In a survey, 24 people were asked how much they spent on their child's last birthday gift. The results were roughly bell-shaped with a mean of $40 and standard deviation of $14. Construct a confidence interval at a 90% confidence level.

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To construct a confidence interval at a 90% confidence level for the mean amount spent on the birthday gift, we can use the following formula:

\[
\text{Confidence Interval} = \text{Mean} \pm z_{\alpha/2} \times \left(\frac{\sigma}{\sqrt{n}}\right)
\]

Where:
- $\text{Mean} = 40$ (the average amount spent)
- $\sigma = 14$ (the standard deviation)
- $n = 24$ (the sample size)
- $z_{\alpha/2}$ is the critical value from the standard normal distribution for a 90% confidence level.

### Step 1: Determine the critical value $z_{\alpha/2}$
For a 90% confidence level, the critical value $z_{\alpha/2}$ is approximately 1.645.

### Step 2: Calculate the standard error
\[
\text{Standard Error (SE)} = \frac{\sigma}{\sqrt{n}} = \frac{14}{\sqrt{24}} \approx \frac{14}{4.899} \approx 2.857
\]

### Step 3: Compute the margin of error
\[
\text{Margin of Error} = z_{\alpha/2} \times \text{SE} = 1.645 \times 2.857 \approx 4.7
\]

### Step 4: Construct the confidence interval
\[
\text{Confidence Interval} = 40 \pm 4.7
\]
So, the 90% confidence interval is approximately:

\[
40 - 4.7 \text{ to } 40 + 4.7 \quad \text{or} \quad (35.3, 44.7)
\]

Thus, the confidence interval at a 90% confidence level is **$35.3 ± $4.7**.

Would you like more details or have any questions about this process?

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Here are 8 related questions you might find helpful:
1. How would the confidence interval change if the sample size was larger?
2. What is the interpretation of the confidence interval in this context?
3. How do we determine the critical value for different confidence levels?
4. What happens to the confidence interval if the standard deviation increases?
5. How would a 95% confidence level affect the interval compared to 90%?
6. Can you construct a confidence interval if the sample size is less than 30?
7. How would the confidence interval differ if the distribution was not bell-shaped?
8. What is the relationship between confidence intervals and hypothesis testing?

**Tip:** Increasing the sample size generally leads to a narrower confidence interval, as it reduces the standard error.

Learn how to construct a 90% confidence interval for the mean amount spent on birthday gifts based on survey data. Includes step-by-step calculation using the confidence interval formula. Suitable for advanced high school or college-level students and anyone interested in statistical analysis.

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