Question

January of 1999, the German Auto Bild magazine randomly tested different tread patterns across all tire brands as their journalists toured from the UK to the Bulgarian coast. They discovered on one leg of the tour that out of 100 tires, or 12%, shortcomings were due to defective sidewall wire. On another leg they found 125 tires, 15% of the defects were due to poor vulcanization.

a) Find an appropriate 95% confidence interval. Explain the difference in outcomes. Let poor vulcanization represent sample #1.

b) Use your confidence interval to explain the difference in outcomes. 2

c) An expert at TUV  thought defects due to wire imperfections were more common than poor vulcanization issues. Reexamine the question in part “b” using a hypothesis test to determine if you support this German regulator’s expert opinion?

Answer 1

a)

Answer:

Explanation:

The 95% Confidence Interval of the difference between two proportions is obtained using the following formula,

For a 95% confidence interval, z critical value is obtained using the standard normal distribution table,

Now,

b)

The 95% confidence interval for the difference in proportions is

Since the 95 confidence interval for the difference in proportions includes zero value, there is not sufficient evidence to conclude that the proportion of shortcomings due to defective sidewall wire on one leg is different from the proportions of defects due to poor vulcanization on another leg.

c)

The hypothesis test is performed in the following steps,

Hypothesis:

The null and alternative hypothesis are,

Where,

This is a right-tailed test

The significance level,

The z-test is used to compare two population proportions.

Test statistic:

The z-statistic is,

Where,

Now,

P-value

The P-value for the z-statistic is obtained from the z distribution table for z = 0.6510

Conclusion:

Since the p-value is greater than 0.05, the null hypothesis is not rejected. Hence there is not sufficient evidence to conclude that the proportion of shortcomings due to defective sidewall wire on one leg is different from the proportions of defects due to poor vulcanization on another leg.

The result is the same hence we cannot support the German regulator’s expert opinion.