Question

A population of values has a normal distribution with μ=134.3μ=134.3 and σ=44.2σ=44.2. You intend to draw a random sample of size n=135n=135.

Find the probability that a single randomly selected value is between 133.2 and 145.
P(133.2 < X < 145) =

Find the probability that a sample of size n=135n=135 is randomly selected with a mean between 133.2 and 145.
P(133.2 < M < 145) =

I tried to solve this problem multiple times but I am truly confused. HELP!

Answer 1

Population mean, µ = 134.3

Population standard deviation, σ = 44.2

Sample size, n = 135

1. Probability that a single randomly selected value is between 133.2 and 145 =

= P(133.2< X <145)

= P( [(133.2-134.3)/44.2 ] < (X-µ)/σ < [(145-134.3)/44.2])

= P(-0.0249< z <0.2421)

= P( z <0.2421 ) - P( z <-0.0249)

Using excel function:

= NORM.S.DIST(0.2421,1) - NORM.S.DIST(-0.0249,1)

= 0.1056

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2. Probability that mean is between 133.2 and 145 =

= P(133.2 < X̅ < 145)

= P( [(133.2-134.3)/(44.2/√135) ] < (X̅ -µ)/(σ/√n) < [(145-134)/(44.2/√135) ])

= P(-0.2892< z <2.8127)

= P( z <2.8127 ) - P( z <-0.2892)

Using excel function:

= NORM.S.DIST(2.8127,1) - NORM.S.DIST(-0.2892,1)

= 0.6113