Thank you for visiting 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. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
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:
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