Question

1) In a recent study of 46 drug addiction patients, 32% admitted to having a relapse in the last three months. Determine the 95% confidence interval for the population.

2)At McDonalds, the automatic soda machine is suppose to dispense 32oz of soda and ice into a large cup. You have received multiple complaints that the machine is not working properly. You need to determine if the machine is working properly so you take a sample of 12 large sodas and find an average of 33.2 oz with a standard deviation of 1.1 oz. Your general manager wants to be 99% confident about the machine’s performance. Determine a confidence interval and test the machine’s effectiveness using the confidence interval.

Answer 1

1)

sample proportion, = 0.32
sample size, n = 45
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.32 * (1 - 0.32)/45) = 0.0695

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, Zc = Z(α/2) = 1.96

Margin of Error, ME = zc * SE
ME = 1.96 * 0.0695
ME = 0.1362

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.32 - 1.96 * 0.0695 , 0.32 + 1.96 * 0.0695)
CI = (0.1838 , 0.4562)

2)

sample mean, xbar = 33.2
sample standard deviation, s = 1.1
sample size, n = 12
degrees of freedom, df = n - 1 = 11

Given CI level is 99%, hence α = 1 - 0.99 = 0.01
α/2 = 0.01/2 = 0.005, tc = t(α/2, df) = 3.106

ME = tc * s/sqrt(n)
ME = 3.106 * 1.1/sqrt(12)
ME = 0.986

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (33.2 - 3.106 * 1.1/sqrt(12) , 33.2 + 3.106 * 1.1/sqrt(12))
CI = (32.21 , 34.19)

Here, we can reject the null hypothesis because confidence interval does not contain 32 and it si effective