Question

The quality-control manager at a light bulb factory needs to determine whether the mean life of a large shipment of light bulbs is equal to 375 hours. The population standard deviation is 100 hours. A random sample of 64 light bulbs indicates a sample mean life of 350 hours. At the 0.05 level of significance, is there evidence that the mean life is different from 375 hours?

a. Yes, there is evidence that the mean life is different from 375 hours.
b. No, there is no evidence that the mean life is different from 375 hours.

Answer 1

There is evidence that the mean life is different from 375 hours

Therefore, correct option is a. Yes, there is evidence that the mean life is different from 375 hours.

Step-by-step explanation:

To determine whether the mean life of the light bulbs is different from 375 hours, we perform a hypothesis test. The null hypothesis is that t
`(\(H_0\))`he mean life is equal to 375 hours, and the alternative hypothesis
`(\(H_1\))`is that the mean life is different from 375 hours.

The test statistic for a sample mean (\(t\)) is calculated using the formula \[ t = \frac{\bar{X} - \mu}{\frac{\sigma}{\sqrt{n}}} \], where \(\bar{X}\) is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.

In this case, with a sample mean of 350 hours, a population standard deviation of 100 hours, and a sample size of 64, we calculate the \(t\)-statistic. We then compare it to the critical \(t\)-value at a 0.05 level of significance for a two-tailed test. If the calculated \(t\)-statistic falls in the rejection region, we reject the null hypothesis, providing evidence that the mean life is different from 375 hours.

Therefore, correct option is a. Yes, there is evidence that the mean life is different from 375 hours.