Question 974790: In the past, 20% of all airline passengers flew first class. In a sample of 15 passengers, 5 flew first class. At α = 0.10, can you conclude that the proportions have changed?
Since sample size is less than 30, so T-test would be used. However, there is no Std Deviation and Sample Mean or 5 is actually the sample mean? I am not sure how to proceed.
H0: p= 0.2, H1: p≠0.2,
T-test: x̅ - μ / (s/ sqrt n)
Answer by rothauserc(4718)   (Show Source):
You can put this solution on YOUR website! This problem is solved using "Sample Distribution of the Proportion"
Use 2-tailed test and the following hypothesis:
Ho: p1 = p2
H1: p1 not = p2
p1 = 0.20 and p2 is determined from the binomial distribution since a passenger is either first class or not
p2 = C(n,r) * p1^r * (1-p1)^n-r
p2 = (15! / (5! * (15-5)!)) * (0.20)^5 * (1-0.20)^(15-5)
p2 = ((15*14*13*12*11) / (5*4*3*2*1)) * 0.00032 * 0.107374182
p2 = 3003 * 0.00032 * 0.107374182 = 0.103182294 approx 0.10
p2 is the sample mean and
sample standard deviation is sqrt(pq/n) = sqrt((0.10*0.90)/15) = 0.077459667
now we want to compare p1 and p2 at alpha = 0.10 (using 2-tailed we have 0.05)
t = (0.20 - 0.10) / (0.077459667 / square root(15)) = 5
now t14 for 0.05 is 2.145
Since the test stat is greater than the critical value, we reject the
null at .05 two-tailed significance level
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