Questions: A major department store chain is interested in estimating the mean amount its credit card customers spent on their first visit to the chain's new store in the mall. Fifteen credit card accounts were randomly sampled and analyzed with the following results: X̄ = 50.50 and s=20. Assuming the distribution of the amount spent on their first visit is normal, what is the shape of the sampling distribution of the sample mean that will be used to create the desired confidence interval for μ ? A. At distribution with 14 degrees of freedom B. Approximately normal with a mean of 50.50 C. A standard normal distribution D. At distribution with 15 degrees of freedom

A major department store chain is interested in estimating the mean amount its credit card customers spent on their first visit to the chain's new store in the mall. Fifteen credit card accounts were randomly sampled and analyzed with the following results: X̄ = 50.50 and s=20. Assuming the distribution of the amount spent on their first visit is normal, what is the shape of the sampling distribution of the sample mean that will be used to create the desired confidence interval for μ ?
A. At distribution with 14 degrees of freedom
B. Approximately normal with a mean of 50.50
C. A standard normal distribution
D. At distribution with 15 degrees of freedom

Transcript text: A major department store chain is interested in estimating the mean amount its credit card customers spent on their first visit to the chain's new store in the mall. Fifteen credit card accounts were randomly sampled and analyzed with the following results: $\bar{X}=\$ 50.50$ and $s=20$. Assuming the distribution of the amount spent on their first visit is normal, what is the shape of the sampling distribution of the sample mean that will be used to create the desired confidence interval for $\mu$ ? A. At distribution with 14 degrees of freedom B. Approximately normal with a mean of $\$ 50.50$ C. A standard normal distribution D. At distribution with 15 degrees of freedom

Solution

Solution Steps

Step 1: Given Information

We are provided with the following data from a sample of credit card customers at a department store:

Sample mean ($ \bar{X} $): \$50.50
Sample standard deviation ($ s $): 20
Sample size ($ n $): 15

Step 2: Determine the Degrees of Freedom

To analyze the sampling distribution of the sample mean, we need to calculate the degrees of freedom, which is given by: \[ \text{Degrees of Freedom} = n - 1 = 15 - 1 = 14 \]

Step 3: Identify the Distribution Type

Since the sample size is $ n = 15 $ and the population from which the sample is drawn is assumed to be normally distributed, the sampling distribution of the sample mean will follow a t-distribution. The t-distribution is appropriate here because the sample size is small ($ n < 30 $) and we are using the sample standard deviation.

Step 4: Conclusion

The shape of the sampling distribution of the sample mean is a t-distribution with 14 degrees of freedom.