Question

in which of the following scenarios can we calculate a confidence interval for the population mean? check all that apply.
a. a random sample of 60 cell phone calls is selected and the mean length of calls is determined to be 3.25 minutes with a standard deviation of 4.2 minutes.
b. a random sample of 15 women athletes is selected and their mean weight is determined to be 136 pounds. female athlete weights are known to be normally distributed with a population standard deviation of 20 pounds.
c. a random sample of 14 cell phone calls is selected and the mean length of calls is determined to be 3.25 minutes with a standard deviation of 4.2 minutes.
d. a random sample of 35 women athletes is selected and their mean weight is determined to be 136 pounds. female athlete weights are known to be normally distributed with a population standard deviation of 20 pounds.

Explanation:

Step1: Recall confidence - interval conditions

To calculate a confidence interval for the population mean, we can use the z - interval when the population standard deviation $\sigma$ is known and the population is normally distributed or the sample size $n\geq30$, and the t - interval when the population standard deviation is unknown and we use the sample standard deviation $s$, with the population being normally distributed or $n\geq30$.

Step2: Analyze Option A

The sample size $n = 60\geq30$. The sample mean $\bar{x}=3.25$ and sample standard deviation $s = 4.2$. We can use the t - distribution to calculate the confidence interval for the population mean.

Step3: Analyze Option B

The population is normally distributed and the population standard deviation $\sigma=20$ is known, and the sample size $n = 15$. We can use the z - distribution to calculate the confidence interval for the population mean.

Step4: Analyze Option C

The sample size $n=14<30$ and we only know the sample standard deviation $s = 4.2$. Without knowing that the population is normally distributed, we cannot calculate a confidence interval for the population mean.

Step5: Analyze Option D

The population is normally distributed and the population standard deviation $\sigma = 20$ is known, and the sample size $n=35\geq30$. We can use the z - distribution to calculate the confidence interval for the population mean.

Answer:

A. A random sample of 60 cell phone calls is selected and the mean length of calls is determined to be 3.25 minutes with a standard deviation of 4.2 minutes.
B. A random sample of 15 women athletes is selected and their mean weight is determined to be 136 pounds. Female athlete weights are known to be normally distributed with a population standard deviation of 20 pounds.
D. A random sample of 35 women athletes is selected and their mean weight is determined to be 136 pounds. Female athlete weights are known to be normally distributed with a population standard deviation of 20 pounds.