Question

The average price of a college math textbook is $171 and the standard deviation is $20. Suppose that 17 textbooks are randomly chosen. Round all answers to 4 decimal places where possible.

A. What is the distribution of ¯x? ¯x ~ N(     ,    )

B. For the group of 17, find the probability that the average price is between $172 and $176.

C. Find the third quartile for the average textbook price for this sample size. $ (round to the nearest cent)

D. For part b), is the assumption that the distribution is normal necessary?   Yes or No

Answer 1

As per Central limit theorem, If X follows normal distribution or the sample size is large , then sampling distribution of sample mean (sample size : n) is also normal with and standard deviation /

A.

X : Price of a a college math textbook with mean = $171 and standard deviation = $20

Assumed X follows normal distribution

Sample size : Number of text books randomly chosen : n= 17

As per Central limit theorem ,

follows normal distribution with mean : and standard deviation:

The Distribution of

B.

For the group of 17, Probability that the average price is between $172 and $176 =

Z-score for 176 = (176-171)/4.8507 = 1.03 ; Z-score for 172 = (172-171)/4.8507= 0.21

From standard normal tables, P(Z1.03) = 0.8485 P(Z0.21) = 0.5832

For the group of 17, Probability that the average price is between $172 and $176 = 0.2653

C.

Third quartile for the average textbook price for this sample size

Q₃ be the third quartile, By definition,

P(XQ₃) = 0.75

Let Z₃ be the z-score for Q₃

Z₃ = (Q₃ - 171)/4.8507 ; Q₃ = 171+4.8507Z₃

P(ZZ₃) = P(XQ₃)  From standard normal tables,

P(Z0.67) = 0.74860.75

Z₃ = 0.67

Q₃ = 171+4.8507Z₃ = 171 + 4.8507 x 0.67 = 174.25

Third quartile for the average textbook price for this sample size = $174.25

D. For part b), is the assumption that the distribution is normal necessary? Yes, as the sample size is small