The quality-control manager at a light bulb factory needs to determine whether the mean life of a large shipment of light bulbs is equal to 375 hours. The population standard deviation is 100 hours. A random sample of 64 light bulbs indicates a sample mean life of 350 hours.a. At the 0.05 level of significance, is there evidence that the mean life is different from 375 hours?b. Compute the p-value and interpret its meaning.c. Construct a 95% confidence interval estimate of the population mean life of the light bulbs.d. Compare the results of (a) and (c). What conclusions do you reach?
The Correct Answer and Explanation is:
- Population standard deviation (σ) = 100 hours
- Sample size (n) = 64
- Sample mean (x̄) = 350 hours
- Hypothesized mean (μ₀) = 375 hours
- Significance level (α) = 0.05
a. Hypothesis Test
Null Hypothesis (H₀): μ = 375
Alternative Hypothesis (H₁): μ ≠ 375
This is a two-tailed z-test.
Calculate the test statistic: z=xˉ−μ0σ/n=350−375100/64=−2512.5=−2.0z = \frac{\bar{x} – \mu_0}{\sigma / \sqrt{n}} = \frac{350 – 375}{100 / \sqrt{64}} = \frac{-25}{12.5} = -2.0
At a 0.05 significance level for a two-tailed test, the critical z-values are ±1.96.
Since -2.0 < -1.96, we reject the null hypothesis.
b. Compute the p-value
Using the standard normal distribution: p-value=2×P(Z<−2.0)=2×0.0228=0.0456p\text{-value} = 2 \times P(Z < -2.0) = 2 \times 0.0228 = 0.0456
Interpretation: The probability of observing a sample mean of 350 or more extreme, assuming the population mean is 375, is about 4.56%. Since this is less than the significance level (0.05), it supports rejecting the null hypothesis.
c. 95% Confidence Interval
CI=xˉ±zα/2×σn=350±1.96×10064=350±24.5CI = \bar{x} \pm z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}} = 350 \pm 1.96 \times \frac{100}{\sqrt{64}} = 350 \pm 24.5 ⇒(325.5,374.5)\Rightarrow (325.5, 374.5)
d. Comparison and Conclusion
From (a), we rejected the null hypothesis, indicating evidence that the mean is different from 375 hours. The confidence interval (325.5 to 374.5) from (c) does not include 375, which supports this conclusion. Both methods point to the same result: the true mean life of the light bulbs is significantly different from 375 hours. The company should investigate possible causes for the reduced average lifespan.
