To calculate the margin of error and construct the confidence interval for the population mean using the Student's t-distribution, follow these steps:

Step 1: Identify a sample statistic. Since we are trying to estimate a population mean, we choose the sample mean (x̄) as the sample statistic. From the problem, we know that x̄ = 36.4.

Step 2: Select a confidence level. In this analysis, the confidence level is defined for us in the problem. We are working with a 98% confidence level.

Step 3: Find the standard deviation or standard error. Since we do not know the standard deviation of the population, we cannot compute the standard deviation of the sample mean. Instead, we compute the standard error (SE). SE of the sample is the standard deviation of the sample divided by the square root of the sample size. In this case, SE = s/√n = 17/√43 = 2.59 (rounded to two decimal places).

Step 4: Find the critical value. The critical value is a factor used to compute the margin of error. Because the standard deviation of the population is unknown, we express the critical value as a t-score rather than a z-score. To find the critical value, we take these steps.
- Compute alpha (α): α = 1 - (confidence level / 100) = 1 - 0.98 = 0.02.
- Find the critical probability (p*): p* = 1 - α/2 = 1 - 0.02/2 = 0.99.
- Find the degrees of freedom (df): df = n - 1 = 43 - 1 = 42.
- Find the critical value. You can use a t-distribution table or an online calculator to find the critical value corresponding to 42 degrees of freedom and a critical probability of 0.99. The critical value is approximately 2.68.

Step 5: Compute the margin of error (ME). ME = critical value * standard error = 2.68 * 2.59 = 6.94 (rounded to two decimal places).

Step 6: Construct the confidence interval. Confidence Interval = x̄ ± ME = 36.4 ± 6.94. So, the confidence interval is (29.46, 43.34).

So, the margin of error is 6.94 and the confidence interval is (29.46, 43.34).

Question

To calculate the margin of error and construct the confidence interval for the population mean using the Student's t-distribution, follow these steps:

Step 1: Identify a sample statistic. Since we are trying to estimate a population mean, we choose the sample mean (x̄) as the sample statistic. From the problem, we know that x̄ = 36.4.

Step 2: Select a confidence level. In this analysis, the confidence level is defined for us in the problem. We are working with a 98% confidence level.

Step 3: Find the standard deviation or standard error. Since we do not know the standard deviation of the population, we cannot compute the standard deviation of the sample mean. Instead, we compute the standard error (SE). SE of the sample is the standard deviation of the sample divided by the square root of the sample size. In this case, SE = s/√n = 17/√43 = 2.59 (rounded to two decimal places).

Step 4: Find the critical value. The critical value is a factor used to compute the margin of error. Because the standard deviation of the population is unknown, we express the critical value as a t-score rather than a z-score. To find the critical value, we take these steps.
- Compute alpha (α): α = 1 - (confidence level / 100) = 1 - 0.98 = 0.02.
- Find the critical probability (p*): p* = 1 - α/2 = 1 - 0.02/2 = 0.99.
- Find the degrees of freedom (df): df = n - 1 = 43 - 1 = 42.
- Find the critical value. You can use a t-distribution table or an online calculator to find the critical value corresponding to 42 degrees of freedom and a critical probability of 0.99. The critical value is approximately 2.68.

Step 5: Compute the margin of error (ME). ME = critical value * standard error = 2.68 * 2.59 = 6.94 (rounded to two decimal places).

Step 6: Construct the confidence interval. Confidence Interval = x̄ ± ME = 36.4 ± 6.94. So, the confidence interval is (29.46, 43.34).

So, the margin of error is 6.94 and the confidence interval is (29.46, 43.34).

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