High School

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

98, 98.4, 96.5, 97.5, 99.8, 96.7, 98.3, 96.4, 97, 98.9

Assume high temperatures are normally distributed. Based on this data, find the 80% confidence interval of the mean high temperature of towns.

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.

Learn more about confidence interval:

brainly.com/question/32546207

#SPJ11