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The quality control manager at a light bulb factory needs to estimate the mean life of a large shipment of light bulbs. The standard deviation is 70 hours. A random sample of 49 light bulbs indicated a sample mean life of 280 hours. Construct a 95% confidence interval estimate for the population mean life of light bulbs in this shipment.

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Final answer:

The 95% confidence interval for the population mean life of the light bulbs in the shipment is (260.4 hours, 299.6 hours), based on the given sample mean and standard deviation.

Step-by-step explanation:

The quality control manager needs to estimate the mean life of a large shipment of light bulbs using a 95% confidence interval. Given that the sampled light bulbs have a mean life of 280 hours with a standard deviation of 70 hours, and a sample size of 49, we first need to determine the z-score that corresponds to a 95% confidence level, which is approximately 1.96. Next, we calculate the standard error of the mean (SEM) by dividing the standard deviation by the square root of the sample size, resulting in SEM = 70 / √49 = 10. Then, we multiply the z-score by the SEM to find the margin of error (MOE): MOE = 1.96 × 10 ≈ 19.6.

Finally, we construct the confidence interval by adding and subtracting the margin of error from the sample mean:

  • Lower bound = 280 - 19.6 = 260.4 hours
  • Upper bound = 280 + 19.6 = 299.6 hours

Therefore, the 95% confidence interval estimate for the population mean life of light bulbs is (260.4 hours, 299.6 hours).

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