The “rule of thumb,” stated on page 497, indicated that we would expect the sampling distribution of p to be approximately normal when

“n  20 and both np and nq were greater than 5.”

What happens when these guidelines are not followed?

a. Use the following set of computer or calculator commands to show you what happens. Try n  15 and p  0.1

(K1  n and K2  p). Do the distributions look normal? Explain what causes the “gaps.” Why do the histograms look alike? Try some different combinations of n (K1) and p (K2):

MINITAB Choose Calc  Random Data  Binomial to simulate a 1000 trials for an n of 15 and a p of 0.5. Divide each generated value by n, forming a column of sample p’s. Calculate a z value for each sample p by using z  (p  p)/

p(1 p)/n. Construct a histogram for the sample p’s and another histogram for the z’s.

Excel Choose Tools  Data Analysis  Random Number Generation

 Binomial to simulate a 1000 trials for an n of 15 and a p of 0.5. Divide each generated value by n, forming a column of sample p’s. Calculate a z value for each sample p by using z  (p  p)/p(1 p)/n. Construct a histogram for the sample p’s and another histogram for the z’s.

TI-83/84 Plus Choose MATH  PRB  7:randBin, then enter n, p, number of trials. The maximum number of elements (trials) in a list is 999. (slow process for large n’s) Divide each generated value by n, forming a list of sample p’s. Calculate a z value for each sample p by using z  (p 

p)/p(1 p)/n. Construct a histogram for the sample p’s and another histogram for the z’s.

b. Try n  15 and p  0.01.

c. Try n  50 and p  0.03.

d. Try n  20 and p  0.2.

e. Try n  20 and p  0.8.

f. What happens when the rule of thumb is not followed?