Provide an appropriate response.

A random sample of 40 students has a test score with [tex]\bar{x}=81.5[/tex]. Assume the population standard deviation is 10.2. Construct the confidence interval for the population mean, [tex]\mu[/tex], if [tex]c = 0.90[/tex].

Select one:
A. [tex](78.8, 84.2)[/tex]
B. [tex](51.8, 92.3)[/tex]
C. [tex](71.8, 93.5)[/tex]
D. [tex](66.3, 89.1)[/tex]



Answer :

To construct the confidence interval for the population mean, we can follow these steps:

1. Identify the given information:
- Sample mean, [tex]\( \bar{x} = 81.5 \)[/tex]
- Population standard deviation, [tex]\( \sigma = 10.2 \)[/tex]
- Sample size, [tex]\( n = 40 \)[/tex]
- Confidence level, [tex]\( c = 0.90 \)[/tex]

2. Determine the critical value (z-score) for the desired confidence level:
- For a 90% confidence level, the z-score can be found using standard z-tables or statistical tools, resulting in approximately [tex]\( z = 1.645 \)[/tex].

3. Calculate the standard error of the mean (SE):
- The standard error is given by:
[tex]\[ SE = \frac{\sigma}{\sqrt{n}} \][/tex]
Substituting the given values:
[tex]\[ SE = \frac{10.2}{\sqrt{40}} \approx 1.613 \][/tex]

4. Compute the margin of error (ME):
- The margin of error is found by multiplying the z-score and the standard error:
[tex]\[ ME = z \times SE \][/tex]
[tex]\[ ME = 1.645 \times 1.613 \approx 2.6528 \][/tex]

5. Calculate the confidence interval:
- The lower bound of the confidence interval is:
[tex]\[ \bar{x} - ME = 81.5 - 2.6528 \approx 78.847 \][/tex]
- The upper bound of the confidence interval is:
[tex]\[ \bar{x} + ME = 81.5 + 2.6528 \approx 84.153 \][/tex]

Based on these calculations, the confidence interval for the population mean is approximately [tex]\( (78.8, 84.2) \)[/tex].

Therefore, the correct answer is:
A. [tex]\( (78.8, 84.2) \)[/tex]