Question

The distribution of the number of people in line at a grocery store checkout has a mean of 3 and a variance of 9. A sample of the numbers of people in 50 grocery store checkout lines is taken.

(a) (3 points) What’s the probability that the sample mean is less than 4.5? Round your answer to four (4) decimal places.

(b) (3 points) What’s the probability that the sample mean is more than 2.25? Round your answer to four (4) decimal places.

(c) (3 points) What’s the probability that the sample mean differs from the population mean by more than 0.75? Round your answer to four (4) decimal places.

Answer 1

This is a normal distribution question with

Sample size (n) = 50

Since we know that

a) x = 4.5

P(x < 4.5)=?

The z-score at x = 4.5 is,

z = 1.1785

This implies that

P(x < 4.5) = P(z < 1.1785) = 0.8807

b) x = 2.25

P(x > 2.25)=?

The z-score at x = 2.25 is,

z = 2.25-3.0/1.2728

z = -0.5893

This implies that

P(x > 2.25) = P(z > -0.5893) = 1 - 0.2778300224356762

P(x > 2.25) = 0.7222

c) x1 = 2.25

x2 = 3.75

P(2.25 < x < 3.75)=?

This implies that

P(2.25 < x < 3.75) = P(-0.5893 < z < 0.5893) = P(Z < 0.5893) - P(Z < -0.5893)

P(2.25 < x < 3.75) = 0.7221699775643238 - 0.2778300224356762

P(2.25 < x < 3.75) = 0.4443

PS: you have to refer z score table to find the final probabilities.

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