Questions: 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 7,469 hours. The population standard deviation is 1,080 hours. A random sample of 81 light bulbs indicates a sample mean life of 7,169 hours. At the 0.05 level of significance, is there evidence that the mean life is different from 7,469 hours? Compute the p-value and interpret its meaning. Construct a 95% confidence interval estimate of the population mean life of the light bulbs. Compare the results of (a) and (c). What conclusions do you reach?

Solution Steps

To determine the standard error \(SE\) of the sample mean, we use the formula:

\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{1080}{\sqrt{81}} = 120.0 \]

We perform a two-tailed hypothesis test to check if the mean life of the light bulbs is different from \( \mu_0 = 7469 \) hours. The test statistic \(Z\) is calculated as follows:

\[ Z = \frac{\bar{x} - \mu_0}{SE} = \frac{7169 - 7469}{120.0} = -2.5 \]

Next, we calculate the p-value for the two-tailed test:

\[ P = 2 \times (1 - T(|z|)) = 0.0124 \]

At a significance level of \( \alpha = 0.05 \), we compare the p-value with \( \alpha \):

Since \( P = 0.0124 < 0.05 \), we reject the null hypothesis. This indicates that there is evidence that the mean life of the light bulbs is different from \( 7469 \) hours.

To construct a 95% confidence interval for the population mean, we use the formula:

\[ \bar{x} \pm z \cdot \frac{\sigma}{\sqrt{n}} = 7169 \pm 1.96 \cdot 120.0 \]

Calculating the bounds:

\[ \text{Lower Bound} = 7169 - 1.96 \cdot 120.0 = 6933.8 \] \[ \text{Upper Bound} = 7169 + 1.96 \cdot 120.0 = 7404.2 \]

Thus, the 95% confidence interval is:

\[ (6933.8, 7404.2) \]

Final Answer