2 Probability and Stats questions!

1st question: suppose a sample of 85 is taken from a population with a mean that is claimed to be 38 and a standard deviation of 9. if this is the actual mean of the population, which of these values would be within the 95% confidence interval for the mean of the sample? Answer choices: A) 28.2 B) 32.2 C) 36.2 D) 40.2

2nd question: suppose a sample of 102 is taken from a population with a mean that is claimed to be 76 and a standard deviation of 20. If this is the actual mean of the population, which of these values would be outside the 95% confidence interval for the mean of the sample? Answer choices: A) 71.8 B) 74.8 C) 76.8 D) 79.8

1) To find the confidence interval

the sample mean x = 38 σ = 9; n = 85;

The confidence level is 95% (CL = 0.95) CL = 0.95

so α = 1 - CL = 0.05

α/2 = 0.025 Z (α/2) = z0.025

The area to the right of Z0.025 is 0.025 and the area to the left of Z0.025 is 1 - 0.025 = 0.975

Z (α/2) = z0.025 = 1.645 This can be found using a computer, or using a probability table for the standard normal distribution.

EBM = (1.645) * (9) / (85^0.5) = 1.6058 x - EBM = 38 - 1.6058 = 36.3941 x + EBM = 38 + 1.6058 = 39.6058

The 95% confidence interval is (36.3941, 39.6058).

The answer is the letter D

The value of 40.2 is within the 95% confidence interval for the mean of the sample

2) To find the confidence interval

the sample mean x = 76 σ = 20; n = 102;

The confidence level is 95% (CL = 0.95) CL = 0.95

so α = 1 - CL = 0.05

α/2 = 0.025 Z (α/2) = z0.025

The area to the right of Z0.025 is 0.025 and the area to the left of Z0.025 is 1 - 0.025 = 0.975

Z (α/2) = z0.025 = 1.645 This can be found using a computer, or using a probability table for the standard normal distribution.

EBM = (1.645) * (20) / (102^0.5) = 3.2575 x - EBM = 76 - 3.2575 = 72.7424 x + EBM = 76 + 3.2575 = 79.2575

The 95% confidence interval is (72.7424,79.2575).

The answer is the letter A and the letter D

The value of 71.8 and 79.8 are outside the 95% confidence interval for the mean of the sample