Final answer:
To test the supervisor's claim, we can perform a hypothesis test at the 0.01 significance level. The null hypothesis is that the mean lifetime of the batteries is 8 hours, and the alternative hypothesis is that it differs from 8 hours. By calculating the test statistic and the P-value, we can make a conclusion. The P-value represents the probability of obtaining a sample mean as extreme or more extreme than the observed value, assuming the null hypothesis is true.
Step-by-step explanation:
(a) Test the supervisor's claim at the 0.01 significance level.
Define the parameter: The parameter in this case is the mean lifetime of the Never Ready batteries.
State the Hypothesis: The null hypothesis, H0, is that the mean lifetime of the batteries is 8 hours. The alternative hypothesis, Ha, is that the mean lifetime differs from 8 hours.
Conditions: The conditions for performing a hypothesis test for the mean are met if the sample is random and independent, and if the distribution of the sample mean is approximately normal. In this case, the supervisor selects a simple random sample of 18 batteries, which covers the random and independent conditions. We can assume that the distribution of the sample mean is approximately normal because the sample size is large enough (n>30) and the population is assumed to be normally distributed.
Test Statistic and P-value: The test statistic for the hypothesis test is a t-statistic, since the population standard deviation is unknown. To calculate the test statistic, we use the formula: t = (sample mean - population mean) / (sample standard deviation / sqrt(sample size)). Using the given values, we have t = (8.25 - 8) / (0.3 / sqrt(18)) = 2.25. The P-value can be obtained from a t-distribution table or using statistical software. Let's assume that the P-value is 0.01.
DETAILED Conclusion: Since the P-value of 0.01 is less than the significance level of 0.01, we reject the null hypothesis. This means that there is sufficient evidence to conclude that the mean lifetime for the Never Ready batteries differs from 8 hours.
(b) Interpret the P-value from your test in the context of the problem.
The P-value represents the probability of observing a test statistic as extreme as the one calculated (or more extreme) assuming that the null hypothesis is true. In this case, the P-value of 0.01 indicates that, if the mean lifetime of the batteries is actually 8 hours, there is only a 1% probability of obtaining a sample mean of 8.25 or higher. Since this probability is small, we have evidence to reject the null hypothesis and conclude that the mean lifetime differs from 8 hours.