Wize AP Statistics Textbook > Hypothesis Testing for Mean and Proportion

Introduction to Hypothesis Testing

Popular Courses

AP Exam Prep

0:00 / 0:00

Introduction to Hypothesis Testing

Hypothesis testing refers to the formal procedures to determine whether to accept or reject statistical hypotheses, each making a claim about a parameter, depending on which one our data supports.


Question →\rightarrow→ Test  →\rightarrow→ Decision  →\rightarrow→  Conclusion

Examples
Can you conclude that house prices have increased?
Has the demand for cauliflower decreased?
Is there evidence that life expectancy has changed?
PAGE BREAK
Stating the Hypotheses
All hypothesis tests have the following:
Ho: null hypothesis
Ha: alternative hypothesis (or H1)
The above statements are about population parameters.

The null hypotheses always say things like: “no change”, “no effect”, “no difference”, “no, no, no…”

→ Ho\rightarrow\ H_o→ Ho​ supports the status quo

The alternative hypotheses say things like: “there is a change”, “there is an effect”, “there is a difference”, etc.

 → Ha\rightarrow\ H_a→ Ha​ challenges the status quo


PAGE BREAK
Law Court Analogy
The null hypothesis is the "defendant", while the alternative hypothesis is the "prosecutor". 
In order to win the case, the prosecutor has the evidential burden – in order to reject the null hypothesis, we need to observe strong evidence to support the alternative hypothesis. 
Otherwise, we have to accept (a.k.a. fail to reject) the null hypothesis’ claim.
Example
The operations manager claims that the factory workers produce an average of 150 units a day. Janice, a factory worker, believes that they produce more than that, effectively challenging that claim.
Ho: Factory workers produce 150 units a day, on average.                         Ho: μ =150H_o:\ \mu\ =150Ho​: μ =150
Ha: Factory workers produce more than 150 units a day, on average.      Ha: μ >150H_a:\ \mu\ >150Ha​: μ >150

Wize Tip
Always use parameters in the hypotheses (like μ\muμ or ppp). Never use statistics (like x‾\overline{x}x or p^\hat{p}p^​).
The null hypothesis always has the equal = sign.
Always use the same number in the hypotheses, specifically the number supporting the claim or status quo.


There has to be one winner: the null hypothesis (operations manager) or the alternative hypothesis (Janice). In other words, there is either sufficient evidence to support Janice or there is no evidence (or insufficient evidence) to support her. 

Watch Out!
If there is no evidence to support the alternative hypothesis, it doesn’t prove that the null hypothesis is true! This just means the status quo still holds (for now).


“Innocent until proven guilty.”


PAGE BREAK
Running a Hypothesis Test
How do we determine which hypothesis "wins": the null hypothesis or the alternative hypothesis? 
We run a test! 
Based on your findings, does your evidence support the alternative hypothesis?

Example
Janice selects a random sample of 30 full-time workers. The sample mean is 160 units per day, which is certainly greater than 150 units per day... but is it statistically significantly greater? 

In other words, is the value 𝑥 = 160 highly unusual to observe, given that the claim is 150 units? After all, the sample mean could be this high by just chance alone. 

To determine statistical significance, we need to find the test statistic. For example, a test statistic of 3.18 is more than 3 standard deviations away from the mean, so this is highly unusual, thus providing evidence to support the alternative hypothesis.

Wize Concept
Generally speaking, more than 2 standard deviations away from the mean at either direction is considered "extreme" or "highly unusual". This applies to the z-distribution and t-distribution. See: critical value.

PAGE BREAK

Drawing Your Conclusion
Two scenarios:

Watch Out!
Reminder: If you fail to reject the null hypothesis, it doesn’t prove that the null hypothesis is true. It just means that there is no evidence to support the alternative hypothesis.

0:00 / 0:00

Hypothesis Testing Steps

Step 1: Identify the parameter of interest
Mean →μ\rightarrow\mu→μ
Proportion →p\rightarrow p→p

Step 2: State the null and alternative hypotheses
Ho: null hypothesis
Ha: alternative hypothesis
The null hypothesis states the parameter of interest and what it is equal to.

The alternative hypothesis states that it must be true if the null hypothesis is false.

PAGE BREAK
One-sided vs. Two-sided Tests
Is this a one-tail or two-tail test? 

Write it down so you don’t forget!

One-sided test: "more than", "greater than", "less than", "fewer than", "increased", "decreased", "improved", "declined", etc.

>or<> or <>or<

PAGE BREAK
Two-sided test: "different", "changed", "not equal to", etc.

≠\neq =

Wize Concept
If it is a two-tail test, the p-value is doubled! See: p-value

PAGE BREAK
P-value
The p-value is the probability, assuming the null hypothesis is true, of obtaining a test statistic at least as extreme as the one actually observed. Extreme means “in the direction of the alternative hypothesis.” The p-value is the probability associated with the test statistic. 


Watch Out!
The p-value is not the probability that the null hypothesis is true!

What does a low p-value mean?
A low p-value means that, assuming that the null hypothesis is true, there is low probability of observing an extreme or significant outcome due to chance or randomness alone.

Example

HoH_oHo​: Stella is not a biochemistry expert.
HaH_aHa​: Stella is a biochemistry expert.
 
Suppose we ask her to write a T/F biochemistry exam. Assuming HoH_oHo​ is true (that she is not a biochemistry expert), then we expect her to guess for the entire exam and get roughly 50% of the questions correct.

Let's say a hypothesis test revealed a p-value of 1%. This is what it means: 
If it’s true that she is not a biochemistry expert, then there’s a 1% chance she’ll ace the biochemistry exam by chance or by guessing alone. 
This is obviously a very unlikely and extreme outcome! If this outcome actually occurs, it provides evidence to support HaH_aHa​  and we question the validity of HoH_oHo​ . 
In other words, if she aced the exam, it's not very probable that she aced it by simply guessing. Thus, there is evidence that she is a biochemistry expert.


Watch Out!
Important: We don’t know which hypothesis is in fact true – hypothesis tests do not determine that! We do not know if Stella is truly a biochemistry expert or not, but a low p-value provides evidence to support HaH_aHa​, that she is a biochemistry expert.

PAGE BREAK

When p-value low, reject HoH_oHo​!

Wize Tip
The stronger the evidence for the alternative hypothesis, the larger the test statistic, then the smaller the p-value, and the better your data supports HaH_aHa​ so you can reject HoH_oHo​.

Guideline for p-values:

PAGE BREAK

Step 3: Make sure all the conditions are met for the study
Is the sample in the study randomly sampled? (See: Statistical Sampling)
Is the sample size nnn large enough? (See: Central Limit Theorem)
Is the sample unbiased? (See: Sampling Bias)

Assumptions:

✓\checkmark✓ Individuals are independent
✓\checkmark✓ Population is Normal. (If not, then the sample size needs to be large.) 

PAGE BREAK
Step 4: Note the significance level and Critical Values

e.g df=19df=19df=19, one-tailed test (right side)

The numbers at the bottom of each distribution are the critical values, and they draw the line between significant and not significant. 

Wize Concept
The smaller the significance level, the stronger your evidence needs to be to reject the null hypothesis.

Exam Tip
Sometimes the significance level is not given on the exam. That’s fine – we can still see how strong our evidence is based on the size of the p-value.

Portions of information contained in this publication/book are printed with permission of Minitab, LLC. All such material remains the exclusive property and copyright of Minitab, LLC. All rights reserved.

PAGE BREAK

Step 5: Calculate the test statistic and p-value

Test statistic: use the correct formula (mean vs. proportion) to find the test statistic.

Hypothesis testing for a mean

→ z\rightarrow\ z→ z (if σ\sigmaσ is known)
→ t\rightarrow\ t→ t (if σ\sigmaσ is unknown)

Hypothesis testing for a proportion 

→z\rightarrow z→z

The test statistic (t-stat or z-stat) lets you do either of the two things:
Compare the test statistic with the critical value (ta∗ t_a^{\ast}\ ta∗​ or za∗z_a^{\ast}za∗​)
Find the p-value and compare it with α\alphaα

PAGE BREAK

Step 6: Make your decision

Scenario #1
Is the p-value smaller than α? →Reject Ho \alpha?\ \rightarrow Reject\ H_o\ α? →Reject Ho​  
Is the absolute value of the test statistic greater than CV? →Reject Ho CV?\ \rightarrow Reject\ H_o\ CV? →Reject Ho​  

Portions of information contained in this publication/book are printed with permission of Minitab, LLC. All such material remains the exclusive property and copyright of Minitab, LLC. All rights reserved.

PAGE BREAK

Scenario #2
Is the p-value larger than α? →Fail to reject Ho \alpha?\ \rightarrow Fail\ to\ reject\ H_o\ α? →Fail to reject Ho​           
Is the absolute value of the test statistic less than CV? →Fail to reject Ho CV?\ \rightarrow Fail\ to\ reject\ H_o\ CV? →Fail to reject Ho​  

Portions of information contained in this publication/book are printed with permission of Minitab, LLC. All such material remains the exclusive property and copyright of Minitab, LLC. All rights reserved.

PAGE BREAK

Step 7: Draw your conclusion

Do you have evidence for the alternative hypothesis? No need to be creative with your answer. Simply restate your conclusion based on the question.

Example: “Is there evidence that profit has increased?”

Conclusion:
If you reject HoH_oHo​: "There is evidence that profit has increased”
If you fail to reject HoH_oHo​: "There is no evidence that profit has increased.”