Let's shake on it: A random sample of 12 -ounce milkshakes from 13 fast-food restaurants had the following number of calories.
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Assume the population standard deviation is $\sigma =92$.
Part 1 of 3
(a) Explain why it is necessary to check whether the population is approximately nomal before constructing a confidence interval.
It is necessary to check whether the population is approximately normal because the sample size is less than or equal to 30
Part 2 of 3
(b) Following is a dotplot of these data. Is it reasonable to assume that the population is approximately normal?
It is $\mathbf{V}$ reasonable to assume that the population is approximately normai.

Part: $2/3$

Part 3 of 3
(c) If appropriate, construct a $99.5\mathrm{\%}$ confidence interval for the mean calorie count for all 12 -ounce milkshakes sold at fast-food restaurants. Round the answers to at least two decimal places.

A $99.5\mathrm{\%}$ confidence interval for the mean calorie count for all 12 -ounce milkshakes sold at fast-food restaurants is $<\mu <◻$.
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Step 1 ：Given the data of calorie counts from 13 different fast-food restaurants, we are asked to construct a 99.5% confidence interval for the mean calorie count for all 12-ounce milkshakes sold at fast-food restaurants. The population standard deviation is given as $\sigma =92$.

Step 2 ：To construct the confidence interval, we need to calculate the sample mean and then use the formula for the confidence interval which is: $$\overline{x}\pm Z_{\alpha /2}\ast \frac{\sigma }{\sqrt{n}}$$ where: $\overline{x}$ is the sample mean, $Z_{\alpha /2}$ is the Z-score for the desired confidence level (for 99.5% confidence level, $Z_{\alpha /2}$ is approximately 2.807), $\sigma$ is the population standard deviation, and $n$ is the sample size.

Step 3 ：First, calculate the sample mean ($\overline{x}$) of the given data. The sample mean is 538.0.

Step 4 ：Next, calculate the margin of error using the formula: $$Z_{\alpha /2}\ast \frac{\sigma }{\sqrt{n}}$$ The margin of error is approximately 71.62.

Step 5 ：Finally, construct the confidence interval by subtracting and adding the margin of error from the sample mean. The lower bound of the confidence interval is $\overline{x}-\text{margin of error}=466.38$ and the upper bound is $\overline{x}+\text{margin of error}=609.62$.

Step 6 ：$\boxed{{\displaystyle \text{Final Answer: A 99.5\% confidence interval for the mean calorie count for all 12 -ounce milkshakes sold at fast-food restaurants is }466.38<\mu <609.62}}$