Question

A laundry detergent company, Prodder and Gamby,designs their production machines to fill cartons to a mean weight of 32 ounces. A sample of cartons is periodically selected

and weighed to determine whether underfilling or overfilling is occurring. If so, the machines need to be recalibrated. One pallet has exactly 49 cartons on it. You randomly select a pallet and measure the weight of each carton and calculate a sample mean of 31.6.

a.Write the null and alternative hypotheses that are most appropriate for this situation. Interpret the alternative hypothesis in words.

b. Past research shows that the population standard deviation is 0.3. Calculate the test statistics for the hypothesis test above.

c.From the pallet you sampled, is there evidence that the machines to be recalibrated at the 10% level? The 5%? The 1%?

(Make sure to show your work)

Answer 1

Part a

Null hypothesis: H₀: The cartons have a mean weight of 32 ounces.

Alternative hypothesis: H_a: The cartons have a mean weight of other than 32 ounces.

H₀: µ = 32 versus H_a: µ ≠ 32

This is a two tailed test.

Here, we select the alternative hypothesis as the mean weight is not equal to 32 ounces because it is given that the sample is checking for under filling and overfilling. In both cases, that is, under filling and over filling, the process is recalibrated.

Part b

We are given σ = 0.3

Xbar = 31.6

n = 49

The test statistic formula is given as below:

Z = (Xbar - µ)/[σ/sqrt(n)]

Z = (31.6 – 32)/[0.3/sqrt(49)]

Z = -0.4/ 0.042857

Z = -9.33333

Test statistic = Z = -9.33333

Part c

From above part, we have test statistic = -9.33333

So, P-value = 0.00 (by using z-table)

P-value < α = 0.10

P-value < α = 0.05

P-value < α = 0.01

So, we reject the null hypothesis at 10%, 5%, and 1% significance level.

There is sufficient evidence to conclude that the machines to be recalibrated at 10%, 5%, and 1% significance level.