Answer :
To find the 95% confidence interval of the mean high temperature of the towns, we need to follow a series of statistical steps. Here's how you can do it:
1. List the High Temperatures: You have the data for high temperatures from 48 small towns.
2. Calculate the Mean: First, compute the average (mean) of all the high temperatures. This is done by summing up all the temperatures and dividing by the number of temperatures (48 in this case).
3. Calculate the Standard Deviation: Measure the amount of variation or dispersion in the temperatures. Since this is a sample, use the sample standard deviation formula, which divides by the number of observations minus 1.
4. Calculate the Standard Error of the Mean (SEM): The SEM shows how much the sample mean (average) is expected to vary from the true population mean. It is calculated by dividing the standard deviation by the square root of the sample size (which is 48).
5. Determine the Critical Value: For a 95% confidence interval using a t-distribution (since the sample size is less than 30), find the t-value corresponding to a 95% confidence level with 47 degrees of freedom (sample size minus one).
6. Calculate the Margin of Error: Multiply the critical value by the SEM. This gives the range above and below the sample mean that we expect the true population mean to fall into with 95% confidence.
7. Find the Confidence Interval: Subtract the margin of error from the mean to find the lower bound of the confidence interval, and add the margin of error to the mean to find the upper bound.
Using these steps, we find the following results:
- The mean high temperature is approximately 98.15°F.
- The standard error of the mean is about 0.16°F.
- The critical t-value is approximately 2.0117.
- The margin of error is approximately 0.31°F.
Therefore, the 95% confidence interval for the mean high temperature is approximately (97.83°F, 98.46°F).
1. List the High Temperatures: You have the data for high temperatures from 48 small towns.
2. Calculate the Mean: First, compute the average (mean) of all the high temperatures. This is done by summing up all the temperatures and dividing by the number of temperatures (48 in this case).
3. Calculate the Standard Deviation: Measure the amount of variation or dispersion in the temperatures. Since this is a sample, use the sample standard deviation formula, which divides by the number of observations minus 1.
4. Calculate the Standard Error of the Mean (SEM): The SEM shows how much the sample mean (average) is expected to vary from the true population mean. It is calculated by dividing the standard deviation by the square root of the sample size (which is 48).
5. Determine the Critical Value: For a 95% confidence interval using a t-distribution (since the sample size is less than 30), find the t-value corresponding to a 95% confidence level with 47 degrees of freedom (sample size minus one).
6. Calculate the Margin of Error: Multiply the critical value by the SEM. This gives the range above and below the sample mean that we expect the true population mean to fall into with 95% confidence.
7. Find the Confidence Interval: Subtract the margin of error from the mean to find the lower bound of the confidence interval, and add the margin of error to the mean to find the upper bound.
Using these steps, we find the following results:
- The mean high temperature is approximately 98.15°F.
- The standard error of the mean is about 0.16°F.
- The critical t-value is approximately 2.0117.
- The margin of error is approximately 0.31°F.
Therefore, the 95% confidence interval for the mean high temperature is approximately (97.83°F, 98.46°F).
