Question

Barron's reported that the average number of weeks an individual is unemployed is 17.5 weeks. Assume that for the population of all unemployed individuals the population mean length of unemployment is 17.5 weeks and that the population standard deviation is 4 weeks. Suppose you would like to select a sample of 45 unemployed individuals for a follow-up study.

(a) Show the sampling distribution of x, the sample mean average for a sample of 45 unemployed individuals.

A bell-shaped curve is above a horizontal axis labeled weeks.

• The horizontal axis ranges from about 43.2 to about 46.8.
• The curve enters the viewing window near 43.2 just above the horizontal axis, curves up to the right, and reaches a maximum near 45.
• The curve then curves down and to the right until it leaves the viewing window at the same height it entered near 46.8.

A bell-shaped curve is above a horizontal axis labeled weeks.

• The horizontal axis ranges from about 15.7 to about 19.3.
• The curve enters the viewing window near 15.7 just above the horizontal axis, curves up to the right, and reaches a maximum near 17.5.
• The curve then curves down and to the right until it leaves the viewing window at the same height it entered near 19.3.

A bell-shaped curve is above a horizontal axis labeled weeks.

• The horizontal axis ranges from about -1.8 to about 1.8.
• The curve enters the viewing window near -1.8 just above the horizontal axis, curves up to the right, and reaches a maximum near 0.
• The curve then curves down and to the right until it leaves the viewing window at the same height it entered near 1.8.

A bell-shaped curve is above a horizontal axis labeled weeks.

• The horizontal axis ranges from about 5.5 to about 29.5.
• The curve enters the viewing window near 5.5 just above the horizontal axis, curves up to the right, and reaches a maximum near 17.5.
• The curve then curves down and to the right until it leaves the viewing window at the same height it entered near 29.5.

(b) What is the probability that a simple random sample of 45 unemployed individuals will provide a sample mean within 1 week of the population mean? (Round your answer to four decimal places.)

(c) What is the probability that a simple random sample of 45 unemployed individuals will provide a sample mean within 1/2 week of the population mean? (Round your answer to four decimal places.)

Answer 1

a) Sampling distribution of x:

µₓ = 17.5

σₓ = σ/√n = 4/√45 = 0.6

- weeks 15.7 16.3 16.9 17.5 18.1 18.7 19.3

b) Probability that sample mean will be within 1 week of the population mean, P(|X̅-µ|< 1) =

= P(-1 < (X̅-µ) < 1)

= P( (-1)/(4/√45) < (X-µ)/(σ/√n) < (1)/(4/√45) )

= P(-1.6771 < z < 1.6771)

= P(z < 1.6771) - P(z < -1.6771)

Using excel function:

= NORM.S.DIST(1.6771, 1) - NORM.S.DIST(-1.6771, 1)

= 0.9065

c) Probability that sample mean will be within 1/2 week of the population mean, P(|X̅-µ|< 0.5) =

P(|X̅-µ|< 0.5) =

= P(-0.5 < (X̅-µ) < 0.5)

= P( (-0.5)/(4/√45) < (X-µ)/(σ/√n) < (0.5)/(4/√45) )

= P(-0.8385 < z < 0.8385)

= P(z < 0.8385) - P(z < -0.8385)

Using excel function:

= NORM.S.DIST(0.8385, 1) - NORM.S.DIST(-0.8385, 1)

= 0.5983