AP Statistics 4.2 Constructing a Confidence Interval for a Population Mean Questions

C

Part (b):
Step 1: Identifies the appropriate confidence interval (by name or by formula) and checks appropriate conditions.

The appropriate procedure is a one-sample t-interval for a population mean.
Conditions: 1. The sample is randomly selected from the population.
2. The population has a normal distribution, or the sample size is large.

The first condition is met because we were told that the crows were randomly selected. The sample size of 23 is not considered large, so we need to examine the sample data to assess whether it is reasonable to assume that the population distribution of lead levels for all crows in this region is normal. The stem-and-leaf plot shows no strong skewness or outliers, so we will consider the second condition to be met.

Step 2: Correct mechanics

A 95\% confidence interval for the population mean $\mu$ is given by: $\bar{x} \pm t^{*} \frac{S}{\sqrt{n}}$. The critical value for 95\% confidence, based on 23-1=22 degrees of freedom, is $t^{*}=2.074$. The 95\% confidence interval for $\mu$ is therefore
$$4.90 \pm 2.074 \times \frac{1.12}{\sqrt{23}} \approx 4.90 \pm 0.484,$$
which is the interval (4.416, 5.384) ppm.

Using the raw data rather than the given summary statistics, the 95\% confidence interval for $\mu$ is (4.411, 5.3803).

Step 3: Interpretation
We can be 95\% confident that the population mean lead level among all crows in this region is between 4.416 and 5.384 parts per million.

No, based on the response to part (d-i), Julio's data would not satisfy the normality condition because neither of the criteria listed are met. Julio only has a sample size of 20, which is less than 30, and Pearson's coefficient of skewness indicates the distribution of sample data is strongly skewed.

Essentially correct (E) if the response satisfies the following five components:

In part (d-i) the response indicates the sample is strongly skewed or makes a statement consistent with the graph in part (c)

In part (d-i) the response uses or refers to the graph or the Pearson coefficient of skewness to make the correct determination of skewness

In part (d-ii) the response states that the condition of normality is not met or makes a statement consistent with the explanation in part (d-i)

In part (d-ii) the response explains the sample size of 20 is less than 30

In part (d-ii) the response explains that the distribution of sample data is strongly skewed or makes a statement consistent with the explanation in part (d-i)

Partially correct (P) if the response does not meet the criteria for E but satisfies three or four of components 1-5 required for E.

Incorrect (I) if the response does not meet the criteria for E or P.

Additional Notes:

- A response that determines the sample is strongly skewed, but says the normality condition is satisfied, should be scored I.

- In part (d-i) a response that indicates the distribution is strongly skewed to the left does not satisfy component 1.

- If a response in part (c-ii) has an x on the curve, the response in part (d-i) must indicate the sample is skewed (not strongly skewed or approximately symmetric), or indicate it is not possible to choose whether it is strongly skewed or approximately symmetric to satisfy component 1.

- In part (d-i) a response that indicates the distribution is skewed, but not strongly skewed, because the Pearson coefficient of skewness is not that far from the curve on the graph satisfies component 1 and component 2.

- In part (d-i) a response that indicates the distribution is skewed, but not strongly skewed, may satisfy components 3-5 if in part (d-ii) the response indicates the conditions are met because the distribution is not very skewed and there are no outliers. The response does not need to indicate the sample size is less than 30.

- In part (d-ii) a response that indicates the sample size is small without explicitly referring to the values 20 and 30 may satisfy component 4.

- The quality of communication, or the number of components satisfied for responses with score P, should be considered if holistic scoring is required.

D