Answer :
To calculate a 99% confidence interval for the mean weight of the product, we will use the formula for a confidence interval when the population standard deviation is known:
Identify the given information:
- Sample mean ([tex]\bar{x}[/tex]) = 75 lb
- Population standard deviation ([tex]\sigma[/tex]) = 100 lb
- Sample size ([tex]n[/tex]) = 50
- Confidence level = 99%
Determine the critical value (( Z^ )) for a 99% confidence level:*
The critical value for a 99% confidence level can be found using a standard normal distribution table or calculator. For a two-tailed test:
[tex]Z^* \approx 2.576[/tex].Calculate the standard error of the mean (SE):
[tex]SE = \frac{\sigma}{\sqrt{n}} = \frac{100}{\sqrt{50}} \approx 14.14[/tex]Calculate the margin of error (ME):
[tex]ME = Z^* \times SE = 2.576 \times 14.14 \approx 36.42[/tex]Find the confidence interval:
The confidence interval is calculated using the sample mean, the margin of error, and the critical value:
[tex]\text{Confidence Interval} = \bar{x} \pm ME = 75 \pm 36.42[/tex]
This means the confidence interval is:
[tex][75 - 36.42, 75 + 36.42] = [38.58, 111.42][/tex]
Therefore, the 99% confidence interval for the mean weight of the product is [tex][38.58, 111.42][/tex] pounds. This interval provides a range where we are 99% confident that the true average weight of the product lies.
