Question

A manufacturer of light bulbs advertises that, on average, its long-life bulb will last more than 5000 hours. To test the claim, a statistician took a random sample of 100 bulbs and measured the amount of time until each bulb burned out. If we assume that the lifetime of this type of bulb has a standard deviation of 400 hours, can we conclude at the 5% significance level that the claim is true?

Answer 1

To test the claim, we can use a hypothesis test. Let's assume that the claim is true (null hypothesis) and that the average lifetime of the bulbs is indeed more than 5000 hours. The alternative hypothesis would be that the average lifetime is less than or equal to 5000 hours. We can then calculate the test statistic using the formula:...

Step-by-step explanation:

To test the claim, we can use a hypothesis test. Let's assume that the claim is true (null hypothesis) and that the average lifetime of the bulbs is indeed more than 5000 hours. The alternative hypothesis would be that the average lifetime is less than or equal to 5000 hours. We can then calculate the test statistic using the formula:

t = (x-bar - μ) / (σ / sqrt(n))

where:

• x-bar is the sample mean of the lifetimes
• μ is the population mean of the lifetimes (5000 hours)
• σ is the standard deviation of the lifetimes (400 hours)
• n is the sample size (100 bulbs)

Using the test statistic, we can then find the p-value, which represents the probability of observing a sample mean as extreme as the one we obtained, assuming that the null hypothesis is true. If the p-value is less than the significance level (0.05), we can reject the null hypothesis and conclude that the claim is not true.