Statistics

Question

statistics

The average number of books read by an American man is for books with a standard deviation of 1.25 it is also known that the number of books read by men is not normally distributed. What is probability that 25 randomly American mean read fewer than 3.7 books on average?

Answer 1

The point of this problem is to decide whether the distribution of the sample mean is approximately normal. If it is then we can estimate the probability using the z-score formula and the probability tables for the standard normal distribution.

 

The big question is whether this is a reasonable approximation. The Central Limit Theorem says that the sampling distribution of any statistic will be normal or nearly normal, if the sample size is large enough. Is a sample size of 25 large enough? Most statisticians say that if the population distribution is roughly bell shaped AND the sample size is at least 30 then the sample distribution will be close enough to normal to give a good approximation of the actual probability. Since 25 is NOT at least 30 AND we don't know if the population distribution is roughly bell shaped it is not advisable to assume that the sample distribution is normal.

 

That said, we can still calculate a probability even though it's quite likely to be way off.

 

First find the standard deviation of the sample mean (also called standard error) which is given by the formula:

 

σ = σp/√n = (1.25)/√25 = 1.25/5 = 0.25 (where σp is the standard deviation of the population and n is the sample size)

 

Convert the raw score (3.7) to a z score

 

z = (x-µ)/σ = (3.7 - 4)/.25 = -0.3/.25 = -1.2

 

Find the probability for this z score using a table or calculator.

 

P(Z<-1.2) = 0.11507