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The weight of a product is measured in pounds. A sample of 50 units is taken from production. The sample yielded a mean of 75 lb. The population standard deviation is 100 lb. Calculate a 99 percent confidence interval.

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:

  1. 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%

  2. 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].

  3. Calculate the standard error of the mean (SE):
    [tex]SE = \frac{\sigma}{\sqrt{n}} = \frac{100}{\sqrt{50}} \approx 14.14[/tex]

  4. Calculate the margin of error (ME):
    [tex]ME = Z^* \times SE = 2.576 \times 14.14 \approx 36.42[/tex]

  5. 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.

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