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A manufacturer claims that the mean lifetime of its light bulbs is 55 months. The standard deviation of these lifetimes is 8 months. One hundred bulbs are selected at random, and their mean lifetime is found to be 54 months. Can we conclude, at the 0.1 level of significance, that the mean lifetime of light bulbs made by this manufacturer differs from 55 months? Perform a two-tailed test.

a. State the null hypothesis and the alternative hypothesis.

b. Determine the type of test statistic to use.

c. Find the value of the test statistic.

d. Find the two critical values.

e. Can we conclude that the mean lifetime of light bulbs made by this manufacturer differs from 55 months?

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Final answer:

a. Null hypothesis: The mean lifetime of light bulbs made by this manufacturer is equal to 55 months. Alternative hypothesis: The mean lifetime of light bulbs made by this manufacturer differs from 55 months. b. Use the Z-test statistic to perform the hypothesis test. c. Calculate the test statistic value using the formula Z = (sample mean - hypothesized mean) / (standard deviation / sqrt(sample size)).

Step-by-step explanation:

a. Null hypothesis: The mean lifetime of light bulbs made by this manufacturer is equal to 55 months. Alternative hypothesis: The mean lifetime of light bulbs made by this manufacturer differs from 55 months.

b. Test statistic: Since the sample size is large (100 bulbs selected), we can use the Z-test statistic to perform the hypothesis test.

c. Test statistic value: To find the test statistic value, we need to calculate the Z-score using the formula Z = (sample mean - hypothesized mean) / (standard deviation / sqrt(sample size)). In this case, the sample mean is 54 months, the hypothesized mean is 55 months, the standard deviation is 8 months, and the sample size is 100. Plugging in these values, we get Z = (54 - 55) / (8 / sqrt(100)) = -1 / 0.8 = -1.25.

d. Critical values: To perform a two-tailed test at the 0.1 level of significance, we need to divide the alpha (0.10) by 2 to get alpha/2 (0.05). Then, we look up the Z-score for alpha/2 in the standard normal distribution table. The critical values are the Z-scores that correspond to the alpha/2 level of significance. In this case, the critical values are -1.645 and 1.645.

e. Conclusion: Since the test statistic value (-1.25) is between the critical values (-1.645 and 1.645), we fail to reject the null hypothesis. Therefore, we cannot conclude, at the 0.1 level of significance, that the mean lifetime of light bulbs made by this manufacturer differs from 55 months.

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