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Answer :
To find the 95% confidence interval for the mean high temperature of the small towns, we follow these steps:
1. Collect the Data: We start with the high temperatures provided for 48 small towns.
2. Calculate the Sample Mean: The sample mean [tex]\(\bar{x}\)[/tex] is the average of all the temperature values. From the data, the sample mean is 98.15°F.
3. Calculate the Sample Standard Deviation: The sample standard deviation [tex]\(s\)[/tex] gives us a measure of the spread of the temperatures around the mean. For this data set, the sample standard deviation is 1.08°F. This calculation uses Bessel's correction, which means dividing by the sample size minus one ([tex]\(n-1\)[/tex]) for an unbiased estimate.
4. Determine the Sample Size: The number of temperatures in our sample is 48, so [tex]\(n = 48\)[/tex].
5. Determine the Confidence Level: We're asked for a 95% confidence interval, so the confidence level is 95%.
6. Find the Critical t-Value: Since the high temperatures are assumed to be normally distributed and the sample size is less than 30, we use the t-distribution. The critical t-value for a 95% confidence level with [tex]\(n - 1 = 47\)[/tex] degrees of freedom is approximately 2.012.
7. Calculate the Margin of Error: The margin of error (ME) is calculated using the formula:
[tex]\[
\text{Margin of Error} = t \times \left(\frac{s}{\sqrt{n}}\right)
\][/tex]
Plugging in the values, we get:
[tex]\[
\text{Margin of Error} = 2.012 \times \left(\frac{1.08}{\sqrt{48}}\right) \approx 0.31
\][/tex]
8. Calculate the Confidence Interval: The confidence interval (CI) is given by:
[tex]\[
\text{CI} = (\bar{x} - \text{ME}, \bar{x} + \text{ME})
\][/tex]
[tex]\[
\text{CI} = (98.15 - 0.31, 98.15 + 0.31) = (97.84, 98.46)
\][/tex]
Therefore, the 95% confidence interval for the mean high temperature of these small towns is [tex]\((97.84, 98.46)\)[/tex].
1. Collect the Data: We start with the high temperatures provided for 48 small towns.
2. Calculate the Sample Mean: The sample mean [tex]\(\bar{x}\)[/tex] is the average of all the temperature values. From the data, the sample mean is 98.15°F.
3. Calculate the Sample Standard Deviation: The sample standard deviation [tex]\(s\)[/tex] gives us a measure of the spread of the temperatures around the mean. For this data set, the sample standard deviation is 1.08°F. This calculation uses Bessel's correction, which means dividing by the sample size minus one ([tex]\(n-1\)[/tex]) for an unbiased estimate.
4. Determine the Sample Size: The number of temperatures in our sample is 48, so [tex]\(n = 48\)[/tex].
5. Determine the Confidence Level: We're asked for a 95% confidence interval, so the confidence level is 95%.
6. Find the Critical t-Value: Since the high temperatures are assumed to be normally distributed and the sample size is less than 30, we use the t-distribution. The critical t-value for a 95% confidence level with [tex]\(n - 1 = 47\)[/tex] degrees of freedom is approximately 2.012.
7. Calculate the Margin of Error: The margin of error (ME) is calculated using the formula:
[tex]\[
\text{Margin of Error} = t \times \left(\frac{s}{\sqrt{n}}\right)
\][/tex]
Plugging in the values, we get:
[tex]\[
\text{Margin of Error} = 2.012 \times \left(\frac{1.08}{\sqrt{48}}\right) \approx 0.31
\][/tex]
8. Calculate the Confidence Interval: The confidence interval (CI) is given by:
[tex]\[
\text{CI} = (\bar{x} - \text{ME}, \bar{x} + \text{ME})
\][/tex]
[tex]\[
\text{CI} = (98.15 - 0.31, 98.15 + 0.31) = (97.84, 98.46)
\][/tex]
Therefore, the 95% confidence interval for the mean high temperature of these small towns is [tex]\((97.84, 98.46)\)[/tex].
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