Answer :
To estimate the population mean [tex]\(\mu\)[/tex] from a sample and calculate a 95% confidence interval, follow these steps:
1. Collect the Sample Data:
You have a sample of data with the following values:
[tex]\[
69.7, 79.8, 61, 69.9, 54.3, 36.7, 80, 101.3, 77.7, 52.4, 67.4, 79.1,
92.3, 94.6, 75.5, 86.4, 42.1, 70.2, 80, 65.4, 85.8, 78.6, 68.7, 78.7,
75.4, 62.1, 91, 108.6, 69.2, 116.8, 42.7, 98.6, 55.9, 56.6, 67.2, 56.9,
85.4, 92.3, 66, 80.2, 66.8, 72.5, 82, 86.3, 67.6, 92.5, 47.9, 97.6,
96.9, 54.5
\][/tex]
2. Calculate the Sample Mean ([tex]\(\bar{x}\)[/tex]):
The average of this sample is calculated to be:
[tex]\[
\bar{x} = 74.742
\][/tex]
3. Calculate the Sample Standard Deviation ([tex]\(s\)[/tex]):
The standard deviation of the sample, which measures how spread out the numbers are, is:
[tex]\[
s = 17.49
\][/tex]
4. Determine the Sample Size ([tex]\(n\)[/tex]):
From the data, the number of observations [tex]\(n\)[/tex] is:
[tex]\[
n = 50
\][/tex]
5. Calculate the Standard Error (SE):
The standard error of the mean is calculated as:
[tex]\[
SE = \frac{s}{\sqrt{n}} = \frac{17.49}{\sqrt{50}} \approx 2.473
\][/tex]
6. Find the Critical Value (t):
For a 95% confidence level and degrees of freedom [tex]\(df = n - 1 = 49\)[/tex], we use the t-distribution to find the critical value. The critical t-value is approximately:
[tex]\[
t \approx 2.009
\][/tex]
(Note: The exact number can vary slightly based on different t-distribution tables or computational tools.)
7. Calculate the Margin of Error (ME):
The margin of error is found with:
[tex]\[
ME = t \times SE \approx 2.009 \times 2.473 \approx 4.97
\][/tex]
8. Determine the Confidence Interval:
The 95% confidence interval for the population mean is:
[tex]\[
\text{Lower bound} = \bar{x} - ME = 74.742 - 4.97 = 69.77
\][/tex]
[tex]\[
\text{Upper bound} = \bar{x} + ME = 74.742 + 4.97 = 79.71
\][/tex]
Thus, the 95% confidence interval for the population mean [tex]\(\mu\)[/tex] is:
[tex]\[
69.77 < \mu < 79.71
\][/tex]
1. Collect the Sample Data:
You have a sample of data with the following values:
[tex]\[
69.7, 79.8, 61, 69.9, 54.3, 36.7, 80, 101.3, 77.7, 52.4, 67.4, 79.1,
92.3, 94.6, 75.5, 86.4, 42.1, 70.2, 80, 65.4, 85.8, 78.6, 68.7, 78.7,
75.4, 62.1, 91, 108.6, 69.2, 116.8, 42.7, 98.6, 55.9, 56.6, 67.2, 56.9,
85.4, 92.3, 66, 80.2, 66.8, 72.5, 82, 86.3, 67.6, 92.5, 47.9, 97.6,
96.9, 54.5
\][/tex]
2. Calculate the Sample Mean ([tex]\(\bar{x}\)[/tex]):
The average of this sample is calculated to be:
[tex]\[
\bar{x} = 74.742
\][/tex]
3. Calculate the Sample Standard Deviation ([tex]\(s\)[/tex]):
The standard deviation of the sample, which measures how spread out the numbers are, is:
[tex]\[
s = 17.49
\][/tex]
4. Determine the Sample Size ([tex]\(n\)[/tex]):
From the data, the number of observations [tex]\(n\)[/tex] is:
[tex]\[
n = 50
\][/tex]
5. Calculate the Standard Error (SE):
The standard error of the mean is calculated as:
[tex]\[
SE = \frac{s}{\sqrt{n}} = \frac{17.49}{\sqrt{50}} \approx 2.473
\][/tex]
6. Find the Critical Value (t):
For a 95% confidence level and degrees of freedom [tex]\(df = n - 1 = 49\)[/tex], we use the t-distribution to find the critical value. The critical t-value is approximately:
[tex]\[
t \approx 2.009
\][/tex]
(Note: The exact number can vary slightly based on different t-distribution tables or computational tools.)
7. Calculate the Margin of Error (ME):
The margin of error is found with:
[tex]\[
ME = t \times SE \approx 2.009 \times 2.473 \approx 4.97
\][/tex]
8. Determine the Confidence Interval:
The 95% confidence interval for the population mean is:
[tex]\[
\text{Lower bound} = \bar{x} - ME = 74.742 - 4.97 = 69.77
\][/tex]
[tex]\[
\text{Upper bound} = \bar{x} + ME = 74.742 + 4.97 = 79.71
\][/tex]
Thus, the 95% confidence interval for the population mean [tex]\(\mu\)[/tex] is:
[tex]\[
69.77 < \mu < 79.71
\][/tex]