The quality-control manager at a compact fluorescent light bulb(CFL)factory needs to determine whether the mean life of a large shipment of CFLs is equal to 7450 hours.The population standard deviation is 990 hours.A random sample of 81 light bulbs indicates a sample mean life of 7,252 hours. At the 0.05 level of significance.is there evidence that the mean life is different from 7450 hours? What is the test statistic?
Question
The quality-control manager at a compact fluorescent light bulb(CFL)factory needs to determine whether the mean life of a large shipment of CFLs is equal to 7450 hours.The population standard deviation is 990 hours.A random sample of 81 light bulbs indicates a sample mean life of 7,252 hours. At the 0.05 level of significance.is there evidence that the mean life is different from 7450 hours? What is the test statistic?
Solution
To answer this question, we will perform a hypothesis test for the population mean.
Step 1: State the null and alternative hypotheses. The null hypothesis (H0) is that the mean life of the light bulbs is 7450 hours. H0: μ = 7450 The alternative hypothesis (H1) is that the mean life of the light bulbs is not 7450 hours. H1: μ ≠ 7450
Step 2: Identify a test statistic. Since the population standard deviation is known, we can use the z-test. The test statistic (z) is calculated as follows:
z = (X̄ - μ) / (σ/√n)
where: X̄ is the sample mean = 7252 hours μ is the population mean = 7450 hours σ is the population standard deviation = 990 hours n is the sample size = 81
Step 3: Calculate the test statistic. z = (7252 - 7450) / (990/√81) = -1.99
Step 4: Determine the p-value. The p-value associated with a z-score of -1.99 (two-tailed test) is approximately 0.046.
Step 5: Compare the p-value to the level of significance. The p-value (0.046) is less than the level of significance (0.05), so we reject the null hypothesis.
Conclusion: There is evidence at the 0.05 level of significance that the mean life of the light bulbs is different from 7450 hours. The test statistic is -1.99.
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