A pharmaceutical company has developed a new drug that is believed to reduce blood pressure more effectively than the existing treatment. To test the effectiveness of the new drug, a random sample of 50 patients with high blood pressure was selected. After taking the new drug for a month, the mean reduction in blood pressure for these patients was found to be 8.5 mm Hg with a standard deviation of 3.2 mm Hg. The company claims that the new drug reduces blood pressure by more than 7 mm Hg. Test this claim at the 5% level of significance.
Question
A pharmaceutical company has developed a new drug that is believed to reduce blood pressure more effectively than the existing treatment. To test the effectiveness of the new drug, a random sample of 50 patients with high blood pressure was selected. After taking the new drug for a month, the mean reduction in blood pressure for these patients was found to be 8.5 mm Hg with a standard deviation of 3.2 mm Hg. The company claims that the new drug reduces blood pressure by more than 7 mm Hg. Test this claim at the 5% level of significance.
Solution
To test this claim, we can use a one-sample t-test. Here are the steps:
Step 1: State the null and alternative hypotheses. The null hypothesis (H0) is that the mean reduction in blood pressure is 7 mm Hg. The alternative hypothesis (H1) is that the mean reduction in blood pressure is more than 7 mm Hg.
H0: μ = 7 H1: μ > 7
Step 2: Calculate the test statistic. The test statistic for a one-sample t-test is calculated as follows:
t = (X̄ - μ0) / (s/√n)
where X̄ is the sample mean, μ0 is the population mean under the null hypothesis, s is the sample standard deviation, and n is the sample size.
t = (8.5 - 7) / (3.2/√50) = 3.35
Step 3: Determine the critical value. The critical value for a one-tailed t-test with 49 degrees of freedom (n-1) at the 5% level of significance is approximately 1.677.
Step 4: Make a decision. Since the test statistic (3.35) is greater than the critical value (1.677), we reject the null hypothesis.
Step 5: Interpret the result. There is sufficient evidence at the 5% level of significance to support the company's claim that the new drug reduces blood pressure by more than 7 mm Hg.
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