Question

A study conducted by the research department of a pharmaceutical company claims that the annual spending (per person) for prescription drugs for allergy relief,

μ1

, is greater than or equal to the annual spending (per person) for non-prescription allergy relief medicine,

μ2

. A health insurance company conducted an independent study and collected data from a random sample of

225

individuals for prescription allergy relief medicine. The sample mean is found to be

17.5

dollars/year, with a sample standard deviation of

5.7

dollars/year. They have also collected data for non-prescription allergy relief medicine. An independent random sample of

285

individuals yielded a sample mean of

18.5

dollars/year, and a sample standard deviation of

4.3

dollars/year. Since the sample size is quite large, it is assumed that the population standard deviation of the sales (per person) for prescription and non-prescription allergy relief medicine can be estimated by using the sample standard deviation values given above. Is there sufficient evidence to reject the claim made by the research department of the company, at the

0.01

level of significance? 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.

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 critical value at the 0.01 level of significance: (Round to at least three decimal places.) Can we reject the claim that the mean spending on prescription allergy relief medication is greater than or equal to the mean spending on non-prescription allergy relief medication? Yes No

Answer 1

Solution :

The null and alternative hypotheses are as follows :

Since, the sample sizes are quite large, therefore we shall use z-test to test the hypothesis. The test statistic is given as follows :

Where, are sample means, are population standard deviations and n1, n2 are sample sizes.

We have,

Since, population standard deviations are not known and sample sizes are quite large, therefore we shall use sample standard deviations as the estimate of population standard deviations.

Hence,

The value of the test statistic is -2.186.

Since, our test is left-tailed test, therefore we shall obtain left-tailed critical z-value at the given significance level. The left-tailed critical z-value at 0.01 significance level is given as follows :

Critical value at the 0.01 significance level is -2.326.

For left-tailed test, we make decision rule as follows :

If value of the test statistic is less than the critical value, then we reject the null hypothesis.

If value of the test statistic is greater than the critical value, then we fail to reject the null hypothesis.

Test statisic = -2.186

Critical value = -2.326

(-2.186 > -2.326)

Since, value of the test statistic is greater than the critical value, therefore we shall be fail to reject the null hypothesis (H​​​​​​0) at 0.01 significance level.

Conclusion : At 0.01 significance level, there is not sufficient evidence to reject the claim that the mean spending on prescription allergy relief medication is greater than or equal to the mean spending on non-prescription allergy relief medication.

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