In a survey, 30 people were asked how much they spent on their child's last birthday gift. The results were roughly bell-shaped with a mean of $\$ 39.4$ and standard deviation of $\$ 10.7$. 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 $\bar{x} \pm$ Error.
Answer
\(\boxed{\text{Therefore, the typical parent would spend between $36.896 and $41.904 on their child's birthday gift with 80% confidence.}}\)
Step 1 ：Given that the mean (\(\bar{x}\)) is $39.4, the standard deviation (\(\sigma\)) is $10.7, and the sample size (\(n\)) is 30. The z-score for an 80% confidence level is approximately 1.282.
Step 2 ：We can calculate the error of the estimate using the formula \(z \frac{\sigma}{\sqrt{n}}\). Substituting the given values, we get an error of approximately $2.504.
Step 3 ：Subtracting and adding this error from the mean, we get the confidence interval. The lower bound of the interval is \(\bar{x} - \text{Error} = 39.4 - 2.504 = 36.896\) and the upper bound is \(\bar{x} + \text{Error} = 39.4 + 2.504 = 41.904\).
Step 4 ：\(\boxed{\text{Therefore, the typical parent would spend between $36.896 and $41.904 on their child's birthday gift with 80% confidence.}}\)
