AP Statistics 4.2 D Calculate an Appropriate Confidence Interval for a Population Mean or Population Mean Difference Questions

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.