Question

Show all work for credit, including calculations and complete parts a)-c) Find all variables( SE, z or t, P-value, p hat, etc)

2) In 1994 I surveyed European and Asian new car owners one year after purchase about consumer satisfaction with the tread war life of their OEM tires to see if they would repurchase the same tire. Of 450 randomly selected Asian car owners, 342 owners said they would buy the same tires again. Of 360 randomly selected car owners, 252 said they would buy again. Use larger proportion p̂1.

a) State the conditions and verify they are met.

b) Find the indicated value for a 95% confidence interval of the difference in consumer satisfaction. Interpret your results.

c) Now perform an appropriate hypothesis test, and state your conclusion.

Answer 1

a)
• The two samples must be independently drawn and reasonably random or subjects were
randomly assigned to two groups.
• The sample sizes must be large enough so that: 1 1 n pˆ , 1 1 n p (1 ) − ˆ , 2 2 n pˆ , 2 2 n p (1 ) − ˆ are all five
or more. (the number of successes and the number of failures must be at least 5) for the
confidence interval.
The sample size must be large enough so that: 1 ˆ c n p , 1(1 ) ˆ c n p − , 2 ˆ c n p , 2 (1 ) ˆ c n p − are all five or
more. (the number of successes and the number of failures must be at least 5) for the
significance test.

b)

Here, , n1 = 450 , n2 = 360
p1cap = 0.76 , p2cap = 0.7

Standard Error, sigma(p1cap - p2cap),
SE = sqrt(p1cap * (1-p1cap)/n1 + p2cap * (1-p2cap)/n2)
SE = sqrt(0.76 * (1-0.76)/450 + 0.7*(1-0.7)/360)
SE = 0.0314

For 0.95 CI, z-value = 1.96
Confidence Interval,
CI = (p1cap - p2cap - z*SE, p1cap - p2cap + z*SE)
CI = (0.76 - 0.7 - 1.96*0.0314, 0.76 - 0.7 + 1.96*0.0314)
CI = (-0.0015 , 0.1215)

c)

p1cap = X1/N1 = 342/450 = 0.76
p1cap = X2/N2 = 252/360 = 0.7
pcap = (X1 + X2)/(N1 + N2) = (342+252)/(450+360) = 0.7333

Below are the null and alternative Hypothesis,
Null Hypothesis, H0: p1 = p2
Alternate Hypothesis, Ha: p1 ≠ p2

Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.76-0.7)/sqrt(0.7333*(1-0.7333)*(1/450 + 1/360))
z = 1.92

P-value Approach
P-value = 0.0549
As P-value >= 0.05, fail to reject null hypothesis.