Question 935955
A mechanical press is used to mold shapes for plastic toys. when the machine is adjusted and working well, it still produces about 8% defective toys. The toys are manufactured in lots of n=100. Let r be a random variable representing the number of defective toys in the lot. Then p(hat) = r/n is the proportion of defective toys in a lot. 
a. Find P(.07<=p(hat) <=.09). Round to the nearest 4 decimal places.

Solution:
.05/100 = .005
The z-score for .065 is {{{(0.065-0.08)/sqrt((0.08)*(1-0.08)/100)}}} =-0.5529
The z-score for .095 is {{{(0.095-0.08)/sqrt((0.08)*(1-0.08)/100)}}} = 0.5529

P(-0.5529 < Z < 0.5529) = {{{highlight(0.4197)}}}  [Answer]

b. Find the probability that between 7 and 9 defective toys produced in the lot n=100 toys. That is, find P(7<=r<=9) remembering to use the continuity correction of +-0.5. 

The z-score for 6.5 is {{{(6.5-8)/sqrt(100*(0.08)*(1-0.08))}}} = -0.5529
The z-score for 9.5 is {{{(9.5-8)/sqrt(100*(0.08)*(1-0.08))}}} = 0.5529

P(-0.5529 < Z < 0.5529) = {{{highlight(0.4197)}}}  [Answer]