The Quality Control Manager At A Light Globe Factory Needs T

The Quality Control Manager At A Light Globe Factory Needs To Estimate

The quality control manager at a light globe factory needs to estimate the mean life of a large shipment of energy-saving compact light globes. The historical standard deviation of light globe life is 250 hours. A random sample of 64 light globes indicates a sample mean life of 7,940 hours.

a) Construct a 95% confidence interval estimate of the population mean life of light globes in this shipment.

b) Do you think that the manufacturer has the right to state that the light globes last on average 8,000 hours? Explain.

c) Must you assume that the population of light globe life is normally distributed? Explain.

d) Suppose that the standard deviation changes to 180 hours. What are your emended answers in part (a) and (b)?

Paper For Above instruction

The estimation of the average lifespan of a batch of light globes is crucial for manufacturers to ensure product reliability and maintain consumer trust. Given the data provided, we will analyze the sample statistics to construct confidence intervals, evaluate the manufacturer’s claims, and consider the assumptions necessary for valid statistical inferences.

In part (a), with a sample size (n) of 64 and a known population standard deviation (σ) of 250 hours, we utilize the z-distribution to construct a 95% confidence interval for the population mean. The sample mean (x̄) is 7,940 hours. The formula for the confidence interval when σ is known is:

CI = x̄ ± Z_α/2 * (σ / √n)

Where Z_α/2 corresponds to the z-score for a 95% confidence level, which is approximately 1.96. Plugging in the numbers:

CI = 7940 ± 1.96 (250 / √64) = 7940 ± 1.96 (250 / 8) = 7940 ± 1.96 * 31.25

Calculating further:

CI = 7940 ± 1.96 * 31.25 ≈ 7940 ± 61.25

Therefore, the 95% confidence interval for the average lifespan of the light globes in this shipment is approximately (7878.75 hours, 8001.25 hours). This interval suggests that, with 95% confidence, the true mean lifespan of the globes lies within this range.

In part (b), the manufacturer claims an average lifespan of 8,000 hours. Since 8,000 hours falls within the calculated confidence interval (7878.75, 8001.25), statistically, there is support for the claim that the true mean lifespan could be around 8,000 hours. However, the point estimate (sample mean) is slightly below 8,000 hours, and the interval's upper bound is just slightly above it. This indicates that while it's plausible that the mean lifespan is 8,000 hours, there remains uncertainty, and additional data could refine this estimate.

Regarding part (c), the normality assumption is important when the sample size is small, as it influences the distribution of the sample mean. However, in this case, the sample size is 64, which is sufficiently large according to the Central Limit Theorem. The CLT states that for large samples, the sampling distribution of the sample mean tends to be approximately normal regardless of the population’s distribution. Therefore, we do not need to assume the population of light globe life is strictly normally distributed because the large sample size allows us to rely on the normal approximation.

Finally, in part (d), when the standard deviation σ is assumed to change to 180 hours, the calculations for the confidence interval need adjustment. Using the same sample mean and sample size:

CI = 7940 ± 1.96 (180 / 8) = 7940 ± 1.96 22.5 ≈ 7940 ± 44.1

This simplifies to an interval of approximately (7895.9 hours, 7984.1 hours). Comparing this to the previous interval, the narrower range reflects reduced uncertainty due to the smaller standard deviation.

For the manufacturer's claim of an 8,000-hour lifespan, since 8,000 hours falls within this narrower interval, it remains consistent with the data. The decreased variability supports a narrower confidence interval, giving more precise information about the mean lifespan.

In conclusion, the analytical process demonstrates how confidence intervals are constructed and interpreted in quality control settings. It also highlights the importance of understanding assumptions and how changes in known population parameters influence inference. Large sample sizes and accurate knowledge of standard deviation are key factors enabling reliable statistical conclusions in manufacturing quality assessments.

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