To determine whether the bags are indeed being under filled by the machinery, we are to test if the mean weight of the bags is less than 50 pounds or not. We thus set up our null and alternative hypotheses as:

H0: mu = 50 vs H1: mu < 50

where mu is the population mean of the bag's weight.

The test statistic is T= (xbar-mu0)/(s/sqrt(n)) ; where xbar = sample mean, mu0 = the hypothesized value of the population mean, n = sample size, s = sample standard deviation, sqrt refers to the square root function. Under H0, T ~ t(n-1)

The decision rule is--we reject H0 if T(observed) < - t(alpha,(n-1)), where t(alpha,(n-1)) is the upper alpha point of the t - distribution with (n-1) degrees of freedom.alpha = level of significance.

The critical value if we work with a significant level = 0.05 is -t(0.05,(n-1)) = -1.729 (Obtained from the probability table of Student's t distribution)

Here, the test statistic is T(observed) = -2.232

Hence, T(observed) < - t(alpha,(n-1)).

Thus, we reject H0 and conclude at a 5% level of significance on the basis of the given sample that there is enough evidence to support the claim that the average value of mean weight of the bags is less than 50 pounds.

Thus, the bags are indeed being under filled and the machinery should be recalibrated. Often times we are not made aware of situations like this until a customer brings it to our attention. As many mentioned, this machine will need to be recalibrated. It's good practice to avoid situations like this and attempt to check periodically.

Thinking of your work experience. Do you think companies are more proactive or reactive in checking areas like this?