Question

1. A mail-order company promises its customers that their orders will be processed and mailed within 24 hours after an order is placed. The quality control department at the company checks from time to time to see if this promise is being kept. Recently the quality control department took a sample of 200 orders and found that 176 of them were processed and mailed within 24 hours of the placement of the orders.
a. What is the point estimate for the proportion of orders processed within 24 hours?
b. Construct a 95% confidence interval for the proportion of orders processed within 24 hours.
c. If the quality control departments wants their margin of error to be 1% for the same confidence level, what sample would they need to collect?
d. The company's president wants to advertise that his company processes 95% of their orders within 24 hours. Construct a hypothesis test (with a significance level of 0.5) to determine if the president's claim is warranted.

Answer 1

Solution-A:

the point estimate for the proportion of orders processed within 24 hours

p^=x/n=176/200=0.88

b. Construct a 95% confidence interval for the proportion of orders processed within 24 hour

z crit for 95%=1.96

95% confidence interval for p is

p^-z*sqrt(p^*(1-p^)/n),p^+z*sqrt(p^*(1-p^)/n)

0.88-1.96*sqrt(0.88*(1-0.88)/200),0.88+1.96*sqrt(0.88*(1-0.88)/200)

0.8349626,0.9250374

we are 95% confident that the proportion of orders processed within 24 hours lies in between 0.8349626 and 0.9250374

c. If the quality control departments wants their margin of error to be 1% for the same confidence level, what sample would they need to collect?

E=0.01

Z crit for 95%=1.96

p^=0.88

n=p^*(1-p^)*(zcrit/E)^2

=(0.88)*(1-0.88)*(1.96/0.01)^2

=4056.73

=4057

sample would they need to collect is n=4057