Question

In investigating several complaints concerning the weight of the "NET WT. 12 OZ." jar of a local brand of peanut butter, the Better Business Bureau selected a sample of 36 jars. The sample showed an average net weight of \(11.92\) ounces and a standard deviation of \(.3\) ounce. Using a \(.01\) level of significance, what would the Bureau conclude about the operation of the local firm?

Short answer

Based on the hypothesis test with a 0.01 level of significance, the Better Business Bureau cannot conclude that the local firm's average peanut butter jar weight is significantly different from 12 ounces. Therefore, the Bureau should consider the operation of the local firm fair in terms of peanut butter jar weight.

Step-by-step solution

State the hypothesis

The null hypothesis (H0) and the alternative hypothesis (H1) are as follows. \(H_0\): The average weight of peanut butter jars (\(μ\)) is 12 ounces. \(H_1\): The average weight of peanut butter jars (\(μ\)) is not 12 ounces.

Determine the level of significance

The level of significance (\(α\)) is given as 0.01.

Calculate the test statistic

The test statistic for the sample is: \[z = \frac{(\bar{x} - μ)}{(\frac{s}{\sqrt{n}})}\] Where \(z\) is the test statistic, \(\bar{x}\) is the sample mean, \(μ\) is the population mean, \(s\) is the sample standard deviation, and \(n\) is the sample size.

Substitute values and calculate the test statistic

Now we will substitute the given values: \(z = \frac{(11.92 - 12)}{(.3/\sqrt{36})}\) Solve for \(z\): \(z = -2\)

Determine the critical values

Since this is a two-tailed test (due to the alternative hypothesis stating that the average weight is "not" 12 ounces), we will have two critical values. We will compare the calculated test statistic (\(z\)) to these critical values. For a level of significance (\(α\)) of 0.01, the critical values are \(z_{0.005}\) = -2.576 and \(z_{0.995}\) = 2.576 (using a z-score table).

Compare the test statistic to the critical values

Now we have to compare the test statistic (\(z\)) with the critical values: -2.576 < -2 < 2.576 The test statistic falls within the range of the critical values, which means we fail to reject the null hypothesis (\(H_0\)).

Conclusion

Based on the results of this hypothesis test at a 0.01 level of significance, the Better Business Bureau does not have enough evidence to conclude that the local firm's average peanut butter jar weight is significantly different from 12 ounces. The Bureau should conclude that the operation of the local firm is fair in terms of peanut butter jar weight.