AP Statistics 4.6 Sampling Distributions for the Difference Between Two Population Means Questions

Calculate and interpret the sampling distribution for the difference between two independent sample means, including centre, spread, and normality.

Exam points

set the mean of x-bar1 minus x-bar2 equal to the corresponding population-mean difference
add the two independent sample-mean variances, then take the square root for the standard deviation
require independent groups and avoid the independent-means spread formula for paired measurements
use both sample sizes of at least 30 to justify an approximately normal difference of sample means

Covered syllabus points

4.6.A—Calculate the mean and standard deviation of a sampling distribution for the difference between two sample means 4.6.B—Justify the appropriateness of conditions for the sampling distribution of the difference between two sample means 4.6.C—Interpret the mean, standard deviation, and probabilities for a sampling distribution for the difference between…

Two students at a large high school, Peter and Rania, wanted to estimate $\mu$ , the mean number of soft drinks that a student at their school consumes in a week. A complete roster of the names and genders for the 2,000 students at their school was available. Peter selected a simple random sample of 100 students. Rania, knowing that 60 percent of the students at the school are female, selected a simple random sample of 60 females and an independent simple random sample of 40 males. Both asked all of the students in their samples how many soft drinks they typically consume in a week.

Peter and Rania conducted their studies as described. Peter used the sample mean $\bar{X}$ as a point estimator for $\mu$ . Rania used $\bar{X}_{\text {overall }}=(0.6) \bar{X}_{\text {female }}+(0.4) \bar{X}_{\text {male }}$ as a point estimator for $\mu$ , where $\bar{X}_{\text {female }}$ is the mean of the sample of 60 females and $\bar{X}_{\text {male }}$ is the mean of the sample of 40 males.

Summary statistics for Peter's data are shown in the table below.

Summary statistics for Rania's data are shown in the table below.

A dotplot of Peter's sample data is given below.

Comparative dotplots of Rania's sample data are given below.

Based on the summary statistics, calculate the estimated standard deviation of the sampling distribution of Rania's point estimator $\bar{X}_{\text {overall }}=(0.6) \bar{X}_{\text {female }}+(0.4) \bar{X}_{\text {male }}$ .

Part (c):
The variance of Rania's estimator is $(0.6)^{2} \operatorname{Var}\left(\bar{X}_{f}\right)+(0.4)^{2} \operatorname{Var}\left(\bar{X}_{m}\right)$ , where $\operatorname{Var}\left(\bar{X}_{f}\right)=\frac{\sigma_{f}^{2}}{n_{f}}$ represents the variance of the point estimator for females and $\operatorname{Var}\left(\bar{X}_{m}\right)=\frac{\sigma_{m}^{2}}{n_{m}}$ represents the variance of the point estimator for males.

The estimated standard deviation is the square root of the variance. Using the respective sample standard deviations $s_{f}$ and $s_{m}$ for the population parameters, Rania's estimate is calculated as:

\sqrt{(0.6)^{2} \frac{s_{f}^{2}}{n_{f}}+(0.4)^{2} \frac{s_{m}^{2}}{n_{m}}}=\sqrt{(0.6)^{2} \frac{(1.80)^{2}}{60}+(0.4)^{2} \frac{(2.22)^{2}}{40}} \approx \sqrt{0.01944+0.01972} \approx 0.198 .

[Maximum number: 1]

Researchers surveyed a random sample of 55 adults in neighborhood A and 64 adults in neighborhood B from populations of over 1,000 adults in each neighborhood. The sampled individuals were asked what their families anticipated spending on Halloween. Let $\bar{x}_{A}$ be the calculated mean spending of the 55 adults in neighborhood A, and let $\bar{x}_{B}$ be the calculated mean spending of the 64 adults in neighborhood B. Which of the following is the best explanation for why the sampling distribution of $\bar{x}_{A}-\bar{x}_{B}$ can be modeled with a normal distribution?

There are at least 30 people in each population.

The sample sizes are sufficiently large.

The population distributions are assumed to be roughly normal.

The sample distributions are assumed to be unimodal and roughly symmetric.

[Maximum number: 1]

Sean and Evan are college roommates who have part-time jobs as servers in restaurants. The distribution of Sean's weekly income is approximately normal with mean $225 and standard deviation $25. The distribution of Evan's weekly income is approximately normal with mean $240 and standard deviation $15. Assuming their weekly incomes are independent of each other, which of the following is closest to the probability that Sean will have a greater income than Evan in a randomly selected week?

0.067

0.159

0.227

0.303

0.354

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Covered syllabus points

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