Question

Given a normal population whose mean is 50 and whose standard deviation is 5, find the probability that a random sample of a. 4 has a mean between 49 and 52. b. 16 has a mean between 49 and 52. c. 25 has a mean between 49 and 52

Answer 1

$\mu=50 \\ \sigma=5$

a. n=4

$P(49<\bar{x}<52) = P(\bar{x}<52) -P(\bar{x}<49) \\ =P(Z< \frac{52-50}{5 / \sqrt{4}}) -P(Z< \frac{49-50}{5 / \sqrt{4}}) \\ =P(Z< 0.8) -P(Z< -0.4) \\ = 0.7881 - 0.3445 \\ =0.4436$

b. n=16

$P(49<\bar{x}<52) = P(\bar{x}<52) -P(\bar{x}<49) \\ =P(Z< \frac{52-50}{5 / \sqrt{16}}) -P(Z< \frac{49-50}{5 / \sqrt{16}}) \\ =P(Z< 1.6) -P(Z< -0.8) \\ = 0.9452 - 0.2118 \\ =0.7334$

c. n=25

$P(49<\bar{x}<52) = P(\bar{x}<52) -P(\bar{x}<49) \\ =P(Z< \frac{52-50}{5 / \sqrt{25}}) -P(Z< \frac{49-50}{5 / \sqrt{25}}) \\ =P(Z< 2) -P(Z< -1) \\ = 0.9772 - 0.1586 \\ =0.8186$