Answer :
To construct a 95% confidence interval estimate of the mean of the population, follow these steps:
1. Identify the Given Information:
- Sample size, [tex]\( n = 62 \)[/tex]
- Population standard deviation, [tex]\( \sigma = 11.4 \)[/tex]
- Sample mean, [tex]\( \bar{x} = 102.1 \)[/tex]
- Confidence level, [tex]\( 95\% \)[/tex]
2. Find the Z-Score for a 95% Confidence Level:
A 95% confidence level means we are looking for the Z-score that leaves 2.5% in each tail of the normal distribution. The Z-score for 95% confidence is approximately [tex]\( 1.96 \)[/tex].
3. Calculate the Standard Error:
The standard error (SE) helps to find how much the sample mean is expected to vary from the true population mean. It is calculated using the formula:
[tex]\[
\text{Standard Error (SE)} = \frac{\sigma}{\sqrt{n}}
\][/tex]
Substituting the given values:
[tex]\[
\text{SE} = \frac{11.4}{\sqrt{62}} \approx 1.45
\][/tex]
4. Calculate the Margin of Error:
The margin of error determines the width of the confidence interval around the sample mean. It is calculated by multiplying the Z-score by the standard error:
[tex]\[
\text{Margin of Error} = Z \times \text{SE} = 1.96 \times 1.45 \approx 2.84
\][/tex]
5. Construct the Confidence Interval:
To find the confidence interval, subtract and add the margin of error from/to the sample mean:
[tex]\[
\text{Lower Bound} = \bar{x} - \text{Margin of Error} = 102.1 - 2.84 \approx 99.3
\][/tex]
[tex]\[
\text{Upper Bound} = \bar{x} + \text{Margin of Error} = 102.1 + 2.84 \approx 104.9
\][/tex]
Therefore, the 95% confidence interval for the mean of the population is [tex]\( 99.3 < \mu < 104.9 \)[/tex].
1. Identify the Given Information:
- Sample size, [tex]\( n = 62 \)[/tex]
- Population standard deviation, [tex]\( \sigma = 11.4 \)[/tex]
- Sample mean, [tex]\( \bar{x} = 102.1 \)[/tex]
- Confidence level, [tex]\( 95\% \)[/tex]
2. Find the Z-Score for a 95% Confidence Level:
A 95% confidence level means we are looking for the Z-score that leaves 2.5% in each tail of the normal distribution. The Z-score for 95% confidence is approximately [tex]\( 1.96 \)[/tex].
3. Calculate the Standard Error:
The standard error (SE) helps to find how much the sample mean is expected to vary from the true population mean. It is calculated using the formula:
[tex]\[
\text{Standard Error (SE)} = \frac{\sigma}{\sqrt{n}}
\][/tex]
Substituting the given values:
[tex]\[
\text{SE} = \frac{11.4}{\sqrt{62}} \approx 1.45
\][/tex]
4. Calculate the Margin of Error:
The margin of error determines the width of the confidence interval around the sample mean. It is calculated by multiplying the Z-score by the standard error:
[tex]\[
\text{Margin of Error} = Z \times \text{SE} = 1.96 \times 1.45 \approx 2.84
\][/tex]
5. Construct the Confidence Interval:
To find the confidence interval, subtract and add the margin of error from/to the sample mean:
[tex]\[
\text{Lower Bound} = \bar{x} - \text{Margin of Error} = 102.1 - 2.84 \approx 99.3
\][/tex]
[tex]\[
\text{Upper Bound} = \bar{x} + \text{Margin of Error} = 102.1 + 2.84 \approx 104.9
\][/tex]
Therefore, the 95% confidence interval for the mean of the population is [tex]\( 99.3 < \mu < 104.9 \)[/tex].
