Answer :
Sure, let's go through the calculation step-by-step to find the 90% confidence interval for the mean high temperature.
1. Collect the data and find the sample mean:
The high temperatures are:
[tex]\[
\{98.1, 96.7, 99.2, 99, 99.3, 99.8, 99.7, 97, 98.5\}
\][/tex]
The sample mean [tex]\(\overline{x}\)[/tex] is calculated as:
[tex]\[
\overline{x} = \frac{\sum x_i}{n} = \frac{98.1 + 96.7 + 99.2 + 99 + 99.3 + 99.8 + 99.7 + 97 + 98.5}{9} = 98.5889 \text{ (rounded to 4 decimal places)}
\][/tex]
2. Calculate the sample standard deviation:
The sample standard deviation (s) is calculated using:
[tex]\[
s = \sqrt{\frac{\sum (x_i - \overline{x})^2}{n-1}}
\][/tex]
After calculating this, the sample standard deviation is approximately:
[tex]\[
s \approx 1.123
\][/tex]
3. Calculate the standard error of the mean:
The standard error (SE) is:
[tex]\[
SE = \frac{s}{\sqrt{n}} = \frac{1.123}{\sqrt{9}} = 0.3743
\][/tex]
4. Determine the t critical value:
For a 90% confidence interval with [tex]\(n - 1 = 8\)[/tex] degrees of freedom, the t critical value can be found using the t-distribution table or statistical software. The t critical value is approximately:
[tex]\[
t \approx 1.860
\][/tex]
5. Calculate the margin of error:
The margin of error (ME) is:
[tex]\[
ME = t \times SE = 1.860 \times 0.3743 = 0.6961
\][/tex]
6. Determine the confidence interval:
The 90% confidence interval for the mean is:
[tex]\[
\left( \overline{x} - ME, \overline{x} + ME \right) = (98.5889 - 0.6961, 98.5889 + 0.6961) = (97.8928, 99.2850)
\][/tex]
So, the 90% confidence interval of the mean high temperature of these towns is:
[tex]\[
(97.89, 99.29)
\][/tex]
I hope this helps! If you have any more questions or need further clarifications, feel free to ask.
1. Collect the data and find the sample mean:
The high temperatures are:
[tex]\[
\{98.1, 96.7, 99.2, 99, 99.3, 99.8, 99.7, 97, 98.5\}
\][/tex]
The sample mean [tex]\(\overline{x}\)[/tex] is calculated as:
[tex]\[
\overline{x} = \frac{\sum x_i}{n} = \frac{98.1 + 96.7 + 99.2 + 99 + 99.3 + 99.8 + 99.7 + 97 + 98.5}{9} = 98.5889 \text{ (rounded to 4 decimal places)}
\][/tex]
2. Calculate the sample standard deviation:
The sample standard deviation (s) is calculated using:
[tex]\[
s = \sqrt{\frac{\sum (x_i - \overline{x})^2}{n-1}}
\][/tex]
After calculating this, the sample standard deviation is approximately:
[tex]\[
s \approx 1.123
\][/tex]
3. Calculate the standard error of the mean:
The standard error (SE) is:
[tex]\[
SE = \frac{s}{\sqrt{n}} = \frac{1.123}{\sqrt{9}} = 0.3743
\][/tex]
4. Determine the t critical value:
For a 90% confidence interval with [tex]\(n - 1 = 8\)[/tex] degrees of freedom, the t critical value can be found using the t-distribution table or statistical software. The t critical value is approximately:
[tex]\[
t \approx 1.860
\][/tex]
5. Calculate the margin of error:
The margin of error (ME) is:
[tex]\[
ME = t \times SE = 1.860 \times 0.3743 = 0.6961
\][/tex]
6. Determine the confidence interval:
The 90% confidence interval for the mean is:
[tex]\[
\left( \overline{x} - ME, \overline{x} + ME \right) = (98.5889 - 0.6961, 98.5889 + 0.6961) = (97.8928, 99.2850)
\][/tex]
So, the 90% confidence interval of the mean high temperature of these towns is:
[tex]\[
(97.89, 99.29)
\][/tex]
I hope this helps! If you have any more questions or need further clarifications, feel free to ask.
