Answer :
To construct a 95% confidence interval estimate of the mean of the population, follow these steps:
1. Identify the Given Values:
- Sample size ([tex]\( n \)[/tex]) = 62
- Population standard deviation ([tex]\( \sigma \)[/tex]) = 11.4
- Sample mean ([tex]\( \bar{x} \)[/tex]) = 102.1
2. Determine the Critical Z-Value:
For a 95% confidence level, the critical Z-value is 1.96. This value corresponds to the standard normal distribution where 95% of the data falls within this Z-score.
3. Calculate the Standard Error:
The standard error (SE) of the sample mean is calculated using the formula:
[tex]\[
SE = \frac{\sigma}{\sqrt{n}}
\][/tex]
where [tex]\( \sigma \)[/tex] is the population standard deviation and [tex]\( n \)[/tex] is the sample size. In this case:
[tex]\[
SE = \frac{11.4}{\sqrt{62}} \approx 1.448
\][/tex]
4. Compute the Margin of Error:
The margin of error (ME) is calculated using the formula:
[tex]\[
ME = Z \times SE
\][/tex]
where [tex]\( Z \)[/tex] is the critical Z-value. In this example:
[tex]\[
ME = 1.96 \times 1.448 \approx 2.838
\][/tex]
5. Determine the Confidence Interval:
The confidence interval is found by subtracting and adding the margin of error from the sample mean:
- Lower Bound: [tex]\( \bar{x} - ME = 102.1 - 2.838 \approx 99.262 \)[/tex]
- Upper Bound: [tex]\( \bar{x} + ME = 102.1 + 2.838 \approx 104.938 \)[/tex]
6. Conclusion:
Therefore, the 95% confidence interval for the population mean is approximately [tex]\( (99.3, 104.9) \)[/tex].
Among the given options, the correct choice that matches this interval is:
[tex]\( 99.3 < \mu < 104.9 \)[/tex]
1. Identify the Given Values:
- Sample size ([tex]\( n \)[/tex]) = 62
- Population standard deviation ([tex]\( \sigma \)[/tex]) = 11.4
- Sample mean ([tex]\( \bar{x} \)[/tex]) = 102.1
2. Determine the Critical Z-Value:
For a 95% confidence level, the critical Z-value is 1.96. This value corresponds to the standard normal distribution where 95% of the data falls within this Z-score.
3. Calculate the Standard Error:
The standard error (SE) of the sample mean is calculated using the formula:
[tex]\[
SE = \frac{\sigma}{\sqrt{n}}
\][/tex]
where [tex]\( \sigma \)[/tex] is the population standard deviation and [tex]\( n \)[/tex] is the sample size. In this case:
[tex]\[
SE = \frac{11.4}{\sqrt{62}} \approx 1.448
\][/tex]
4. Compute the Margin of Error:
The margin of error (ME) is calculated using the formula:
[tex]\[
ME = Z \times SE
\][/tex]
where [tex]\( Z \)[/tex] is the critical Z-value. In this example:
[tex]\[
ME = 1.96 \times 1.448 \approx 2.838
\][/tex]
5. Determine the Confidence Interval:
The confidence interval is found by subtracting and adding the margin of error from the sample mean:
- Lower Bound: [tex]\( \bar{x} - ME = 102.1 - 2.838 \approx 99.262 \)[/tex]
- Upper Bound: [tex]\( \bar{x} + ME = 102.1 + 2.838 \approx 104.938 \)[/tex]
6. Conclusion:
Therefore, the 95% confidence interval for the population mean is approximately [tex]\( (99.3, 104.9) \)[/tex].
Among the given options, the correct choice that matches this interval is:
[tex]\( 99.3 < \mu < 104.9 \)[/tex]
