Question

Construct the indicated confidence interval for the population mean μ using the t-distribution. Assume the population is normally distributed. c=0.95, x= 12.2, s= 3.0, n=8

Answer 1

To construct the confidence interval for the population mean using the t-distribution, calculate the standard error (SE), determine the critical value (CV), multiply the SE by the CV to get the margin of error (ME), and form the confidence interval as (x - ME, x + ME)

Step-by-step explanation:

To construct a confidence interval for the population mean using the t-distribution, we need the sample mean (x), the sample standard deviation (s), the sample size (n), and the desired confidence level (c).

Given the values:

• x = 12.2
• s = 3.0
• n = 8
• c = 0.95 (corresponding to a 95% confidence level)

We can calculate the confidence interval as follows:

1. Calculate the standard error (SE) using the formula: SE = s / sqrt(n).
2. Determine the critical value (CV) by finding the t-value for a (1 - c) / 2 confidence level and n - 1 degrees of freedom.
3. Multiply the SE by the CV to get the margin of error (ME): ME = CV * SE.
4. The confidence interval is then (x - ME, x + ME).

Plugging in the values:

1. SE = 3.0 / sqrt(8) ≈ 1.06
2. CV = t-value for (1 - 0.95) / 2 and 8 - 1 degrees of freedom ≈ 2.306
3. ME = 2.306 * 1.06 ≈ 2.447
4. The confidence interval is (12.2 - 2.447, 12.2 + 2.447), which simplifies to (9.753, 14.647).