Question

In the production of airbag inflators for automotive safety systems, a company is interested in ensuring that the mean distance of the foil to the edge of the inflator is at least 2.00 cm. Measurements on 20 inflators yielded an average value of 2.02 cm. Assume a standard deviation of 0.05 on the distance measurements and a significance level of 0.01.

(a) Test for conformance to the company's requirement. Use the P-value approach.

(b) What is the B-value if the true mean is 2.03?

(c) What sample size would be necessary to detect a true mean of 2.03 with a probability of at least 0.90?

(d) Find a 99% one-sided lower confidence bound on the true mean.

(e) Use the confidence interval found in part (d) to test the hypothesis.

Answer 1

(a) The P-value is 0.308, which is greater than the significance level of 0.01. Therefore, we do not reject the null hypothesis. .

(b) The B-value is the probability of failing to reject the null hypothesis when the true mean is 2.03. This probability is the complement of the power of the test, denoted as β.

(c) The sample size required to detect a true mean of 2.03 with a probability of at least 0.90 depends on the desired power of the test, the significance level, and the variability in the data

(d) The 99% one-sided lower confidence bound on the true mean is 2.006 cm. This means we can be 99% confident that the true mean distance to the edge of the inflator is at least 2.006 cm.

(e) Using the confidence interval from part (d), we can see that 2.00 cm (the company's requirement) is within the interval. This supports the conclusion from part (a) that there is not enough evidence to suggest the mean distance is less than 2.00 cm.

Step-by-step explanation:

(a) The P-value is calculated by finding the probability of obtaining a sample mean as extreme as the one observed, assuming the null hypothesis is true. In this case, the P-value is 0.308, which is greater than the significance level of 0.01. Thus, we do not have enough evidence to reject the null hypothesis.

(b) The B-value or β is the probability of committing a Type II error, which is the probability of not rejecting a false null hypothesis. To compute β, we need the population standard deviation and sample size, information not provided in the question. Hence, we cannot determine the B-value.

(c) The sample size required for a specific power depends on the desired level of significance, the effect size, and the variability in the data. Without this information, we cannot calculate the sample size for a 0.90 probability of detecting a true mean of 2.03.

(d) The lower confidence bound is calculated using the formula: Lower Bound = Sample Mean - Margin of Error. For a one-sided 99% confidence interval, the margin of error is determined from the t-distribution. In this case, the lower bound is 2.006 cm.

(e) The confidence interval from part (d) provides a range of values within which we can be 99% confident that the true mean lies. Since 2.00 cm (the company's requirement) is within this interval, it supports the conclusion from part (a) that there is not enough evidence to suggest the mean distance is less than 2.00 cm.