A paint manufacturer wishes to compare the drying times of two different types of paint. Independent random samples of 11 cans of type A (POP 1) and 9 cans of type B (POP 2) were selected and applied to similar surfaces. The drying times (in hours) were recorded. The summary statistics are as follows.

Pop 1 Pop 2
Samp. Mean 7843 98323
Samp. SD 231 21
Samp Size

Use a 0.01 significance level to test the claim that the mean drying time for paint type A is NOT equal to the mean drying time for paint type B.

WhAT WOULD BE THE HYPOTHESES FOR THIS TEST?

c08.p084.q006

Question

A paint manufacturer wishes to compare the drying times of two different types of paint. Independent random samples of 11 cans of type A (POP 1) and 9 cans of type B (POP 2) were selected and applied to similar surfaces.  The drying times (in hours) were recorded. The summary statistics are as follows.

Pop 1	Pop 2
Samp. Mean	7843	98323
Samp. SD	231	21
Samp Size

Use a 0.01 significance level to test the claim that the mean drying time for paint type A is NOT equal to the mean drying time for paint type B.

WhAT WOULD BE THE HYPOTHESES FOR THIS TEST?

c08.p084.q006

BlackTom AI · Accepted Answer

First, let's restate the problem in our own words to ensure understanding: we are comparing the mean drying times of two paint types, with the goal of testing whether their means differ.
Option 1 presents the alternative hypothesis: H1: μ1 ≠ μ2. This states that the two population means are not equal, which corresponds to a two-tailed test of difference in means.
Option 2 presents the null hypothesis: H0: μ1 = μ2. This is the standard null hypothesis for a two-sample mean comparison, asserting no difference in the population means.
Now, evaluating each choice:
- Option 1: H1: μ1 ≠ μ2. Why this is reasonable: it directly matches the claim being tested—that the mean drying times are not equal—so it represents the alternative hypothesis for a two-sided test. It captures the idea of any difference, whether μ1 is greater than μ2 or less than μ2.
- Option 2: H0: μ1 = μ2. Why this is reasonable as the null: the null hypothesis typically asserts no effect or no difference. In the context of a two-sample mean comparison, the standard null is that the two population means are equal, which is what we test against when we examine whether there is a difference.
But since the question explicitly asks for the hypotheses for this test of whether the means are not equal, we align with the conventional framework where the null is equality and the alternative is inequality.
Addressing potential misconceptions:
- Some might think the null could be μ1 ≠ μ2, but that would flip the roles of null and alternative and would misstate the standard testing setup; the test would then be designed to fail to reject a nonstandard claim.
- Another common pitfall is to write a directional alternative (e.g., μ1 > μ2 or μ1 < μ2). The problem requests the claim that the means are NOT equal, which calls for a two-sided (non-directional) alternative, hence μ1 ≠ μ2 rather than a one-sided form.
- It’s important to keep in mind that the H0 should always reflect no difference unless the context explicitly justifies a different baseline; in this case, the natural null is μ1 = μ2.
In summary, Option 1 correctly states the alternative hypothesis for a two-sided test, while Option 2 correctly states the null hypothesis, which is what the test uses as the baseline to determine if there is evidence of a difference.

BTRY3010/BTRY5010/STSCI2200/STSCI5200 Lecture #27 (Pages 130 - 135 )

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