A major electronics store chain is interested in estimating the average 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:= $50.50 and s2 = 400. A 95% confidence interval for the average amount the credit card customers spent on their first visit to the chain's new store in the mall is: Group of answer choices 1.$50.50 ± $9.09. 2.$50.50 ± $10.12. 3.$50.50 ± $11.08. 4.None of these choices.

To calculate the 95% confidence interval for the average amount spent, we'll use the following formula for the confidence interval:

\[
\text{Confidence Interval} = \bar{x} \pm t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}
\]

Where:
- $\bar{x}$ is the sample mean.
- $t_{\alpha/2}$ is the critical value from the t-distribution for a 95% confidence level and $n-1$ degrees of freedom.
- $s$ is the sample standard deviation.
- $n$ is the sample size.

Given:
- $\bar{x} = 50.50$
- $s^2 = 400$, so $s = \sqrt{400} = 20$
- $n = 15$
- Degrees of freedom $df = n - 1 = 14$
- For a 95% confidence level and 14 degrees of freedom, the $t$ value $t_{\alpha/2}$ is approximately 2.145.

Now, we can plug in these values into the formula:

\[
\text{Margin of Error} = t_{\alpha/2} \cdot \frac{s}{\sqrt{n}} = 2.145 \cdot \frac{20}{\sqrt{15}} \approx 2.145 \cdot 5.165 = 11.08
\]

Thus, the 95% confidence interval is:

\[
50.50 \pm 11.08
\]

So the correct answer is:

**3. $50.50 ± $11.08**

Would you like more details or have any questions?

Here are some related questions to expand your understanding:

1. How do you determine the critical value for a given confidence level?
2. What is the difference between a t-distribution and a normal distribution?
3. Why do we use the t-distribution instead of the z-distribution for small sample sizes?
4. How does increasing the sample size affect the width of the confidence interval?
5. What happens to the confidence interval if the confidence level is increased to 99%?
6. What assumptions are necessary to construct a confidence interval using the t-distribution?
7. How would the confidence interval change if the sample standard deviation was larger?
8. What is the interpretation of a 95% confidence interval in the context of this problem?

**Tip:** When calculating confidence intervals for a mean with a small sample size, always use the t-distribution to account for the additional variability due to the smaller sample size.

Learn how to calculate a 95% confidence interval for the average amount credit card customers spent on their first visit to a new store. The sample mean is $50.50 with a sample standard deviation of $20 from a sample size of 15. The computed confidence interval is $50.50 ± $11.08.

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