In a survey, 10 people were asked how much they spent on their child's last birthday gift. The results were roughly bell-shaped with a mean of $\$ 40.7$ and standard deviation of $\$ 9.6$. Construct a confidence interval at a $99 \%$ confidence level. Express your answer in the format of $\bar{x} \pm$ Error.

Step 1 ：We are given that in a survey, 10 people were asked how much they spent on their child's last birthday gift. The results were roughly bell-shaped with a mean of $40.7 and standard deviation of $9.6. We are asked to construct a confidence interval at a 99% confidence level.

Step 2 ：The formula for the confidence interval is \(\bar{x} \pm z \frac{\sigma}{\sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(z\) is the z-score corresponding to the desired confidence level, \(\sigma\) is the standard deviation, and \(n\) is the sample size.

Step 3 ：In this case, \(\bar{x} = \$ 40.7\), \(\sigma = \$ 9.6\), and \(n = 10\).

Step 4 ：The z-score for a 99% confidence level is approximately 2.576. This value can be found in a standard z-table or using a calculator that can compute it.

Step 5 ：We calculate the error term, which is \(z \frac{\sigma}{\sqrt{n}}\). Substituting the given values, we get an error of approximately \$7.82.

Step 6 ：Thus, the 99% confidence interval for the amount spent on their child's last birthday gift is \(\$ 40.7 \pm \$ 7.82\). So, we are 99% confident that the true mean amount spent on their child's last birthday gift is between \(\$ 32.88\) and \(\$ 48.52\).

Step 7 ：\(\boxed{\text{Final Answer: The 99% confidence interval for the amount spent on their child's last birthday gift is } \$ 40.7 \pm \$ 7.82. \text{So, we are 99% confident that the true mean amount spent on their child's last birthday gift is between } \$ 32.88 \text{ and } \$ 48.52.}\)