The sampling distribution of x is Standard error (SE) = o / √n = 14 / √49 = 2
(a) The sampling distribution of the sample mean x from a simple random sample can be approximated by a normal distribution when the sample size is sufficiently large (n ≥ 30), regardless of the distribution of the population. The mean of the sampling distribution of x is equal to the population mean u, which is 76 in this case. The standard deviation of the sampling distribution of x, also known as the standard error of the mean, is equal to the population standard deviation o divided by the square root of the sample size:
Standard error (SE) = o / √n = 14 / √49 = 2
(b) To find P(x > 78.1), we need to standardize the value of 78.1 using the sampling distribution parameters (u = 76 and SE = 2) and then find the area under the standard normal distribution curve to the right of the standardized value. Let z be the standardized value:
z = (78.1 - 76) / 2 = 1.05
Using a standard normal distribution table or calculator, we find the area corresponding to z = 1.05 is approximately 0.8531. Therefore, P(x > 78.1) ≈ 1 - 0.8531 = 0.1469.
(c) To find P(x < 71.7), we again standardize the value of 71.7 and find the area under the standard normal distribution curve to the left of the standardized value:
z = (71.7 - 76) / 2 = -2.15
Using the standard normal distribution table or calculator, we find the area corresponding to z = -2.15 is approximately 0.0158. Therefore, P(x < 71.7) ≈ 0.0158.
(d) To find P(74 < x < 78.1), we standardize both values and find the area between the standardized values:
z1 = (74 - 76) / 2 = -1.00
z2 = (78.1 - 76) / 2 = 1.05
Using the standard normal distribution table or calculator, we find the area corresponding to z = -1.00 is approximately 0.1587, and the area corresponding to z = 1.05 is approximately 0.8531. Therefore, P(74 < x < 78.1) ≈ 0.8531 - 0.1587 = 0.6944.