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Answer :
To construct a 95% confidence interval estimate of the mean of the population, we'll follow these steps using the given data:
1. Identify the Given Information:
- Sample size ([tex]\(n\)[/tex]) = 62
- Population standard deviation ([tex]\(\sigma\)[/tex]) = 11.4
- Sample mean ([tex]\(\bar{x}\)[/tex]) = 102.1
- Confidence level = 95%
2. Find the Z-Score for the Confidence Level:
- For a 95% confidence level, we determine the critical z-score. The z-score corresponds to the cumulative probability closest to 0.975 (since 0.95/2 + 0.50 = 0.975). The z-score for a 95% confidence level is approximately 1.96.
3. Calculate the Standard Error (SE):
- The standard error is calculated using the formula:
[tex]\[
\text{Standard Error (SE)} = \frac{\sigma}{\sqrt{n}}
\][/tex]
- Substituting the given values:
[tex]\[
\text{SE} = \frac{11.4}{\sqrt{62}}
\][/tex]
4. Calculate the Margin of Error (ME):
- The margin of error is found by multiplying the z-score by the standard error:
[tex]\[
\text{Margin of Error (ME)} = z \times \text{SE}
\][/tex]
- Using z ≈ 1.96, calculate the margin of error.
5. Determine the Confidence Interval:
- The confidence interval is determined by adding and subtracting the margin of error from the sample mean:
[tex]\[
\text{Lower bound} = \bar{x} - \text{ME}
\][/tex]
[tex]\[
\text{Upper bound} = \bar{x} + \text{ME}
\][/tex]
6. Result:
- After completing the calculations, the 95% confidence interval for the population mean is approximately:
[tex]\[
99.3 < \mu < 104.9
\][/tex]
By following these steps, we establish the confidence interval estimate for the mean using the given data, verifying that the correct interval is [tex]\(99.3 < \mu < 104.9\)[/tex].
1. Identify the Given Information:
- Sample size ([tex]\(n\)[/tex]) = 62
- Population standard deviation ([tex]\(\sigma\)[/tex]) = 11.4
- Sample mean ([tex]\(\bar{x}\)[/tex]) = 102.1
- Confidence level = 95%
2. Find the Z-Score for the Confidence Level:
- For a 95% confidence level, we determine the critical z-score. The z-score corresponds to the cumulative probability closest to 0.975 (since 0.95/2 + 0.50 = 0.975). The z-score for a 95% confidence level is approximately 1.96.
3. Calculate the Standard Error (SE):
- The standard error is calculated using the formula:
[tex]\[
\text{Standard Error (SE)} = \frac{\sigma}{\sqrt{n}}
\][/tex]
- Substituting the given values:
[tex]\[
\text{SE} = \frac{11.4}{\sqrt{62}}
\][/tex]
4. Calculate the Margin of Error (ME):
- The margin of error is found by multiplying the z-score by the standard error:
[tex]\[
\text{Margin of Error (ME)} = z \times \text{SE}
\][/tex]
- Using z ≈ 1.96, calculate the margin of error.
5. Determine the Confidence Interval:
- The confidence interval is determined by adding and subtracting the margin of error from the sample mean:
[tex]\[
\text{Lower bound} = \bar{x} - \text{ME}
\][/tex]
[tex]\[
\text{Upper bound} = \bar{x} + \text{ME}
\][/tex]
6. Result:
- After completing the calculations, the 95% confidence interval for the population mean is approximately:
[tex]\[
99.3 < \mu < 104.9
\][/tex]
By following these steps, we establish the confidence interval estimate for the mean using the given data, verifying that the correct interval is [tex]\(99.3 < \mu < 104.9\)[/tex].
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