Question

In a random sample of 85 automobile engine crankshaft bearings, 10 have a surface finish roughness that exceeds the specifications. Do these data present strong evidence that the proportion of crankshaft bearings exhibiting excess surface roughness exceeds 0.10?

a. State and test the appropriate hypothesis using α =0.05.

b. If it is really the situation that p = 0.15, how likely is itthat the test procedure in part (a) will reject the nullhypothesis?

c. If p = 0.15, how large would the sample size have to be for usto have a probability of correctly rejecting the null hypothesis of0.9?

Answer 1

To determine if the proportion of crankshaft bearings with excess surface roughness exceeds 0.10, a hypothesis test with alpha = 0.05 is conducted. The test involves calculating a test statistic and comparing the resulting p-value to alpha to decide whether to reject or not reject the null hypothesis. Furthermore, power analysis can be used to find the appropriate sample size needed to achieve a high probability of correctly rejecting the null hypothesis, if the true proportion is indeed higher than hypothesized.

Step-by-step explanation:

Hypothesis Testing for Proportion

To address the question regarding whether the proportion of crankshaft bearings with excess surface roughness exceeds 0.10, we can perform a hypothesis test with alpha = 0.05. The null hypothesis H0 would state that the true proportion p is equal to 0.10, and the alternative hypothesis H1 would state that the true proportion is greater than 0.10. Using the sample data provided, a test statistic and p-value can be calculated and compared to alpha to make a decision.

a. Hypothesis Test

The test statistic for the proportion is calculated using the formula for the z-test. Given the sample size (n = 85) and number of defects (x = 10), we can find the test statistic and resulting p-value. If the p-value is less than alpha, we reject the null hypothesis, indicating strong evidence that the proportion exceeds 0.10.

b. Probability of Correct Rejection with p = 0.15

If the true proportion is p = 0.15, we can calculate the power of the test, which is the probability that the test will correctly reject the null hypothesis. This involves finding the p-value corresponding to the test statistic under this true proportion and comparing it to alpha.

c. Sample Size for High Probability of Correct Rejection

To achieve a high probability of correctly rejecting the null hypothesis (0.9 or 90%), we need to determine an appropriate sample size. This can be done using power analysis, which considers the desired power level, the true proportion (p = 0.15), and the significance level (alpha = 0.05).

Answer 2

To test the hypothesis that the proportion of crankshaft bearings with excess surface roughness exceeds 0.10 at alpha level 0.05, we compare the calculated p-value from sample data with alpha. If the true proportion is 0.15, the power of the test represents the likelihood of rejecting the null correctly. The necessary sample size to achieve a power of 0.9 is determined via power analysis.

Step-by-step explanation:

To answer whether the data provides strong evidence that the proportion of crankshaft bearings exhibiting excess surface roughness exceeds 0.10, we need to conduct a hypothesis test.

Part a: Hypothesis Test

We are testing the null hypothesis H0: p = 0.10 against the alternative hypothesis Ha: p > 0.10. Using an alpha level of 0.05 and the sample data (10 out of 85 exceed specifications), we calculate the test statistic and the corresponding p-value, and then compare the p-value to our alpha level. If the p-value is lower than 0.05, we reject the null hypothesis, indicating that there is sufficient evidence to conclude that the proportion exceeding specifications is greater than 0.10.

Part b: Probability of Rejecting the Null

With the true proportion p=0.15, we would calculate the power of the test, which is the probability of correctly rejecting the null hypothesis. If p indeed equals 0.15, the test must be sensitive enough to detect a deviation from the null hypothesis of p=0.10 to reject it.

Part c: Sample Size Requirement

To achieve a high power, such as 0.9, we would need to calculate the required sample size using power analysis. The sample size is calculated based on the assumed true proportion, the desired power, and the alpha level we are using for the hypothesis test.