4. A random sample of 60 suspension helmets used by motorcycle riders and race-c

Question

4. A random sample of 60 suspension helmets used by motorcycle riders and race-car drivers was subjected to an impact test, and on 26 of these helmets some damage was observed A. Compute a 95% confidence interval for the true proportion of helmets of this type that would show damage from this impact test. B. In past samples, it has been found that at most 40% of helmets suffered damage as a result of this test. How many helmets must be tested to be 95% confident that the true value of p is within 2% of our point estimate? 5. A tire manufacturer is considering a newly designed tread pattern for its all-weather tires. Tests have indicated that these tires will provide better gas mileage and longer tread life. The last remaining test is for braking effectiveness. The company hopes the tire will allow a car travelling at 100km/h to come to a complete stop not more than an average of 38 metres after brakes are applied. They will adopt the new tread pattern unless there is strong evidence that the tires do not meet this objective. The results for 10 stops on a test track were 39.3, 39.0, 39.6, 40.2, 4., 37.5, 3, 38.1, 39.0, and 39.6. Should the company adopt the new treat patter? Test at ? = 0.01. State any assumptions you make for the test to be valid. Use the p- value and the rejection region methods. 6. A company with a fleet of 150 cars found that the emissions systems of 17 out of the 45 they tested failed to meet the pollution control guidelines. Is there strong evidence that more than 25% of the fleet might be out ofcompliance? Test at ? 0.05. Use the p-value and the rejection region methods. 7. Give the formula for the appropriate test statistic, if any, for the following hypothesis testing situations. If the situation is not suitable for a hypothesis test, simply state that fact. A. Ho , 20,s-45, population normally distributed B. H0 : ?=?), n = 80, ?-29, population not normally distributed ?. H,) : A6, n 36, ?-25, population not normally distributed D. Ho : ?-?-n-| 5, s-36. population not normally distributed E. HO : -A6, n 10, ?-16, population normally distributed F. Ho : ?-?), n = 60, ? = 81 , population nomally distributed G. Ho: y, n 200, s 25, population not normally distributed

Explanation / Answer

Solution:-

6)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: P < 0.25
Alternative hypothesis: P > 0.25

Note that these hypotheses constitute a one-tailed test.  

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method, shown in the next section, is a one-sample z-test.

Analyze sample data. Using sample data, we calculate the standard deviation (S.D) and compute the z-score test statistic (z).

S.D = sqrt[ P * ( 1 - P ) / n ]

S.D = 0.03536
z = (p - P) / S.D

z = 3.61

where P is the hypothesized value of population proportion in the null hypothesis, p is the sample proportion, and n is the sample size.

Since we have a one-tailed test, the P-value is the probability that the z-score is greater than 3.61.

Thus, the P-value = less than 0.001.

Interpret results. Since the P-value (almost 0) is less than the significance level (0.05), we have to reject the null hypothesis.

From the above test we have sufficient evidence in the favor of the claim that more than 25% of the fleet might be out of compliance.

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