AP Statistics 4.6 Difference of Means Overview

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.
Describe a method Peter could have used to select a simple random sample of 100 students from the school.

Peter and Rania conducted their studies as described. Peter used the sample mean X as a point estimator for µ. Rania used X_overall =(0.6) X_female +(0.4) X_male as a point estimator for µ, where X_female is the mean of the sample of 60 females and X_male is the mean of the sample of 40 males.

Summary statistics for Peter's data are shown in the table below.
\begin{tabular}{|l|l|l|l|}
\hline Variable & N & Mean & Standard
Deviation \\
\hline Number of
soft drinks & 100 & 5.32 & 4.13 \\
\hline
\end{tabular}

Summary statistics for Rania's data are shown in the table below.
Based on the summary statistics, calculate the estimated standard deviation of the sampling distribution (sometimes called the standard error) of Peter's point estimator X.

Summary statistics for Rania's data are shown in the table below.
\begin{tabular}{|l|l|l|l|l|}
\hline Variable & Gender & N & Mean & Standard
Deviation \\
\hline \multirow{2}{*}{Number of
soft drinks} & Female & 60 & 2.90 & 1.80 \\
\hline & Male & 40 & 7.45 & 2.22 \\
\hline
\end{tabular}

A dotplot of Peter's sample data is given below.
Based on the summary statistics, calculate the estimated standard deviation of the sampling distribution of Rania's point estimator X_overall =(0.6) X_female +(0.4) X_male.

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

Comparative dotplots of Rania's sample data are 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 $\bar{X}_{\text {overall }}=(0.6) \bar{X}_{\text {female }}+(0.4) \bar{X}_{\text {male }}$.

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?

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?