AP Statistics 3.7 Carrying Out a Test for a Population Proportion Study Notes - New Syllabus

3.7.A.1 Calculating the Test Statistic for a One-Sample z-Test for a Population Proportion

After verifying that all conditions for a hypothesis test are satisfied, the next step is to calculate the test statistic.

• The test statistic measures how many standard errors the observed sample proportion is from the hypothesized population proportion stated in the null hypothesis.
• If the null hypothesis is true, the test statistic follows an approximately standard Normal distribution.

A test statistic close to 0 indicates that the sample proportion is close to the hypothesized population proportion.

A test statistic with a large positive or negative value indicates that the sample result is farther from the hypothesized value and provides stronger evidence against the null hypothesis.

Formula for the Test Statistic

\(z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}\)

Where:

• \(\hat{p}\) = Sample proportion
• \(p_0\) = Hypothesized population proportion from the null hypothesis
• \(n\) = Sample size
• \(z\) = Test statistic

Understanding the Formula

• The numerator, \(\hat{p}-p_0\), measures how far the sample proportion is from the hypothesized population proportion.
• The denominator, \(\sqrt{\dfrac{p_0(1-p_0)}{n}}\), is the standard error calculated under the assumption that the null hypothesis is true.
• The test statistic tells us how many standard errors the observed sample proportion is above or below the hypothesized proportion.

Interpreting the Test Statistic

Example 1

A company claims that 60% of its customers renew their memberships.

A random sample of 200 customers is selected, and 140 renew their memberships.

Step 1: Calculate the sample proportion.

\(\hat{p}=\dfrac{140}{200}=0.70\)

Step 2: Identify the hypothesized proportion.

\(p_0=0.60\)

Step 3: Calculate the standard error.

\(\sqrt{\dfrac{0.60(0.40)}{200}}=\sqrt{0.0012}\approx0.0346\)

Step 4: Calculate the test statistic.

\(z=\dfrac{0.70-0.60}{0.0346}\approx2.89\)

Interpretation:

The sample proportion is approximately 2.89 standard errors above the hypothesized population proportion.

Example 2

A school claims that 75% of students participate in extracurricular activities.

A random sample of 160 students is selected, and 108 participate.

Step 1:

\(\hat{p}=\dfrac{108}{160}=0.675\)

Step 2:

\(p_0=0.75\)

Step 3:

\(SE=\sqrt{\dfrac{0.75(0.25)}{160}}\approx0.0342\)

Step 4:

\(z=\dfrac{0.675-0.75}{0.0342}\approx-2.19\)

Interpretation:

The sample proportion is approximately 2.19 standard errors below the hypothesized population proportion.

Important AP Exam Notes

• Use the hypothesized population proportion \(p_0\) when calculating the standard error for a hypothesis test.
• The test statistic measures the number of standard errors between \(\hat{p}\) and \(p_0\).
• A positive test statistic means \(\hat{p}>p_0\).
• A negative test statistic means \(\hat{p}<p_0\).
• If the null hypothesis is true, the test statistic follows an approximately standard Normal distribution.

Common AP Exam Mistakes

3.7.A.2 Distribution of the Test Statistic Under the Null Hypothesis

When performing a one-sample z-test for a population proportion, the test statistic is calculated under the assumption that the null hypothesis (\(H_0\)) is true.

If all required conditions are satisfied, the distribution of the test statistic under the null hypothesis, called the null distribution, can be approximated by a probability model, such as the standard Normal distribution.

This allows statisticians to calculate probabilities (p-values) and determine whether the observed sample result is unusual under the null hypothesis.

Null Distribution

The null distribution is the probability distribution of the test statistic that would occur if the null hypothesis were true.

For a one-sample z-test for a population proportion, the null distribution is approximately a

Standard Normal Distribution

provided that the Randomization, 10% (if needed), and Normality (Large Counts) Conditions are satisfied.

Characteristics of the Null Distribution

Example 1

A manufacturer claims that 80% of its products meet quality standards.

A hypothesis test is performed, and the calculated test statistic is

\(z=1.82\)

Since all conditions are satisfied, the null distribution of the test statistic is approximately a standard Normal distribution.

The p-value is found by locating 1.82 on the standard Normal distribution.

Example 2

A school claims that 65% of students participate in extracurricular activities.

A hypothesis test produces

\(z=-2.10\)

Because the conditions for a one-sample z-test are satisfied, the sampling distribution of the test statistic under the null hypothesis is approximately standard Normal.

The probability of observing a test statistic of -2.10 or more extreme is determined from the standard Normal distribution.

Why Use the Standard Normal Distribution?

• It provides a probability model for the test statistic when the null hypothesis is true.
• It allows us to calculate the p-value.
• It helps determine whether the observed sample result is unusual.
• It provides the basis for deciding whether to reject or fail to reject the null hypothesis.

Important AP Exam Notes

• The null distribution assumes that the null hypothesis is true.
• For a one-sample z-test for a population proportion, the null distribution is approximately standard Normal when the required conditions are met.
• The standard Normal distribution has a mean of 0 and a standard deviation of 1.
• The p-value is calculated from this null distribution.
• If the required conditions are not satisfied, the standard Normal model should not be used.

Common AP Exam Mistakes

3.7.A.3 Calculating the p-value Using the Standard Normal Distribution

Once the test statistic has been calculated, the next step in a hypothesis test is to determine the p-value.

For a one-sample z-test for a population proportion, the p-value is found using the standard Normal distribution.

The p-value is the probability of obtaining a test statistic that is as extreme as or more extreme than the observed test statistic, assuming the null hypothesis is true.

On the AP Statistics Exam, the p-value may be found using:

• A standard Normal (z) table.
• A graphing calculator.
• Statistical software or other approved technology.

Finding the p-value

Example 1: Right-Tailed Test

A one-sample z-test produces

\(z=2.15\)

with the alternative hypothesis

\(H_a:p>p_0\)

Using the standard Normal distribution:

\(P(Z\ge2.15)\approx0.0158\)

Therefore,

p-value \(=0.0158\)

Example 2: Left-Tailed Test

A hypothesis test produces

\(z=-1.73\)

with

\(H_a:p<p_0\)

Using the standard Normal distribution:

\(P(Z\le-1.73)\approx0.0418\)

Therefore,

p-value \(=0.0418\)

Example 3: Two-Tailed Test

A hypothesis test produces

\(z=2.05\)

with

\(H_a:p\neq p_0\)

Step 1: Find one tail.

\(P(Z\ge2.05)\approx0.0202\)

Step 2: Double the probability.

p-value \(=2(0.0202)=0.0404\)

Using Technology

Graphing calculators and statistical software can calculate p-values directly from the test statistic.

Using technology reduces rounding error and is the preferred method on the AP Statistics Exam.

Important AP Exam Notes

• The p-value is found using the standard Normal distribution after calculating the test statistic.
• The direction of the alternative hypothesis determines whether to use the left tail, right tail, or both tails.
• For a two-sided test, remember to include both tails of the distribution.
• The p-value may be calculated using a z-table or technology.
• Always report the p-value in the context of the hypothesis test.

Common AP Exam Mistakes