Answer :
To construct a 95% confidence interval estimate of the mean of the population, we are provided with the following information:
- Sample size ([tex]\(n\)[/tex]) = 62
- Population standard deviation ([tex]\(\sigma\)[/tex]) = 11.4
- Sample mean ([tex]\(\bar{x}\)[/tex]) = 102.1
Here's how to determine the confidence interval:
1. Identify the Z-value for a 95% confidence level:
- For a 95% confidence level, the Z-value is approximately 1.96. This is derived from standard normal distribution tables.
2. Calculate the margin of error:
- The margin of error (E) is given by the formula:
[tex]\[
E = Z \times \left(\frac{\sigma}{\sqrt{n}}\right)
\][/tex]
- Plugging in the values, the margin of error is approximately 2.84.
3. Calculate the confidence interval:
- The confidence interval is calculated using the formula:
[tex]\[
\text{Confidence interval} = \left(\bar{x} - E, \, \bar{x} + E\right)
\][/tex]
- Substituting the values, the confidence interval is approximately [tex]\((99.3, 104.9)\)[/tex].
Thus, the 95% confidence interval estimate for the mean of the population is [tex]\(99.3 < \mu < 104.9\)[/tex]. This corresponds to the option: [tex]\(99.3 < \mu < 104.9\)[/tex].
- Sample size ([tex]\(n\)[/tex]) = 62
- Population standard deviation ([tex]\(\sigma\)[/tex]) = 11.4
- Sample mean ([tex]\(\bar{x}\)[/tex]) = 102.1
Here's how to determine the confidence interval:
1. Identify the Z-value for a 95% confidence level:
- For a 95% confidence level, the Z-value is approximately 1.96. This is derived from standard normal distribution tables.
2. Calculate the margin of error:
- The margin of error (E) is given by the formula:
[tex]\[
E = Z \times \left(\frac{\sigma}{\sqrt{n}}\right)
\][/tex]
- Plugging in the values, the margin of error is approximately 2.84.
3. Calculate the confidence interval:
- The confidence interval is calculated using the formula:
[tex]\[
\text{Confidence interval} = \left(\bar{x} - E, \, \bar{x} + E\right)
\][/tex]
- Substituting the values, the confidence interval is approximately [tex]\((99.3, 104.9)\)[/tex].
Thus, the 95% confidence interval estimate for the mean of the population is [tex]\(99.3 < \mu < 104.9\)[/tex]. This corresponds to the option: [tex]\(99.3 < \mu < 104.9\)[/tex].
