High School

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The high temperatures (in degrees Fahrenheit) of a random sample of 10 small towns are:

[tex]\[
\begin{array}{|r|}
\hline
97.5 \\
\hline
96.5 \\
\hline
99.1 \\
\hline
99.5 \\
\hline
98.2 \\
\hline
99.7 \\
\hline
96.7 \\
\hline
97.0 \\
\hline
97.1 \\
\hline
99.6 \\
\hline
\end{array}
\][/tex]

Assume high temperatures are normally distributed. Based on this data, find the [tex]\(99\%\)[/tex] confidence interval of the mean high temperature of the towns. 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).

[tex]\[99\% \text{ C.I.} = (\square, \square)\][/tex]

The answer should be obtained without any preliminary rounding. However, the critical value may be rounded to 3 decimal places.

Answer :

To find the 99% confidence interval for the mean high temperature of the towns, we'll follow these steps:

1. Collect the Sample Data:
The high temperatures recorded for the 10 small towns are:
- 97.5, 96.5, 99.1, 99.5, 98.2, 99.7, 96.7, 97.0, 97.1, 99.6

2. Calculate the Sample Mean:
The sample mean is calculated by summing all the temperatures and dividing by the sample size (10).
[tex]\[
\text{Sample Mean} = \frac{97.5 + 96.5 + 99.1 + 99.5 + 98.2 + 99.7 + 96.7 + 97.0 + 97.1 + 99.6}{10} = 98.09
\][/tex]

3. Calculate the Sample Standard Deviation:
The sample standard deviation is a measure of the amount of variation or dispersion in a set of values. It accounts for n-1 degrees of freedom for a sample.
- This calculation gives us a sample standard deviation of approximately 1.29.

4. Calculate the Standard Error:
The standard error (SE) is calculated by dividing the standard deviation by the square root of the sample size.
[tex]\[
\text{Standard Error} = \frac{1.29}{\sqrt{10}} \approx 0.41
\][/tex]

5. Find the Critical Value:
For a 99% confidence interval, we need the critical value associated with this confidence level. Using the standard normal distribution, the critical value (z-score) is approximately 2.576.

6. Calculate the Margin of Error:
The margin of error is the product of the critical value and the standard error.
[tex]\[
\text{Margin of Error} = 2.576 \times 0.41 \approx 1.05
\][/tex]

7. Determine the Confidence Interval:
Using the sample mean and the margin of error, we determine the confidence interval by adding and subtracting the margin of error from the sample mean.
- Lower limit: [tex]\( 98.09 - 1.05 = 97.04 \)[/tex]
- Upper limit: [tex]\( 98.09 + 1.05 = 99.14 \)[/tex]

Therefore, the 99% confidence interval for the mean high temperature of the towns is [tex]\((97.04, 99.14)\)[/tex].

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