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At Farmer's Dairy, a machine is set to fill 32 -ounce milk cartons. However, this machine does not put exactly 32 ounces of milk into each carton; the amount varies slightly from carton to carton. It is known that when the machine is working properly, the mean net weight of these cartons is 32 ounces. The standard deviation of the amounts of milk in all such cartons is always equal to \(.15\) ounce. The quality control department takes a sample of 25 such cartons every week, calculates the mean net weight of these cartons, and makes a \(99 \%\) confidence interval for the population mean. If either the upper limit of this confidence interval is greater than \(32.15\) ounces or the lower limit of this confidence interval is less than \(31.85\) ounces, the machine is stopped and adjusted. A recent sample of 25 such cartons produced a mean net weight of \(31.94\) ounces. Based on this sample, will you conclude that the machine needs an adjustment? Assume that the amounts of milk put in all such cartons have a normal distribution.

Step by step solution

01

Calculate the Margin of error

The margin of error for a 99% confidence interval can be calculated using the formula \(E = Z*\frac{\sigma}{\sqrt{n}}\) where Z is the Z-value for the desired confidence level (which is 2.576 for 99% confidence level), \( \sigma \) is standard deviation (which is 0.15), and n is the sample size (which is 25). Therefore, \(E = 2.576*\frac{0.15}{\sqrt{25}} = 0.077 \) ounces.

02

Calculate the Confidence Interval

The confidence interval is calculated as \( \bar{X} \pm E \) where \( \bar{X} \) is the sample mean (which in this case is 31.94 ounces), and E is the margin of error. Therefore, the confidence interval is \(31.94 \pm 0.077\) i.e., it ranges from \(31.863\) ounces to \(32.017\) ounces.

03

Decision Rule

A decision whether the machine needs adjustment is made based on whether the lower limit of the confidence interval is less than 31.85 ounces or the upper limit is greater than 32.15 ounces. Since this is not the case — our confidence interval ranges from 31.863 ounces to 32.017 ounces, which lies within the set range — we conclude that based on this sample, the machine does not need adjustment.

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