Answer :
To find the 99% confidence interval of the mean high temperature for the given sample of small towns, we'll follow these steps:
1. List the given temperatures:
- 99.5, 97.3, 99.4, 96.4, 99.0, 97.2, 97.4
2. Calculate the sample size ([tex]\(n\)[/tex]):
- There are 7 towns, so [tex]\(n = 7\)[/tex].
3. Find the sample mean ([tex]\(\bar{x}\)[/tex]):
- The mean of these temperatures is approximately 98.03.
4. Calculate the sample standard deviation (s):
- The standard deviation is approximately 1.24.
5. Set the significance level ([tex]\(\alpha\)[/tex]):
- For a 99% confidence interval, [tex]\(\alpha = 1 - 0.99 = 0.01\)[/tex].
6. Determine the critical t-value:
- Since the sample size is small (less than 30), we use the t-distribution.
- With [tex]\(n - 1 = 6\)[/tex] degrees of freedom and a confidence level of 99%, the critical t-value is approximately 3.707.
7. Calculate the margin of error (E):
- [tex]\( E = \text{critical t-value} \times \left(\frac{s}{\sqrt{n}}\right) \)[/tex]
- [tex]\( E \approx 3.707 \times \left(\frac{1.24}{\sqrt{7}}\right) \approx 1.74\)[/tex]
8. Determine the confidence interval:
- Lower limit = [tex]\(\bar{x} - E \approx 98.03 - 1.74 = 96.29\)[/tex]
- Upper limit = [tex]\(\bar{x} + E \approx 98.03 + 1.74 = 99.77\)[/tex]
Therefore, the 99% confidence interval for the mean high temperature of the towns is approximately [tex]\((96.29, 99.77)\)[/tex]. This means we are 99% confident that the true mean high temperature lies within this interval.
1. List the given temperatures:
- 99.5, 97.3, 99.4, 96.4, 99.0, 97.2, 97.4
2. Calculate the sample size ([tex]\(n\)[/tex]):
- There are 7 towns, so [tex]\(n = 7\)[/tex].
3. Find the sample mean ([tex]\(\bar{x}\)[/tex]):
- The mean of these temperatures is approximately 98.03.
4. Calculate the sample standard deviation (s):
- The standard deviation is approximately 1.24.
5. Set the significance level ([tex]\(\alpha\)[/tex]):
- For a 99% confidence interval, [tex]\(\alpha = 1 - 0.99 = 0.01\)[/tex].
6. Determine the critical t-value:
- Since the sample size is small (less than 30), we use the t-distribution.
- With [tex]\(n - 1 = 6\)[/tex] degrees of freedom and a confidence level of 99%, the critical t-value is approximately 3.707.
7. Calculate the margin of error (E):
- [tex]\( E = \text{critical t-value} \times \left(\frac{s}{\sqrt{n}}\right) \)[/tex]
- [tex]\( E \approx 3.707 \times \left(\frac{1.24}{\sqrt{7}}\right) \approx 1.74\)[/tex]
8. Determine the confidence interval:
- Lower limit = [tex]\(\bar{x} - E \approx 98.03 - 1.74 = 96.29\)[/tex]
- Upper limit = [tex]\(\bar{x} + E \approx 98.03 + 1.74 = 99.77\)[/tex]
Therefore, the 99% confidence interval for the mean high temperature of the towns is approximately [tex]\((96.29, 99.77)\)[/tex]. This means we are 99% confident that the true mean high temperature lies within this interval.
