Question

GMU officials are trying to determine if they are using the correct number of campus shuttles for the number of individuals who use them. If the proportion of individuals associated with GMU (including faculty, staff, and students) who use the shuttles is significantly different from 0.28, the officials believe they will have to either remove or add a shuttle to the fleet. In a random sample of 444 people taken from the population of all individuals associated with GMU (including faculty, staff, and students) it was discovered that 123 of these individuals use the shuttle.

a) Check the three conditions of the Central Limit Theorem that allow you to use the oneproportion confidence interval using one complete sentence for each condition. Show work for the numerical calculation.

b) Construct a 99% confidence interval to estimate the population proportion of these individuals who use the shuttle system. Calculate this “by hand” using the formula and showing your work (please type your work, no images accepted here). Round your confidence limits to four decimals.

Explain the connection between the confidence interval and the hypothesis test in this problem (discuss this in relation to the decision made from your hypothesis test and connect it to the confidence interval you constructed in part (b)). Answer this question in one to two sentences.

Answer 1

a)

• The sample must be reasonably random
• The sample must be less than 10% of the population
• The sample must be large enough so that:

n * pcap and n (1 - pcap) > =10

b)

z value at 99% = 2.576

pcap = 123/444 = 0.2770

CI = pcap +/- z *sqrt(pcap *(1-pcap)/n)
= 0.2770 +/- 2.576 * sqrt(0.2770 *(1-0.2770)/444)
= (0.2223 , 0.3317)

c)

Here, Fail to reject H0 as confidenc einterval contain hypothesised value 0.28