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Answer :
To find the 99% confidence interval for the mean high temperature of the towns based on the given data, we follow these steps:
1. Collect the data: We have the high temperatures of 12 small towns. The temperatures are:
- 99, 97.6, 96.3, 97.4, 98.7, 96.7, 98.9, 97.5, 99.9, 99.8, 99.1, 97.2
2. Calculate the sample mean: Add up all the temperature values and divide by the number of towns (12) to find the average temperature. This gives us the sample mean.
3. Calculate the sample standard deviation: This is a measure of the variation or dispersion of the temperatures. The formula involves finding the square root of the average of the squared differences from the mean.
4. Calculate the standard error of the mean: The standard error is found by dividing the sample standard deviation by the square root of the number of observations (12 in this case).
5. Determine the z-score for a 99% confidence level: The z-score corresponds to the probability required for a 99% confidence interval, which you can find in a standard normal distribution table. For a 99% confidence level, the z-score is typically around 2.576.
6. Calculate the margin of error: Multiply the z-score by the standard error of the mean. This value tells us how far the sample mean could reasonably differ from the true population mean.
7. Determine the confidence interval: Subtract the margin of error from the sample mean for the lower bound, and add the margin of error to the sample mean for the upper bound.
Following these steps accurately, we find that the 99% confidence interval for the mean high temperature is approximately (97.28, 99.07). This interval suggests that we are 99% confident that the true average high temperature of these towns is between 97.28°F and 99.07°F.
1. Collect the data: We have the high temperatures of 12 small towns. The temperatures are:
- 99, 97.6, 96.3, 97.4, 98.7, 96.7, 98.9, 97.5, 99.9, 99.8, 99.1, 97.2
2. Calculate the sample mean: Add up all the temperature values and divide by the number of towns (12) to find the average temperature. This gives us the sample mean.
3. Calculate the sample standard deviation: This is a measure of the variation or dispersion of the temperatures. The formula involves finding the square root of the average of the squared differences from the mean.
4. Calculate the standard error of the mean: The standard error is found by dividing the sample standard deviation by the square root of the number of observations (12 in this case).
5. Determine the z-score for a 99% confidence level: The z-score corresponds to the probability required for a 99% confidence interval, which you can find in a standard normal distribution table. For a 99% confidence level, the z-score is typically around 2.576.
6. Calculate the margin of error: Multiply the z-score by the standard error of the mean. This value tells us how far the sample mean could reasonably differ from the true population mean.
7. Determine the confidence interval: Subtract the margin of error from the sample mean for the lower bound, and add the margin of error to the sample mean for the upper bound.
Following these steps accurately, we find that the 99% confidence interval for the mean high temperature is approximately (97.28, 99.07). This interval suggests that we are 99% confident that the true average high temperature of these towns is between 97.28°F and 99.07°F.
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