Question

For questions that ask you to conduct an appropriate hypothesis test, please show all of the following:

1. 1) A statement of the null and alternative hypotheses in words or symbols

2. 2) The calculation of an appropriate test statistic (show your work)

   For test statistics involving the t-distribution: round to 3 decimal places.

   For test statistics involving the Normal distribution: round to 2 decimal places.

3. 3) Calculation of the p-value or critical value, as appropriate (show your work)

4. 4) Conclusion: do you reject the null hypothesis? Why or why not?

5. 5) Interpretation: one sentence explaining what the conclusion means in the context of

   the original research question.

A major keyboard manufacturer has a line of keyboards designed for apartment dwellers. These keyboards need to be light enough to be carried up flights of stairs. The lead engineer wants to use a new type of material. The engineer claims that the new keyboards will be lighter than the old keyboards.

They take a sample of 4 keyboards manufactured using the old material and compute an average weight of 21 kg with a standard deviation of 1 kg.

They take a sample of 8 keyboards manufactured using the new material and compute an average weight of 17 kg with a standard deviation of 2 kg.

1. Conduct an appropriate hypothesis test using the p-value method. Use the old material as population 1. [8 marks]

2. How much evidence is there against the null hypothesis in part (a)? [1 mark]

3. Explain what a Type II Error means in this context. [1 mark]

Answer 1

Claim:  The engineer claims that the new keyboards will be lighter than the old keyboards.

Old material as population 1 and new material as population 2.

New keyboards are lighter means their mean weight is less than old ones.

That is

The claim contains strictly more than sign so it comes under the alternative hypothesis.

(1) The null and alternative hypothesis

H₀: The new keyboards are not lighter than old keyboards that is

H₁: The new keyboards are lighter than old keyboards that is

(2) Here the sample sizes for both samples are small and the population variances are unknown, so t-test would be applicable.

Assume population variances are equal.

The formula of t-test statistics is,

Where

t-test statistics = 3.710

(3) P-value

Here the alternative hypothesis contains greater than sign, so the test is a right-tailed test.

Alpha is not given so it should be 0.05

Degrees of freedom = n1 + n2 - 2 = 4 + 8 - 2 = 10

The P-value using excel is =tdist(test statistics, degrees of freedom, the tail of the test) which is =tdist(3.710, 10, 1) gives 0.002

P-value = 0.002

(4) Conclusion:

The decision rule is,

If P-value > alpha then fail to reject the null hypothesis otherwise reject the null hypothesis.

Here P-value (0.002) is not greater than alpha (0.05) so reject the null hypothesis.

Reject the null means in favor of alternative hypothesis that is claim.

(5) Interpretation: Reject the null hypothesis, that is there is sufficient evidence to support the engineer claim that new keyboards will be lighter than old keyboards.

Type II error:

Type II error is the probability of failing to reject the null hypothesis when actually it is false.

That is the new keyboards will not be lighter than the old keyboards but actually they are lighter than old ones.