AP Statistics 3.6 p-Values Study Notes - New Syllabus

3.6.A.1 Interpreting the p-value for a One-Sample z-Test for a Population Proportion

After calculating the test statistic, the next step in a hypothesis test is to determine the p-value.

The p-value is calculated under the assumption that the null hypothesis (\(H_0\)) is true.

It represents the probability of obtaining a test statistic that is as extreme as or more extreme than the one observed, in the direction specified by the alternative hypothesis.

• A small p-value indicates that the observed sample result would be unlikely if the null hypothesis were true, providing stronger evidence against \(H_0\).
• A large p-value indicates that the observed result is reasonably likely if the null hypothesis is true, providing little evidence against \(H_0\).

Definition of the p-value

Finding the p-value

The method used to calculate the p-value depends on the alternative hypothesis.

Example 1: Right-Tailed Test

A hypothesis test produces a test statistic of

\(z=2.10\)

with the alternative hypothesis

\(H_a:p>p_0\)

The p-value is the probability of obtaining a z-score of 2.10 or greater.

\(P(z\ge2.10)\)

Example 2: Left-Tailed Test

A hypothesis test produces

\(z=-1.85\)

with the alternative hypothesis

\(H_a:p<p_0\)

The p-value is

\(P(z\le-1.85)\)

Example 3: Two-Tailed Test

A hypothesis test produces

\(z=2.25\)

with the alternative hypothesis

\(H_a:p\neq p_0\)

The p-value is the combined probability in both tails.

\(P(z\le-2.25)+P(z\ge2.25)\)

Interpreting the Size of the p-value

Important AP Exam Notes

• The p-value is always calculated assuming the null hypothesis is true.
• The p-value measures the probability of obtaining results as extreme as or more extreme than the observed result.
• The direction of the alternative hypothesis determines whether the p-value comes from the left tail, right tail, or both tails.
• A small p-value provides stronger evidence against the null hypothesis.
• The p-value is not the probability that the null hypothesis is true.

Common AP Exam Mistakes

3.6.A.2 Interpreting the p-value from a Simulated Null Distribution

Sometimes, instead of using the standard Normal distribution, the null distribution of the test statistic is generated through simulation.

In this case, the p-value is calculated as the proportion of simulated test statistics that are as extreme as or more extreme than the observed test statistic, in the direction specified by the alternative hypothesis.

Rather than calculating probabilities from the Normal distribution, you count the simulated results that are at least as extreme as the observed statistic and divide by the total number of simulations.

Formula for a Simulated p-value

\(\text{p-value}=\dfrac{\text{Number of simulated statistics at least as extreme as the observed statistic}}{\text{Total number of simulated statistics}}\)

Determining the p-value from a Simulation

Example 1: Right-Tailed Test

A simulation generates 5,000 test statistics.

The observed test statistic is

\(z=2.10\)

Among the simulated statistics, 95 are at least 2.10.

Calculate the p-value.

\(\text{p-value}=\dfrac{95}{5000}=0.019\)

Interpretation:

Assuming the null hypothesis is true, about 1.9% of simulated samples produced a test statistic at least as large as the observed value.

Example 2: Left-Tailed Test

A simulation generates 10,000 test statistics.

The observed test statistic is

\(z=-1.85\)

Among the simulated values, 412 are at or below -1.85.

Calculate the p-value.

\(\text{p-value}=\dfrac{412}{10000}=0.0412\)

Example 3: Two-Tailed Test

A simulation produces 8,000 test statistics.

The observed statistic is

\(z=2.30\)

There are

• 74 simulated statistics less than or equal to -2.30.
• 78 simulated statistics greater than or equal to 2.30.

Calculate the p-value.

\(\text{p-value}=\dfrac{74+78}{8000}=\dfrac{152}{8000}=0.019\)

Important AP Exam Notes

• When using simulations, the p-value is a proportion, not a Normal probability.
• The p-value includes simulated statistics that are as extreme as or more extreme than the observed statistic.
• The direction of the alternative hypothesis determines which simulated values are counted.
• For a two-sided test, include both tails of the simulated null distribution.
• A smaller simulated p-value provides stronger evidence against the null hypothesis.

Common AP Exam Mistakes

3.6.A.3 Interpreting the p-value in Context

When interpreting a p-value, always begin by stating that it is calculated assuming the null hypothesis is true.

• This means we assume that the true population proportion is equal to the value stated in the null hypothesis.
• The p-value represents the probability of obtaining a test statistic that is as extreme as or more extreme than the observed test statistic, in the direction of the alternative hypothesis, if the null hypothesis is true.

A complete AP Statistics interpretation should include:

• That the null hypothesis is assumed to be true.
• The population proportion stated in the null hypothesis.
• The probability of obtaining the observed result (or one more extreme).
• The context of the study.

General Interpretation Template

Assuming that the true population proportion is \(p_0\), the p-value is the probability of obtaining a sample result (or test statistic) as extreme as or more extreme than the one observed simply due to random sampling.

Example

A company claims that 70% of its customers are satisfied with its service.

A hypothesis test produces a p-value of

\(0.018\)

Interpretation:

Assuming that the true proportion of all customers who are satisfied is 0.70, there is a 1.8% chance of obtaining a sample result at least as extreme as the one observed due to random sampling alone.

3.6.A.4 Interpreting a Small p-value

A small p-value indicates that the observed sample result would be unusual if the null hypothesis were true.

• Therefore, a small p-value provides convincing statistical evidence in favor of the alternative hypothesis.
• The smaller the p-value, the stronger the statistical evidence against the null hypothesis.

Relationship Between the p-value and Evidence

Example

A hypothesis test produces

\(p\text{-value}=0.003\)

Since the p-value is very small, the observed sample result would be highly unusual if the null hypothesis were true.

Therefore, there is very convincing statistical evidence supporting the alternative hypothesis.

3.6.A.5 Interpreting a Large p-value

A large p-value indicates that the observed sample result would not be unusual if the null hypothesis were true.

• Therefore, the sample data do not provide convincing statistical evidence for the alternative hypothesis.
• However, a large p-value does not prove that the null hypothesis is true.

Instead, it means that the sample data are consistent with the null hypothesis, and there is insufficient evidence to conclude otherwise.

Example

A hypothesis test produces

\(p\text{-value}=0.41\)

Since the p-value is large, the observed sample result is not unusual if the null hypothesis is true.

Therefore, there is not convincing statistical evidence supporting the alternative hypothesis.

This does not mean that the null hypothesis has been proven true.

Important AP Exam Notes

• Always interpret the p-value by stating that the null hypothesis is assumed to be true.
• A small p-value provides stronger evidence against the null hypothesis.
• The smaller the p-value, the more convincing the evidence for the alternative hypothesis.
• A large p-value means the data are consistent with the null hypothesis.
• A large p-value does not prove that the null hypothesis is true.
• The p-value measures the strength of the evidence in the sample data, not the probability that a hypothesis is true.

Common AP Exam Mistakes