A survey by American Express Spending reported that the average amount spent on a wedding gift for a close family member is $166. A random sample of 45 people who purchased wedding gifts for a close family member spent an average of $160.50. Assume that the population standard deviation is $38. Use a 95% confidence interval to test the validity of this report and choose the one statement that is correct.
A)Because this confidence interval does not include $166, the report by American Express Spending is not validated.
B)Because this confidence interval does include $166, the report by American Express Spending is not validated.
C)Because this confidence interval does include $166, the report by American Express Spending is validated.
D)Because this confidence interval does not include $166, the report by American Express Spending is validated.

Question

Quizplus · Accepted Answer

To test the validity of the report, we calculate the 95% confidence interval for the mean amount spent on a wedding gift using the sample data. The formula for a confidence interval is $\bar{x} \pm z\frac{\sigma}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $z$ is the z-score corresponding to the confidence level (for 95%, $z = 1.96$), $\sigma$ is the population standard deviation, and $n$ is the sample size.Plugging in the given values: $160.50 \pm 1.96\frac{38}{\sqrt{45}}$.Calculating the margin of error: $1.96 \times \frac{38}{\sqrt{45}} \approx 11.07$.Thus, the 95% confidence interval is $160.50 \pm 11.07$, which is approximately $149.43$ to $171.57$.Since $166 falls within this interval, it supports the validity of the American Express Spending report.

A Survey by American Express Spending Reported That the Average