Final answer:
To calculate the 95% confidence interval for the population mean, you need to calculate the sample mean, standard error, and margin of error. The sample mean is 75.8, the standard error is 2.100, and the margin of error is 4.7482. Therefore, the 95% confidence interval for the population mean is (70.05, 81.55).
Step-by-step explanation:
To calculate the 95% confidence interval for the population mean, we first need to calculate the sample mean and the standard error. Let's calculate step by step:
Step 1: Calculate the sample mean:
Sample mean = (65 + 71 + 80 + 76 + 78 + 82 + 68 + 72 + 65 + 81) / 10 = 75.8
Step 2: Calculate the standard error:
Standard error = sqrt(variance / sample size) = sqrt(44.1 / 10) = 2.100
Step 3: Calculate the margin of error:
Margin of error = critical value * standard error. Since we want a 95% confidence interval, the critical value corresponding to the sample size of 10 is 2.262.
Margin of error = 2.262 * 2.100 = 4.7482
Step 4: Calculate the confidence interval:
Confidence interval = sample mean ± margin of error = 75.8 ± 4.7482 = (70.0518, 81.5482)
Therefore, the 95% confidence interval for the population mean is (70.05, 81.55), which is closest to option (d) (68.58,79.42).