Question

-           For a random variable that is normally distributed, with μ = 80 and σ = 10, determine the probability that a simple random sample of 25 items will have a mean that is

a.         greater than 78.

b.         between 79 and 85.

c.          less than 85.

please show step by step resolution.

thanks

Answer 1

Let X ~ N(80, 10).

Let M be the sample mean of 25 items.

Thus, E(M) = 80, s.d.(M) = 10/ = 2.

Hence, M ~ N(80, 2) i.e. (M - 80)/2 ~ N(0,1).

a. P(M > 78) = P[(M - 80)/2 > (78 - 80)/2]

= P[(M - 80)/2 > -1] = 1 - P[(M - 80)/2 < -1] = 1 - (-1)

= 1 - 0.1587 = 0.8413. (Ans).

b. P(79 < M < 85) = P[(79 - 80)/2 < (M - 80)/2 < (85 - 80)/2]

= P[-0.5 < (M - 80)/2 < 2.5] = (2.5) - (-0.5)

= 0.9938 - 0.3085 = 0.6853. (Ans).

c. P(M > 85) = 1 - P(M < 85) = 1 - P[(M - 80)/2 < (85 - 80)/2]

= 1 - P[(M - 80)/2 < 2.5] = 1 - (2.5) = 1 - 0.9938 = 0.0062.

(Ans).