Question

The life in hours of a 75-watt light bulb is known to be normally distributed with σ=25 hours. A random sample of 20 bulbs has a mean life of

Xˉ=1014 hours. Construct a 95% two-sided confidence interval on the mean life.

Answer 1

The 95% confidence interval for the mean life of the 75-watt light bulb is [956.38 hours, 1071.62 hours].

Step-by-step explanation:

To calculate the confidence interval, we'll use the formula:

`\[ \bar{X} \pm Z_(\alpha/2) \cdot (\sigma)/(√(n)) \]`

where
`\( \bar{X} \)` is the sample mean (1014 hours),
`\( \sigma \)` is the standard deviation (25 hours), n is the sample size (20), and
`\( Z_(\alpha/2) \)` is the critical value for a 95% confidence interval, which is 1.96.

Substituting the values into the formula:

`\[ \text{Margin of Error} = Z_(\alpha/2) \cdot (\sigma)/(√(n)) = 1.96 \cdot (25)/(√(20)) \ = 29.31 \]`

The confidence interval is calculated as:

`\[ \text{Confidence Interval} = \bar{X} \pm \text{Margin of Error} = 1014 \pm 29.31 = [956.38, 1071.62] \]`

Therefore, we are 95% confident that the true mean life of 75-watt light bulbs lies between 956.38 hours and 1071.62 hours based on the given sample. This range provides a level of confidence that the actual mean life of the bulbs falls within this interval, accounting for sampling variability.

This calculation assumes that the sample is representative of the population of 75-watt light bulbs and that the distribution is approximately normal. The confidence interval gives us a range within which we can reasonably expect the true population mean to be, given the sample mean and the variability of the data.