Question

A manufacturer of light bulbs advertises that, on average, its long-life bulb will last more than 4700 hours. To test this claim, a statistician took a random sample of 97 bulbs and measured the amount of time until each bulb burned out. The mean lifetime of the sample of bulbs is 4757 hours and has a standard deviation of 400 hours. Can we conclude with 90% confidence that the claim is true? Fill in the requested information below.

A. The value of the standardized test statistic is [Enter the value].

B. The rejection region for the standardized test statistic is [Enter the rejection region in interval notation].

C. The p-value is [Enter the p-value].

D. Your decision for the hypothesis test is:

A. Reject H1.
B. Do Not Reject H1.
C. Do Not Reject H0.
D. Reject H0.

Answer 1

To test the manufacturer's claim, we can conduct a hypothesis test using the sample mean and standard deviation. The calculated z value falls in the rejection region, and the p-value is less than 0.05. Therefore, we reject the null hypothesis and can conclude that the claim is true at a 90% confidence level.

Step-by-step explanation:

To test the claim that the average lifetime of the manufacturer's long-life bulb is more than 4700 hours, we can conduct a hypothesis test using the sample mean and standard deviation.

A. To find the value of the standardized test statistic, we use the formula: z = (sample mean - population mean) / (standard deviation / sqrt(sample size)). Plugging in the values from the question, we get z = (4757 - 4700) / (400 / sqrt(97)) = 2.83.

B. The rejection region for a one-tailed test with 90% confidence is z > 1.645. Since our calculated z value is greater than 1.645, the test statistic falls in the rejection region.

C. The p-value is the probability of obtaining a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true. Since the test is one-tailed and our calculated z value is greater than the critical value, the p-value is less than 0.05.

D. Based on the calculated z value falling in the rejection region and the p-value being less than 0.05, we reject the null hypothesis (H0) and conclude that there is evidence to support the claim that the average lifetime of the long-life bulb is more than 4700 hours at a 90% confidence level.