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.
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