion sections of your research paper , dissertation or thesis .

In the results section you should give a brief summary of the data and a summary of the results of your statistical test (for example, the estimated difference between group means and associated p -value). In the discussion , you can discuss whether your initial hypothesis was supported by your results or not.

In the formal language of hypothesis testing, we talk about rejecting or failing to reject the null hypothesis. You will probably be asked to do this in your statistics assignments.

However, when presenting research results in academic papers we rarely talk this way. Instead, we go back to our alternate hypothesis (in this case, the hypothesis that men are on average taller than women) and state whether the result of our test did or did not support the alternate hypothesis.

If your null hypothesis was rejected, this result is interpreted as “supported the alternate hypothesis.”

These are superficial differences; you can see that they mean the same thing.

You might notice that we don’t say that we reject or fail to reject the alternate hypothesis . This is because hypothesis testing is not designed to prove or disprove anything. It is only designed to test whether a pattern we measure could have arisen spuriously, or by chance.

If we reject the null hypothesis based on our research (i.e., we find that it is unlikely that the pattern arose by chance), then we can say our test lends support to our hypothesis . But if the pattern does not pass our decision rule, meaning that it could have arisen by chance, then we say the test is inconsistent with our hypothesis .

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.

• Normal distribution
• Descriptive statistics
• Measures of central tendency
• Correlation coefficient

Methodology

• Cluster sampling
• Stratified sampling
• Types of interviews
• Cohort study
• Thematic analysis

Research bias

• Implicit bias
• Cognitive bias
• Survivorship bias
• Availability heuristic
• Nonresponse bias
• Regression to the mean

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

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Unit 12: Significance tests (hypothesis testing)

About this unit.

Significance tests give us a formal process for using sample data to evaluate the likelihood of some claim about a population value. Learn how to conduct significance tests and calculate p-values to see how likely a sample result is to occur by random chance. You'll also see how we use p-values to make conclusions about hypotheses.

The idea of significance tests

• Simple hypothesis testing (Opens a modal)
• Idea behind hypothesis testing (Opens a modal)
• Examples of null and alternative hypotheses (Opens a modal)
• P-values and significance tests (Opens a modal)
• Comparing P-values to different significance levels (Opens a modal)
• Estimating a P-value from a simulation (Opens a modal)
• Using P-values to make conclusions (Opens a modal)
• Simple hypothesis testing Get 3 of 4 questions to level up!
• Writing null and alternative hypotheses Get 3 of 4 questions to level up!
• Estimating P-values from simulations Get 3 of 4 questions to level up!

Error probabilities and power

• Introduction to Type I and Type II errors (Opens a modal)
• Type 1 errors (Opens a modal)
• Examples identifying Type I and Type II errors (Opens a modal)
• Introduction to power in significance tests (Opens a modal)
• Examples thinking about power in significance tests (Opens a modal)
• Consequences of errors and significance (Opens a modal)
• Type I vs Type II error Get 3 of 4 questions to level up!
• Error probabilities and power Get 3 of 4 questions to level up!

Tests about a population proportion

• Constructing hypotheses for a significance test about a proportion (Opens a modal)
• Conditions for a z test about a proportion (Opens a modal)
• Reference: Conditions for inference on a proportion (Opens a modal)
• Calculating a z statistic in a test about a proportion (Opens a modal)
• Calculating a P-value given a z statistic (Opens a modal)
• Making conclusions in a test about a proportion (Opens a modal)
• Writing hypotheses for a test about a proportion Get 3 of 4 questions to level up!
• Conditions for a z test about a proportion Get 3 of 4 questions to level up!
• Calculating the test statistic in a z test for a proportion Get 3 of 4 questions to level up!
• Calculating the P-value in a z test for a proportion Get 3 of 4 questions to level up!
• Making conclusions in a z test for a proportion Get 3 of 4 questions to level up!

Tests about a population mean

• Writing hypotheses for a significance test about a mean (Opens a modal)
• Conditions for a t test about a mean (Opens a modal)
• Reference: Conditions for inference on a mean (Opens a modal)
• When to use z or t statistics in significance tests (Opens a modal)
• Example calculating t statistic for a test about a mean (Opens a modal)
• Using TI calculator for P-value from t statistic (Opens a modal)
• Using a table to estimate P-value from t statistic (Opens a modal)
• Comparing P-value from t statistic to significance level (Opens a modal)
• Free response example: Significance test for a mean (Opens a modal)
• Writing hypotheses for a test about a mean Get 3 of 4 questions to level up!
• Conditions for a t test about a mean Get 3 of 4 questions to level up!
• Calculating the test statistic in a t test for a mean Get 3 of 4 questions to level up!
• Calculating the P-value in a t test for a mean Get 3 of 4 questions to level up!
• Making conclusions in a t test for a mean Get 3 of 4 questions to level up!

More significance testing videos

• Hypothesis testing and p-values (Opens a modal)
• One-tailed and two-tailed tests (Opens a modal)
• Z-statistics vs. T-statistics (Opens a modal)
• Small sample hypothesis test (Opens a modal)
• Large sample proportion hypothesis testing (Opens a modal)

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Chapter 9 Hypothesis Testing

• KEY TERMS AND CONCEPTS
• SCIENTISTS IN ACTION

What is a Hypothesis?

Null and Alternative Hypotheses

Critical Region, Critical Values and Significance Level

Types of Hypothesis Testing

Decision Making: P -value Method

Decision Making: Traditional Method

Hypothesis: Accept or Fail to Reject?

Errors In Hypothesis Tests

Testing a Claim about Population Proportion

Testing a Claim about Mean: Known Population SD

Testing a Claim about Mean: Unknown Population SD

Testing a Claim about Standard Deviation

Novel Object Recognition and Object Location Behavioral Testing in Mice on a Budget

Use of Galvanic Skin Responses, Salivary Biomarkers, and Self-reports to Assess Undergraduate Student Performance During a Laboratory Exam Activity

Meta-analysis of Voxel-Based Neuroimaging Studies using Seed-based d Mapping with Permutation of Subject Images (SDM-PSI)

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• Published: 06 June 2024

Frontogenesis at Jovian high latitudes

• Lia Siegelman   ORCID: orcid.org/0000-0003-3330-082X 1 &
• Patrice Klein   ORCID: orcid.org/0000-0002-3089-3896 2 , 3  

Nature Physics ( 2024 ) Cite this article

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• Fluid dynamics
• Giant planets

Frontogenesis is the process by which fronts, separating fluids of different temperatures, are generated in the terrestrial atmosphere and oceans. It is of interest to consider whether this process appears in other planetary atmospheres. Here we analyse infra-red images taken by the Juno spacecraft at the Jovian poles and reveal ubiquitous vortices with a diameter of hundreds to thousands of kilometres and filaments with a width of tens of kilometres embedded in between the vortices. Our analysis shows that the filaments are dynamically active, reminiscent of terrestrial frontogenesis. Furthermore, our results indicate that Jovian frontogenesis acts in concert with moist convection, with the former mechanism favouring the development of cyclones and the latter the development of anti-cyclones. Even though moist convection is the main driver, frontogenesis accounts for a quarter of the total transfer of potential to kinetic energy, enhancing the upscale transfer of energy to larger vortices. Frontogenesis contributes 40% to the vertical heat transport, efficiently redistributing heat from Jupiter’s interior to its tropopause. This study highlights the broad range of interacting scales from tens to thousands of kilometres and the diverse physical mechanisms active at Jovian high latitudes.

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Moist convection drives an upscale energy transfer at Jovian high latitudes

Enhanced upward heat transport at deep submesoscale ocean fronts

Global upper-atmospheric heating on Jupiter by the polar aurorae

Data availability.

JIRAM data are available at the Planetary Data System (PDS) online ( https://atmos.nmsu.edu/PDS/data/PDS4/juno_jiram_bundle/data_calibrated/ ). Data products used in this study are the same as used in ref. 4 . They include calibrated, geometrically controlled, radiance data mapped onto an orthographic projection centred on the north pole and velocity vectors derived from the radiance data. Brightness maps and velocity vectors can be found in the supplementary data sets of ref. 4 .

Code availability

The code is available at https://github.com/liasiegelman/JIRAM/blob/main/frontogenesis.py .

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Acknowledgements

We thank T. Ewald for processing the infra-red images taken by JIRAM used in this study and W. Young for insightful discussions. L.S. is supported by the National Science Foundation (OCE-1657041). P.K. acknowledges funding from JPL/NASA.

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Lia Siegelman

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Patrice Klein

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L.S. and P.K. led the data analysis and data interpretation and wrote the manuscript.

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Extended data

Extended data fig. 1 cumulative integrals of χ..

Same as Fig. 4 for \(\tilde{{\chi }^{+}}/\overline{{\chi }^{+}}\) (red curve) and \(\tilde{{\chi }^{-}}/\overline{{\chi }^{-}}\) (blue curve). The dotted line shows the fractional area as a function of \({{{{\rm{d}}}}}_{\max }/{{{\rm{d}}}}\) , with d the vorticity-strain JDF and \({{{{\rm{d}}}}}_{\max }\) its maximum, χ + are updrafts and χ − downdrafts. \(\overline{{\chi }^{+}}\) and \(\overline{{\chi }^{-}}\) have equal magnitude of 0.3 f . The shaded envelopes indicate the range of results obtained with the different low-pass Butterworth filters (see error section in the Methods ). This figure confirms the asymmetry observed in the strain-vorticity space (Fig. 3c ). For instance, scales larger than 230 km, which occupy half of the domain, capture a little over 50% of the χ + but only 10% of the χ − . Similarly, scales smaller than 150 km, which occupy twenty percent of the domain, capture almost 60% of the χ − but only about 20% of χ + . In summary, positive divergence is localized in a large fraction of the physical space and corresponds to large length scales, whereas negative divergence is scattered throughout the domain, occupies a small fraction of the physical space, and corresponds to small length scales.

Extended Data Fig. 2 Spectrum of ζ and χ.

Wavenumber spectrum of vorticity ζ used in this paper (thick black curve) and wavenumber spectra of divergence χ (thin colored curves) derived after filtering the wind velocities derived from feature tracking with a low-pass Butterworth filter of order n and wavelength cutoff w cutoff . The spectra of ζ is used a an upper bound for the derivation of χ . χ used in the study corresponds to the low-pass Butterworth filter of order 1 and cutoff wavelength of 250 km (green curve, see error section in the Methods ).

Extended Data Fig. 3 Error propagation and sensitivity analysis.

Error propagation analysis in wind velocities derived from feature tracking. Each panel shows the weighted JDF of χ /f in vorticity-strain space, with χ the divergence derived after application of a low-pass Butterworth of order n and cutoff wavelength w cutoff and f the Coriolis parameter. The dashed lines are the ∣ ζ ∣  =  σ lines, with ζ the vorticity and σ strain. a ) n = 5, w cutoff  = 50 km; b ) n = 5, w cutoff  = 100 km; c ) n = 1, w cutoff  = 250 km, d ) n = 1, w cutoff  = 300 km, e ) n = 2, w cutoff  = 300 km, f ) n = 2, w cutoff  = 500 km. Panel c corresponds to Fig. 3c in the main text. The dispersion in panel a indicates that scales ≤ 50 km contain too much noise. All other panels yield comparable results indicating that the effective resolution of the divergence is between 50 and 100 km (see error section in the Methods for more details).

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Siegelman, L., Klein, P. Frontogenesis at Jovian high latitudes. Nat. Phys. (2024). https://doi.org/10.1038/s41567-024-02516-x

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This package implements several hypothesis tests in Julia.

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