Answer to Question #330031 in Statistics and Probability for Rose

Question

an electrical company claims that an average life of bulbs it manufactures is 1200 hours with standard deviation of 250 hours.If a random sample of 100 bulbs is chosen, what is the probability of the sample mean will be:

a.Greater than 1150 hours?

b.Less than 1250 hours?

c.Between 1150 and 1250 hours?​

Answer 1

$\mu=1200 \\ \sigma = 250 \\ n=100$

a.

$P(\bar{x} > 1150) = 1 -P(\bar{x} < 1150) \\ = 1 -P(Z< \frac{1150-1200}{250 / \sqrt{100}} )\\ = 1 -P(Z< -2) \\ = 1 -0.0227 \\ = 0.9773$

b.

$P(\bar{x} < 1250) = P(Z < \frac{1250-1200}{250 / \sqrt{100}}) \\ = P(Z< 2) \\ = 0.9772$

c.

$P(1150 < \bar{x} < 1250) = P(\bar{x} < 1250) -P(\bar{x} < 1150) \\ = P(Z < \frac{1250-1200}{250 / \sqrt{100}}) -P(Z < \frac{1150-1200}{250 / \sqrt{100}}) \\ = P(Z< 2) -P(Z< -2) \\ = 0.9772 -0.0227 \\ = 0.9545$