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Answer :
To find the 80% confidence interval for the mean high temperature of the towns, follow these steps:
1. Identify the Data:
- The high temperatures in degrees Fahrenheit are: 97.8, 96.6, 97.1, 97.6, 97.2, 98.6, 96.3, 96.8, and 98.7.
2. Calculate the Sample Mean:
- Add all the temperatures together and divide by the number of data points (9 in this case).
3. Calculate the Sample Standard Deviation:
- Find the deviation of each temperature from the sample mean, square these deviations, sum them up, divide by the number of observations minus one (to get the sample variance), and take the square root to find the sample standard deviation.
4. Identify the Confidence Level:
- We are given an 80% confidence level.
5. Find the t-Critical Value:
- Since the sample size is small (n = 9), use the t-distribution. Look for the t-value corresponding to an 80% confidence level with 8 degrees of freedom (n - 1).
6. Calculate the Margin of Error:
- Use the formula: Margin of Error = (t-critical value) × (sample standard deviation / √n).
7. Calculate 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, you find that the 80% confidence interval for the mean high temperature is approximately (97.02, 97.8). This means we are 80% confident that the true average high temperature of these towns lies within this interval.
1. Identify the Data:
- The high temperatures in degrees Fahrenheit are: 97.8, 96.6, 97.1, 97.6, 97.2, 98.6, 96.3, 96.8, and 98.7.
2. Calculate the Sample Mean:
- Add all the temperatures together and divide by the number of data points (9 in this case).
3. Calculate the Sample Standard Deviation:
- Find the deviation of each temperature from the sample mean, square these deviations, sum them up, divide by the number of observations minus one (to get the sample variance), and take the square root to find the sample standard deviation.
4. Identify the Confidence Level:
- We are given an 80% confidence level.
5. Find the t-Critical Value:
- Since the sample size is small (n = 9), use the t-distribution. Look for the t-value corresponding to an 80% confidence level with 8 degrees of freedom (n - 1).
6. Calculate the Margin of Error:
- Use the formula: Margin of Error = (t-critical value) × (sample standard deviation / √n).
7. Calculate 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, you find that the 80% confidence interval for the mean high temperature is approximately (97.02, 97.8). This means we are 80% confident that the true average high temperature of these towns lies within this interval.
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