Question

A leading magazine (like Barron's) reported at one time that the average number of weeks an individual is unemployed is 15.3 weeks. Assume that for the population of all unemployed individuals the population mean length of unemployment is 15.3 weeks and that the population standard deviation is 8 weeks. Suppose you would like to select a random sample of 169 unemployed individuals for a follow-up study.

Find the probability that a single randomly selected value is between 13.8 and 14.5.
P(13.8 < X < 14.5) = __________

Find the probability that a sample of size n=169 is randomly selected with a mean between 13.8 and 14.5.
P(13.8 < M < 14.5) = ___________

Enter your answers as numbers accurate to 4 decimal places.

Answer 1

a)

Here, μ = 15.3, σ = 8, x1 = 13.8 and x2 = 14.5. We need to compute P(13.8<= X <= 14.5). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (13.8 - 15.3)/8 = -0.19
z2 = (14.5 - 15.3)/8 = -0.1

Therefore, we get
P(13.8 <= X <= 14.5) = P((14.5 - 15.3)/8) <= z <= (14.5 - 15.3)/8)
= P(-0.19 <= z <= -0.1) = P(z <= -0.1) - P(z <= -0.19)
= 0.4602 - 0.4247
= 0.0355

b)

Here, μ = 15.3, σ = 0.6154, x1 = 13.8 and x2 = 14.5. We need to compute P(13.8<= X <= 14.5). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (13.8 - 15.3)/0.6154 = -2.44
z2 = (14.5 - 15.3)/0.6154 = -1.3

Therefore, we get
P(13.8 <= X <= 14.5) = P((14.5 - 15.3)/0.6154) <= z <= (14.5 - 15.3)/0.6154)
= P(-2.44 <= z <= -1.3) = P(z <= -1.3) - P(z <= -2.44)
= 0.0968 - 0.0073
= 0.0895