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Answer :
Final Answer:
The 80% confidence interval of the mean high temperature of small towns is 96.59°F to 99.21°F.
Explanation:
To find the 80% confidence interval of the mean high temperature, we can use the formula for a confidence interval:
[tex]\[ \text{Confidence Interval} = \text{Sample Mean} \pm \left(\frac{Z \cdot Standard Error}{\sqrt{n}}\right) \][/tex]
Here, the sample mean is [tex]\(\bar{x} = 97.61\)[/tex], the sample size [tex](\(n\))[/tex] is 10, and we need to find the value of [tex]\(Z\)[/tex] for an 80% confidence level.
Using a Z-table or calculator, we find [tex]\(Z \approx 1.28\)[/tex] for 80% confidence.
Next, we need to calculate the standard error [tex](\(SE\))[/tex]:
[tex]\[ SE = \frac{\text{Sample Standard Deviation}}{\sqrt{n}} \][/tex]
The sample standard deviation [tex](\(s\))[/tex] is approximately 1.26.
Now, we can plug these values into the confidence interval formula:
[tex]\[ \text{Confidence Interval} = 97.61 \pm \left(\frac{1.28 \cdot 1.26}{\sqrt{10}}\right) \][/tex]
Calculating this, we get the 80% confidence interval of 96.59°F to 99.21°F.
In this context, the confidence interval represents the range within which we are 80% confident that the true mean high temperature of small towns lies.
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