Question

A company claims that a new manufacturing process changes the mean amount of aluminum needed for cans and therefore changes the weight. Independent random samples of aluminum cans made by the old process and the new process are taken. The summary statistics are given below. Is their evidence the 5% significance level (or 95 confidence level) to support the claim that the mean weight for all old cans is different than the mean weight for all new cans ?Justify fully!

The old process had a sample size of 25 with a mean of 0.509 and a standard deviation of 0.019. The new process has a sample size of 25 with a mean of 0.495 and a standard deviation of 0.021.

For any Hypothesis Test make sure to state Ho, Ha, Test statistic, p-value, whether you reject Ho, and your conclusion in the words of the claim.

For any confidence interval make sure that you interpret the interval in context, in addition to using it for inference.

  

Answer 1

Claim: The mean weight for all old cans is different than the mean weight for all new cans that is

The null and alternative hypotheses are,

Test statistics:

Population standard deviations are unknown, t-test would be applicable.

The formula of t-test statistics:

Where,

Test statistics = 2.472

p-value:

The alternative hypothesis contains not equal to sign, the test is a two-tailed test.

Degrees of freedom = n1 + n2 - 2 = 48

Using excel the p-value is = T.DIST.2T(test statistics, degrees of freedom) = T.DIST.2T(2.472, 48) = 0.0170

p-value = 0.0170

Alpha = significance level = 5% = 0.05

Decision rule: If P-value > alpha then fails to reject the null hypothesis otherwise reject the null hypothesis.

P-value 0.0170 is not greater than alpha 0.05, so reject the null hypothesis.

Conclusion: Reject the null hypothesis H0 that is there is sufficient evidence to support the claim that the mean weight for all old cans is different than the mean weight for all new cans.

If the test is a two-tailed test and the confidence interval is found which is also two-tailed.

If 0 falls between the confidence interval then fail to reject the null hypothesis and if 0 does not fall in the interval then reject the null hypothesis.

The decision of a two-tailed test can be taken from the confidence interval too.