Question

Hypothesis testing is used in business to test assumptions and theories. These assumptions are tested against evidence provided by actual, observed data. A statistical hypothesis is a statement about the value of a population parameter that we are interested in. Hypothesis testing is a process followed to arrive at a decision between 2 competing, mutually exclusive, collective exhaustive statements about the parameter’s value.

Consider the following scenario: An industrial seller of grass seeds packages its product in 50-pound bags. A customer has recently filed a complained alleging that the bags are underfilled. A production manager randomly samples a batch and measures the following weights:

Weight, (lbs)

45.6     49.5

47.7     46.7

47.6     48.8

50.5     48.6

50.2     51.5

46.9     50.2

47.8     49.9

49.3     49.8

53.1     49.3

49.5     50.1

To determine whether the bags are indeed being underfilled by the machinery, the manager must conduct a test of mean with a significance level α = 0.05.

In a minimum of 175 words, respond to the following:

• State appropriate null (Ho) and alternative (H1) hypotheses.
• What is the critical value if we work with a significant level α = 0.05?
• What is the decision rule?
• Calculate the test statistic.
• Are the bags indeed being underfilled?
• Should machinery be recalibrated?

Answer 1

Solution: To determine whether the bags are indeed being underfilled 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 populaion 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 α = 0.05 is -t(0.05,(n-1)) =  -1.729133 (Obtained from the probability table of Student's t distribution)

Here, the test statistic is T(observed) = -2.231851

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 underfilled and the machinery should be recalibrated.