Question

Pharmacologist Dr. Finch was asked by a drug company to compare the bioavailability of two brands of aspirin (brands A and B for simplicity). She randomly chose 10 healthy male subjects and asked each to take 3 pills of each brand on two separate days. The 1-hour urine concentrations (mg%) of aspirin for each subject on both occasions were carefully measured and tabulated as follows.

      Subject ID      Aspirin A 1-hour             Aspirin B 1-hour

	concentration	concentration
1	15	13
2	26	20
3	13	9
4	27	21
5	17	17
6	20	22
7	18	11
8	7	6
9	24	22
10	12	8

1. Let µ_A and µ_B be the mean 1-hour concentration of aspirin A and B, respectively. Write down Dr. Finch’s research objective in terms of H₀ and H_A.
2. Perform an appropriate test at the 5% significance level.
3. If a 95% CI for the difference µ_A − µ_B is desired, would the number zero within or without this CI? Briefly explain why or why not.

Answer 1

i.

Null and Alternative Hypotheses

ii.

The following table is obtained:

	Sample 1	Sample 2	Difference = Sample 1 - Sample 2
	15	13	2
	26	20	6
	13	9	4
	27	21	6
	17	17	0
	20	22	-2
	18	11	7
	7	6	1
	24	22	2
	12	8	4
Average	17.9	14.9	3
St. Dev.	6.471	6.226	2.906
n	10	10	10

For the score differences, we have

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μD​ = 0

Ha: μD​ ≠ 0

This corresponds to a two-tailed test, for which a t-test for two paired samples be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the degrees of freedom are df=9.

Hence, it is found that the critical value for this two-tailed test is tc​=2.262, for α=0.05 and df=9.

The rejection region for this two-tailed test is R={t:∣t∣>2.262}.

(3) Test Statistics

The t-statistic is computed as shown in the following formula:

(4) The decision about the null hypothesis

Since it is observed that ∣t∣=3.265>tc​=2.262, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0098, and since p=0.0098<0.05, it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population mean μA​ is different than μB​, at the 0.05 significance level.

iii.

No, it would not contain 0. This is because we have rejected the null hypothesis that the mean difference is 0, and therefore 0 would not be contained in the interval.

{The actual 95% CI is 0.921<μD​<5.079}

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