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Answer :
To construct a [tex]$95\%$[/tex] confidence interval for the population mean, we follow these steps:
1. Calculate the standard error (SE) of the sample mean:
[tex]$$
SE = \frac{\sigma}{\sqrt{n}} = \frac{11.4}{\sqrt{62}} \approx 1.4478.
$$[/tex]
2. Determine the critical value for a [tex]$95\%$[/tex] confidence level. For a normal distribution, this value is [tex]$z^* \approx 1.96$[/tex].
3. Compute the margin of error (ME):
[tex]$$
ME = z^* \times SE = 1.96 \times 1.4478 \approx 2.8377.
$$[/tex]
4. Establish the lower and upper bounds of the confidence interval:
[tex]$$
\text{Lower bound} = \bar{x} - ME = 102.1 - 2.8377 \approx 99.2623,
$$[/tex]
[tex]$$
\text{Upper bound} = \bar{x} + ME = 102.1 + 2.8377 \approx 104.9377.
$$[/tex]
Thus, the [tex]$95\%$[/tex] confidence interval for the population mean is approximately:
[tex]$$
99.3 < \mu < 104.9.
$$[/tex]
This matches the third option.
1. Calculate the standard error (SE) of the sample mean:
[tex]$$
SE = \frac{\sigma}{\sqrt{n}} = \frac{11.4}{\sqrt{62}} \approx 1.4478.
$$[/tex]
2. Determine the critical value for a [tex]$95\%$[/tex] confidence level. For a normal distribution, this value is [tex]$z^* \approx 1.96$[/tex].
3. Compute the margin of error (ME):
[tex]$$
ME = z^* \times SE = 1.96 \times 1.4478 \approx 2.8377.
$$[/tex]
4. Establish the lower and upper bounds of the confidence interval:
[tex]$$
\text{Lower bound} = \bar{x} - ME = 102.1 - 2.8377 \approx 99.2623,
$$[/tex]
[tex]$$
\text{Upper bound} = \bar{x} + ME = 102.1 + 2.8377 \approx 104.9377.
$$[/tex]
Thus, the [tex]$95\%$[/tex] confidence interval for the population mean is approximately:
[tex]$$
99.3 < \mu < 104.9.
$$[/tex]
This matches the third option.
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