Q.9 - The Practice of Statistics for AP

Question

Doing homework A school newspaper article claims that $60\%$ of the students at a large high school did all their assigned homework last week. Some skeptical AP Statistics students want to investigate whether this claim is true, so they choose an SRS of $100$ students from the school to interview. What values of the sample proportion pˆ would be consistent with the claim that the population proportion of students who completed all their homework is $p$  $0.60$? To find out, we used Fathom software to simulate choosing $250$ SRSs of size $n=100$ students from a population in which $p=0.60$. The figure below is a dotplot of the sample proportion $pˆ$ of students who did all their homework.

(a) Is this the sampling distribution of $pˆ$? Justify your answer. 

(b) Describe the distribution. Are there any obvious outliers? 

(c) Suppose that $45$ of the $100$ students in the actual sample say that they did all their homework last week. What would you conclude about the newspaper article’s claim? Explain. 

Step-by-Step Solution

Answer

a). No, this is not a sampling distribution of $p^$.

b). Yes, there are $4$ outliers.

c). The claim appears to be false.

Part (a) Step 1:Given Information

A school newspaper article claims that $60\%$ of the students at a large high school did all their assigned homework last week. Some skeptical AP Statistics students want to investigate whether this claim is true, so they choose an SRS of $100$ students from the school to interview.

Part (a) Step 2: Explanation

No, because the dot plot contains the results of $250$ simple random samples of size $100$, while the sampling distribution should contain the results of all possible samples of size $100$.

Part (b) Step 1: Given Information

A school newspaper article claims that $60\%$ of the students at a large high school did all their assigned homework last week. Some skeptical AP Statistics students want to investigate whether this claim is true, so they choose an SRS of $100$ students from the school to interview 

Part (b) Step 2: Explanation

Shape: Roughly unimodal and symmetric, because the highest peak is roughly in the middle of the histogram.

Center: The highest peak in the histogram is at about $0.60$, thus the distribution is centered at $0.60$.

Spread: The data values appear to vary from $0.47$ to $0.74$.

Outliers are dots that are separated from the other dots in the dotplot by a gap.

Then we note that there are $4$ outliers (two on each side of the dotplot): $0.47,0.48,0.73,0.74$.

Part (c) Step 1: Given Information

A school newspaper article claims that $60\%$ of the students at a large high school did all their assigned homework last week. Some skeptical AP Statistics students want to investigate whether this claim is true, so they choose an SRS of $100$ students from the school to interview.

Part (c) Step 2: Explanation

Claim: $p=0.60$ or $60\%$ of students did their homework last week.

The sample proportion is the number of successes divided by the sample size:

$$\begin{gathered} \hat{p}=\frac{x}{n} \\ =\frac{45}{100} \\ =0.45 \end{gathered}$$

In the dot plot we note that there are no dots above $0.45$ nor to its left, this means that it is very unusual to obtain a sample proportion of $0.45$ if the population proportion is $0.60$.

Then it appears that the claim is false.