Question

For a z-value of -2.47, compute a p-value for an upper-tail test, a lower-tail test, AND a two-tail test.

To test that the mean lifetime of light bulbs is no more than 900 hours (population is normally distributed and population standard deviation is 30), a random sample of 49 bulbs is tested, yielding a sample mean of 910 hours. What conclusion would you reach if alpha = .05? Show all the steps of hypothesis testing. You must state your conclusion in terms of the context of the mean lifetime of light bulbs.

Answer 1

To compute the p-value for an upper-tail test, find the area to the right of the given z-value. For a lower-tail test, find the area to the left. For a two-tail test, double the area to the right. Conduct the hypothesis test by setting up null and alternative hypotheses, calculating the test statistic, and comparing it to the critical value. If the test statistic is greater than the critical value, reject the null hypothesis.

Step-by-step explanation:

To compute the p-value for an upper-tail test, we need to find the area under the normal distribution curve to the right of the given z-value.

Using a standard normal table or a calculator, we find that the area to the right of -2.47 is approximately 0.9933. Therefore, the p-value for the upper-tail test is 1 - 0.9933 = 0.0067.

For a lower-tail test, the p-value is the area under the normal distribution curve to the left of the given z-value. In this case, the p-value for the lower-tail test is also 0.0067.

For a two-tail test, the p-value is twice the area under the normal distribution curve to the right of the absolute value of the given z-value. Therefore, the p-value for the two-tail test is 2 * 0.0067 = 0.0134.

To conduct the hypothesis test for the mean lifetime of light bulbs, we can set up the null and alternative hypotheses as follows:

Null hypothesis (H0): The mean lifetime of light bulbs is 900 hours.

Alternative hypothesis (Ha): The mean lifetime of light bulbs is less than 900 hours.

Next, we calculate the test statistic using the formula:

t = (sample mean - population mean) / (population standard deviation / √sample size)

Plugging in the given values, we have:

t = (910 - 900) / (30 / √49) = 2.3333

Looking up the critical value for alpha = 0.05 and degrees of freedom (df) = sample size - 1 = 48 in the t-distribution table, we find it to be approximately 1.677.

Since the test statistic (2.3333) is greater than the critical value (1.677), we reject the null hypothesis.

Therefore, we conclude that there is sufficient evidence to suggest that the mean lifetime of light bulbs is less than 900 hours.