Question

Suppose the length of textbooks in a library follows a bimodal distribution with a little right skewness (very mild). The mean of this distribution is 512 pages with a standard deviation of 390 pages.

For each of the following i) draw a picture. ii) label the picture with 2 axes (underneath). iii) label the shorthand for the new distribution. iv) Find the z-score. v) Find the answer.  

a1) What is the probability that a random sample of 36 textbooks has an average of 445.2 pages or less?

a2) What is the probability that a random sample of 49 textbooks has an average higher than 613.3 pages?

a3) What is the probability that a random sample of 30 textbooks has an average number of pages between 400 and 500?

a4) Do you think you could answer a1 - a4 with a sample of just 2 textbooks? Why or why not?

Answer 1

Please note I have used z-distribution table for calculating p-value so answers may be less accurate because z value has to be rounded off to 2 decimal places. Use of calculator provides more accuracy. There may be slight difference in answer. I have provided both answers as nothing is mentioned which method has to be used.

Answer a1)

Mean of Sampling distribution μ_x̄ = Population mean = 512

Standard deviation of sampling distribution σ_x̄ = σ/sqrt(n) = 390/sqrt(36) = 390/6 = 65

The probability that a random sample of 36 textbooks has an average of 445.2 pages or less is 0.1515

Please note that answer with calculator will be  0.1520

Answer a2)

Mean of Sampling distribution μ_x̄ = Population mean = 512

Standard deviation of sampling distribution σ_x̄ = σ/sqrt(n) = 390/sqrt(49) = 390/7 = 55.7143

The probability that a random sample of 49 textbooks has an average higher than 613.3 pages is 0.0344

Please note that answer with calculator will be  0.0345

Answer a3)

Mean of Sampling distribution μ_x̄ = Population mean = 512

Standard deviation of sampling distribution σ_x̄ = σ/sqrt(n) = 390/sqrt(30) = 390/5.4772 = 71.2039

Probability that a random sample of 30 textbooks has an average no. of pages between 400 & 500 is 0.3743

Please note that answer with calculator will be  0.3752

Answer a4)

No, we could not answer a1 - a4 with a sample of just 2 textbooks. This is because as per central limit theorem for sampling distribution to be assumed normal, sample size should be 30 or more.