Question

A paint manufacturer fills cans of paint using a machine that has been calibrated to fill the cans to contain an average of 1 gallon (128 ounces) each. To test whether their machine has come out of calibration, the manufacturer takes a random sample of 25 cans and finds that they average 128.2 ounces with a standard deviation of 2 ounces. Is this strong evidence that the filling machine is set too high and thus is no longer calibrated properly? a). Give the correct null and alternative hypotheses.

b). If the null hypothesis is true, the sampling distribution has a mean: a standard deviation: and c). The standard score for the data: d). Find the P-value for this significance test: e). Do you reject or do you not reject the null hypothesis? f). Choose the correct final conclusion in the context of the problem. Level of significance is 0.05. A). There is not enough evidence to suggest that the machine is set too high and not calibrated properly. B). There is enough evidence to suggest that the machine is set too high and is not calibrated properly.

Answer 1

a) Null hypothesis: The filling machine is calibrated properly and fills cans to contain an average of 128 ounces.

Alternative hypothesis: The filling machine is set too high and fills cans to contain an average of more than 128 ounces.

b) If the null hypothesis is true, the sampling distribution has a mean of 128 ounces, a standard deviation of 2 ounces, and follows a normal distribution.

c) The standard score for the data is (128.2 - 128) / (2 / sqrt(25)) = 1.

d) The P-value for this significance test can be found using a one-sample t-test with 24 degrees of freedom (df = n-1). Using a t-distribution table or a calculator, the P-value for a one-tailed test with a t-score of 1 and 24 degrees of freedom is approximately 0.16.

e) Since the P-value is greater than the level of significance (0.05), we do not reject the null hypothesis.

f) The correct final conclusion in the context of the problem is: There is not enough evidence to suggest that the machine is set too high and not calibrated properly.