Answer to Question #342323 in Statistics and Probability for dave

Question #342323

2. The population mean length of a chocolate wafer stick is 86 mm long with a standard deviation of 6 mm. A sample of 20 sticks are taken. What is the probability that

a. the sample mean is not less than 84 mm?

b. the sample mean is not more than 87 mm?

c. the sample mean differs from the population mean by at most 4 mm?


1
Expert's answer
2022-05-20T08:18:08-0400

Let "X=" the sample mean: "X\\sim N(\\mu, \\sigma^2\/n)."

Given "\\mu=86mm, \\sigma=6mm, n=20."

a.


"P(X\\ge84)=1-P(Z<\\dfrac{84-86}{6\/\\sqrt{20}})"

"\\approx 1-P(Z\\le-1.490712)\\approx0.9320"

b.


"P(X\\le87)=P(Z\\le\\dfrac{87-86}{6\/\\sqrt{20}})"

"\\approx P(Z\\le0.745356)\\approx0.7720"

c.


"P(82<X<90)"

"=P(Z<\\dfrac{90-86}{6\/\\sqrt{20}})-P(Z<\\dfrac{82-86}{6\/\\sqrt{20}})"

"\\approx P(Z<2.981424)-P(Z<-2.981424)"

"\\approx0.99857-0.00143\\approx0.9971"


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