Score: $50 / 100$
$6 / 10$ answered
Question 7

You must estimate the mean temperature (in degrees Fahrenheit) with the following sample temperatures
\begin{tabular}{|r|}
\hline 28 \\
\hline 22.1 \\
\hline 17.2 \\
\hline 17.7 \\
\hline 17.8 \\
\hline
\end{tabular}

Find the $99 \%$ confidence interval. Enter your answer as an open-interval (i.e., parentheses) accurate to two decimal places (because the sample data are reported accurate to one decimal place).
\[
99 \% \text { C.I. }=
\]

Answer should be obtained without any preliminary rounding.
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Step 1 ：Calculate the mean (average) of the sample data. The mean is calculated by adding all the numbers in the data set and then dividing by the number of values in the set. \(\text{Mean} = \frac{{28 + 22.1 + 17.2 + 17.7 + 17.8}}{5} = 20.56\)

Step 2 ：Calculate the standard deviation of the sample data. The standard deviation is a measure of how spread out the numbers in the data set are. It is calculated by taking the square root of the variance. First, calculate the variance. The variance is the average of the squared differences from the mean. \(\text{Variance} = \frac{{(28 - 20.56)^2 + (22.1 - 20.56)^2 + (17.2 - 20.56)^2 + (17.7 - 20.56)^2 + (17.8 - 20.56)^2}}{4} = 16.802\). Then, take the square root of the variance to get the standard deviation. \(\text{Standard deviation} = \sqrt{16.802} = 4.10\)

Step 3 ：Calculate the 99% confidence interval. The 99% confidence interval is calculated using the formula: \(\text{Confidence interval} = \text{mean} ± (Z-score * (\text{standard deviation} / \sqrt{n}))\). The Z-score for a 99% confidence interval is 2.576 (from the Z-table). \(\text{Confidence interval} = 20.56 ± (2.576 * (4.10 / \sqrt{5})) = 20.56 ± 4.71\)

Step 4 ：So, the 99% confidence interval is \(\boxed{(15.85, 25.27)}\). This means that we are 99% confident that the true mean temperature is between 15.85 and 25.27 degrees Fahrenheit.