Two-Proportions: Test Statistic & P-Value

Key Takeaways — How to Calculate the Test Statistic & P-Value for a Two-Proportion Significance Test

Calculating the Test Statistic and P-Value for a Difference in Proportions

• Once conditions are met, we calculate a standardized test statistic (z) for the difference in sample proportions.

• The test statistic standardizes the observed difference using the combined (pooled) proportion, p̂_c.

• The null hypothesis assumes no difference, so the observed difference is compared to 0.

• The P-value is found using the standard normal distribution and depends on the direction of the alternative hypothesis.

• Calculator results may differ slightly from hand calculations due to rounding, but conclusions should be the same.

Calculating the Test Statistic and P-Value for a Two-Proportion Significance Test

Difference in Population Proportions 

Hi everyone. In this video, we’re going to talk about how to calculate the standardized test statistic and the P-value for a significance test about a difference in population proportions.

If the conditions are met, we can proceed with the calculations to test the null and alternative hypotheses.

Standardizing the Difference in Sample Proportions

To perform the test, we need to standardize the difference in sample proportions to obtain a z statistic.

Assuming the conditions are met, the standardized test statistic is calculated as:

z = (p̂₁ − p̂₂ − 0) / standard error

We subtract 0 because under the null hypothesis, we are assuming there is no difference between the population proportions.

Standard Error Using the Combined Proportion

The standard error for this test uses the combined (pooled) sample proportion, p̂_c.

The combined proportion is calculated as:

p̂_c = (x₁ + x₂) / (n₁ + n₂)

where:

• x₁ and x₂ are the number of successes in each sample

• n₁ and n₂ are the sample sizes

The standard error is then calculated using:

standard error = √[ p̂_c (1 − p̂_c) (1/n₁ + 1/n₂) ]

This works because p̂_c represents the proportion of successes assuming the null hypothesis is true.

Finding the P-Value

After calculating the z statistic, we find the P-value by calculating the probability of getting a z statistic as extreme or more extreme than the observed value, in the direction specified by the alternative hypothesis, using the standard normal distribution.

Notes About the Formula Sheet

The specific formula for the standardized test statistic for a difference in population proportions is not explicitly given on the AP Statistics formula sheet.

However:

• The general z-test structure is provided

• The standard error formula is provided

You must know when to use each formula and how to put the pieces together.

Example: McDonald’s vs. Wendy’s Drive-Through Accuracy

Let’s return to the McDonald’s and Wendy’s drive-through example.

Before doing the full hypothesis test, we want to explain why the sample result gives some evidence for the alternative hypothesis.

The observed difference in sample proportions is:

p̂_M − p̂_W = 0.059

This value is not equal to 0, so it does provide some evidence in favor of the alternative hypothesis. However, this alone does not tell us whether the evidence is convincing, which is why we must perform the full hypothesis test.

Calculating the Test Statistic

First, we calculate the combined proportion:

p̂_c = (145 + 139) / (159 + 163) = 284 / 322 ≈ 0.882

Now we calculate the z statistic:

z = (0.059 − 0) / √[ 0.882 (1 − 0.882) (1/159 + 1/163) ] ≈ 1.64

Calculating the P-Value

To find the P-value, we can use either Table A or a calculator.

Using a Calculator

We use the normal CDF:

• Lower bound: 1.64

• Upper bound: a very large number

• Mean: 0

• Standard deviation: 1

Because this is a two-sided test, we multiply the result by 2.

Using Table A

From Table A, a z-score of 1.64 gives an area of 0.9495 to the left.

We subtract from 1 to find the upper-tail probability, then multiply by 2 to account for both tails.

In both methods, the P-value is approximately 0.18.

Conclusion of the Test

At a significance level of α = 0.05, the P-value of 0.18 is greater than α, so we fail to reject the null hypothesis.

There is not convincing evidence of a difference in the population proportions of accurate drive-through orders at McDonald’s and Wendy’s.

Interpreting the Normal Distribution

This result can also be visualized using the standard normal distribution. Because this is a two-sided test, the P-value represents the probability in both tails of the distribution.

Although the probability is relatively small, it is not small enough to rule out variation due to random sampling.

Using the Calculator for a Two-Proportion Z Test

To perform this test on a calculator:

1. Go to STAT

2. Select TESTS

3. Choose 2-PropZTest

Enter:

• x₁ = 145, n₁ = 159

• x₂ = 139, n₂ = 163

• Alternative: not equal

Select Calculate.

The calculator provides:

• The z statistic

• The P-value

• p̂₁, p̂₂, and the combined proportion p̂_c

The calculator uses the symbol p̂ for the combined proportion.

Important Notes

• Calculator values may differ slightly from hand calculations due to rounding.

• The calculator can also display a graph of the standard normal distribution with the shaded P-value area.

• Because this test often leads to calculation errors, it is recommended to use the calculator on the AP Statistics exam.

• Even when using technology, you must still show all four steps: State, Plan, Do, and Conclude.

• In the Do step, always report the z statistic and the P-value.

That’s it for this video. I hope this was helpful. Take care.