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To estimate how much a typical parent would spend on their child’s birthday gift using an 80% confidence level, we will apply the formula for a confidence interval for the mean. The formula is:

\[
\bar{x} \pm E
\]

Where:
- $\bar{x}$ is the sample mean,
- $E$ is the margin of error, calculated as:
  \[
  E = z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}
  \]
  Where:
  - $z_{\alpha/2}$ is the critical value corresponding to the desired confidence level,
  - $\sigma$ is the sample standard deviation,
  - $n$ is the sample size.

### Given values:
- $\bar{x} = 30.9$,
- $\sigma = 11.4$,
- $n = 29$,
- Confidence level = 80%.

### Step 1: Find the critical value $z_{\alpha/2}$ for an 80% confidence level
For an 80% confidence level, the area in each tail is $(1 - 0.80)/2 = 0.10$. From the standard normal table, the critical value $z_{\alpha/2} \approx 1.282$.

### Step 2: Calculate the margin of error $E$
\[
E = 1.282 \times \frac{11.4}{\sqrt{29}} \approx 1.282 \times 2.115 \approx 2.711
\]

### Step 3: Construct the confidence interval
\[
\bar{x} \pm E = 30.9 \pm 2.711
\]
So, the confidence interval is:

\[
30.9 \pm 2.711
\]

Thus, the estimated amount a typical parent would spend on their child’s birthday gift is $30.900 \pm 2.711$ dollars.

Would you like more details or have any questions?

Here are 5 related questions:
1. What does the margin of error represent in this context?
2. How does the confidence level affect the width of the confidence interval?
3. What happens to the confidence interval if the sample size increases?
4. How is the standard deviation used in constructing the confidence interval?
5. Can we use this method if the sample size were smaller?

**Tip:** Increasing the sample size reduces the margin of error, resulting in a narrower confidence interval.

In a survey, 29 people were asked how much they spent on their child’s last birthday gift. The results were roughly bell-shaped with a mean of $30.9 and standard deviation of $11.4. Estimate how much a typical parent would spend on their child’s birthday gift (use a 80% confidence level). Give your answers to 3 decimal places. Express your answer in the format of x̄ ± E.

Estimate the amount a typical parent would spend on a child's birthday gift using an 80% confidence interval. The survey results with a mean of $30.9, a standard deviation of $11.4, and sample size of 29 are used to calculate the margin of error and construct the interval.

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