Question

A random selection of volunteers at a research institute have been exposed to a typical cold virus. After they started to have cold symptoms,

20

of them were given multivitamin tablets daily which contain

1

gram of vitamin C and various other vitamins and minerals. The remaining

20

volunteers were given tablets only containing

4

grams of vitamin C. For each individual, the length of time taken to recover from cold is recorded. At the end of the experiment the following data are obtained:

	Days to recover from cold
Treated with multivitamin	4.1, 6.2, 8.8, 6.7, 6.2, 3.9, 4.5, 5.6, 5.3, 6.3, 2.9, 4.5, 6.3, 4.4, 5.5, 3, 6, 5.7, 9.5, 4.9
Treated with Vitamin C	5.4, 4.9, 5.6, 4.6, 7.5, 5.6, 6.8, 6.3, 3.4, 4.6, 4.6, 6.4, 4.1, 4.2, 3.9, 5.7, 3.8, 4.4, 3.6, 4.7

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It is known that the population standard deviation of recovery time from cold is

1.8

days when treated with multivitamin, and the population standard deviation of recovery time from cold is

1.5

days when treated with vitamin C tablets. It is also known that both populations are approximately normally distributed. The researchers claim that the mean recovery time,

μ1

, of the patients treated with multivitamin is less than or equal to the mean recovery time

μ2

, of the patients who are treated with vitamin C tablets. At the

0.1

level of significance, is there enough evidence to reject this claim? Perform a one-tailed test. Then fill in the table below.

Carry your intermediate computations to at least three decimal places and round your answers as specified in the table. (If necessary, consult a list of formulas.)

The null hypothesis: H0: The alternative hypothesis: H1: The type of test statistic: (Choose one)ZtChi squareF The value of the test statistic: (Round to at least three decimal places.) The p-value: (Round to at least three decimal places.) Can we reject the researchers' claim that the mean recovery time when treated with multivitamin is less than or equal to the mean recovery time when treated with vitamin C only? Yes No

Answer 1

Solution :

The null and alternative hypotheses would be as follows :

Type of test statistic : Z test

Test statistic :

The test statistic is given as follows :

Where, x̅_{​​​​​1}, x̅_{​​​​​2} are sample means, n_{​​​​​​1}, n_{​​​​​​2} are sample sizes and σ_{​​​​​​1}, σ_{​​​​​​2} are population standard deviations.

We have following values :

The value of the test statistic is 0.9734.

P-value :

Since, our test is right-tailed test, therefore we shall obtain right-tailed p-value for the test statistic. The right-tailed p-value is given as follows :

P-value = P(Z > value of the test statistic)

P-value = P(Z > 0.9734)

P-value = 0.1652

The p-value is 0.1652.

Decision :

Since, p-value is greater than the significance level of 0.1, therefore we shall be fail to reject the null hypothesis at 0.1 significance level.

Conclusion :

At 0.1 significance level, we can not reject the researchers' claim that the mean recovery time when treated with multivitamin is less than or equal to the mean recovery time when treated with vitamin C only.

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