3.7 - Carrying Out a Test for a Population Proportion

Overview

The test statistic

• What it measures: how many standard deviations the sample proportion is from the null value. It matches the general form on the AP formula sheet: standardized test statistic $= \frac{\text{statistic} - \text{parameter}}{\text{standard deviation of statistic}}$.

• Why $p_0$ is in the denominator: a test assumes $H_0$ is true, so it uses the standard deviation the sampling distribution of $\hat{p}$ would have if $p = p_0$. (A confidence interval uses $\hat{p}$ instead, because there's no null value to assume.)

• What the sign tells you: a positive $z$ means $\hat{p}$ is above $p_0$, and a negative $z$ means it's below.

The null distribution as a normal model

Finding the $p$-value

• Normal table: the table gives the area to the left of $z$. For a right tail, subtract from 1. For a two-sided test, double the tail area.

• normalcdf: on a TI-84, normalcdf(lower, upper, 0, 1) gives the area between two $z$-values. Use -1E99 or 1E99 for a tail.

• The whole test at once: STAT, TESTS, 1-PropZTest. Enter $p_0$, $x$ (the number of successes), $n$, and the direction of $H_a$. It reports $z$, the $p$-value, and $\hat{p}$.

The significance level

Making the decision

Writing the conclusion

• If $p$-value $\leq \alpha$: "Because the $p$-value of ___ is less than $\alpha =$ ___, we reject $H_0$. There is convincing evidence that the true proportion of [population] who [response] is [greater than / less than / different from] $p_0$."

• If $p$-value $> \alpha$: "Because the $p$-value of ___ is greater than $\alpha =$ ___, we fail to reject $H_0$. There is not convincing evidence that the true proportion of [population] who [response] is [greater than / less than / different from] $p_0$."

The four steps

1. State: the hypotheses, with the parameter defined in context, and the significance level $\alpha$.

2. Plan: name the procedure (here, a one-sample $z$-test for a population proportion) and check each condition in context: randomization, 10%, and normality (Topic 3.5).

3. Do: calculate the test statistic and the $p$-value, showing the formula with your numbers plugged in.

4. Conclude: compare the $p$-value to $\alpha$, make the decision, and state the conclusion in context, in terms of $H_a$.

Example 1: Customer satisfaction (AP-style)

• Randomization: the data come from a random sample of 400 customers.

• 10%: $400 \leq 10\%$ of all customers, since 10% of 50,000 is 5,000.

• Normality: $np_0 = 400(0.80) = 320 \geq 10$ and $n(1 - p_0) = 400(0.20) = 80 \geq 10$.

Example 2: Has support changed? (two-sided)

• Randomization: random sample of 250 residents.

• 10%: 250 is less than 10% of the residents of a large city.

• Normality: $250(0.45) = 112.5 \geq 10$ and $250(0.55) = 137.5 \geq 10$.

Example 3: Finishing the bus test from Topic 3.5

Common mistakes

• Using $\hat{p}$ in the standard deviation of the test statistic. A test assumes $H_0$, so it uses $p_0$.

• Writing hypotheses or conclusions about $\hat{p}$. Hypotheses and conclusions are about the parameter $p$, the true proportion for the whole population.

• Saying "accept $H_0$" or "this proves $H_a$." Use "fail to reject $H_0$" and "convincing evidence."

• Comparing $z$ to $\alpha$. Compare the $p$-value to $\alpha$.

• A conclusion with no context or no link to the $p$-value. Say how the $p$-value compares to $\alpha$, then state the conclusion about the population in the problem's words.

• Picking $\alpha$ after seeing the $p$-value. The significance level is set before the data are collected.