In a survey, 12 people were asked how much they spent on their child's last birthday gift. The results were roughly bell-shaped with a mean of $48 and standard deviation of $19. Construct a confidence interval at a 80% confidence level.

To construct an 80% confidence interval for the mean amount spent on the birthday gift, we follow the same process as before but adjust the values for this specific problem.

### Step 1: Identify the given information
- Mean ($\bar{x}$) = $48
- Standard deviation ($\sigma$) = $19
- Sample size ($n$) = 12
- Confidence level = 80%

### Step 2: Determine the critical value $z_{\alpha/2}$
For an 80% confidence level, the critical value $z_{\alpha/2}$ is approximately 1.282.

### Step 3: Calculate the standard error (SE)
\[
\text{Standard Error (SE)} = \frac{\sigma}{\sqrt{n}} = \frac{19}{\sqrt{12}} \approx \frac{19}{3.464} \approx 5.485
\]

### Step 4: Compute the margin of error
\[
\text{Margin of Error} = z_{\alpha/2} \times \text{SE} = 1.282 \times 5.485 \approx 7.031
\]

### Step 5: Construct the confidence interval
\[
\text{Confidence Interval} = 48 \pm 7.031
\]
So, the 80% confidence interval is approximately:

\[
48 - 7.031 \text{ to } 48 + 7.031 \quad \text{or} \quad (40.969, 55.031)
\]

Thus, the confidence interval at an 80% confidence level is **$40.97 ± $7.03**.

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 does an 80% confidence interval compare to a 90% or 95% interval?
2. What assumptions are necessary for the validity of this confidence interval?
3. How would the interval change if the standard deviation were smaller?
4. What is the impact of using a smaller sample size on the confidence interval?
5. Can you explain how the critical value is chosen for different confidence levels?
6. What is the interpretation of this confidence interval in the context of the survey?
7. How would the confidence interval differ if the data were skewed?
8. How does the width of the confidence interval relate to the confidence level?

**Tip:** Lowering the confidence level (e.g., from 90% to 80%) results in a narrower confidence interval, meaning you're less certain but more precise.

Learn how to construct an 80% confidence interval for survey data with a mean of $48 and standard deviation of $19. This example covers calculating standard error, margin of error, and interpreting confidence intervals.

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