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(20 points) A student takes the temperatures of n randomly selected people. Assume that the temperatures follow a normal distribution, that the sample mean of the temperatures was 98.3 and that the sample standard deviation of the temperatures was 0.3127 (a) If n=50, Find 90% confidence interval for μ. (b) If n=8, Find a 90% confidence interval for μ.

Answer :

For (a) when n = 50, the 90% confidence interval for μ is approximately 98.2274 to 98.3726, and for (b) when n = 8, the 90% confidence interval for μ is approximately 98.118 to 98.482.

(a) To find the 90% confidence interval for μ when n = 50, we will use the formula:

Confidence interval = sample mean ± (critical value) * (standard deviation / sqrt(n))

1. First, we need to find the critical value. Since the sample size is large (n > 30) and the temperatures are assumed to follow a normal distribution, we can use the Z-distribution.
- Look up the critical value for a 90% confidence level in the Z-table. It is approximately 1.645.

2. Next, substitute the values into the formula:
- Sample mean = 98.3
- Standard deviation = 0.3127
- Sample size (n) = 50

Confidence interval = 98.3 ± (1.645 * (0.3127 / sqrt(50)))

3. Calculate the confidence interval:
- Confidence interval = 98.3 ± (1.645 * (0.3127 / 7.071))

Simplifying further:
- Confidence interval = 98.3 ± (1.645 * 0.0442)

Finally:
- Confidence interval = 98.3 ± 0.0726

The 90% confidence interval for μ is approximately 98.2274 to 98.3726.

(b) To find the 90% confidence interval for μ when n = 8, we will use the same formula as above:

Confidence interval = sample mean ± (critical value) * (standard deviation / sqrt(n))

1. Again, we need to find the critical value from the Z-table for a 90% confidence level. It is still approximately 1.645.

2. Substitute the values into the formula:
- Sample mean = 98.3
- Standard deviation = 0.3127
- Sample size (n) = 8

Confidence interval = 98.3 ± (1.645 * (0.3127 / sqrt(8)))

3. Calculate the confidence interval:
- Confidence interval = 98.3 ± (1.645 * (0.3127 / 2.828))

Simplifying further:
- Confidence interval = 98.3 ± (1.645 * 0.1106)

Finally:
- Confidence interval = 98.3 ± 0.182

The 90% confidence interval for μ is approximately 98.118 to 98.482.

Learn more about confidence interval from this link

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