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This problem is based on information taken from The Merck Manual (a reference manual used in most medical and nursing schools). Diltiazem is a commonly prescribed drug for hypertension. However, diltiazem causes headaches in about 12% of patients using the drug. It is hypothesized that regular exercise might help reduce the headaches. If a random sample of 211 patients using diltiazem exercised regularly and only 19 had headaches, would this indicate a reduction in the population proportion of patients having headaches? Use a 1% level of significance. (a) What is the level of significance? State the null and alternate hypotheses. Ho: p = 0.12; Hip < 0.12 Ho: p = 0.12; Hip > 0.12 Ho: p < 0.12; Hip - 0.12 - F: p = 0.12; Fi: p + 0.12 Hu: P = 0.12; Hi: P = 0.12 (b) What sampling distribution will you use? The Student's t, since np > 5 and ng > 5. The standard normal, since np > 5 and ng > 5. The Student's t, since np < 5 and ng < 5. The standard normal, since np < 5 and ng < 5. What is the value of the sample test statistic? (Round your answers to two decimal places.) (c) Find the p-value of the test statistic. (Round your answers to four decimal places.) Sketch the sampling distribution and show the area corresponding to the P-value. WebAssign Plot -3 -2 -1 0 2 (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level a? At the u = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant. At the a = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant. At the a-0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the a = 0.01 level, we fail to reject the null hypothesis and condude the data are not statistically significant (e) Interpret your conclusion in the context of the application. There is sufficient evidence at the 0.01 level to conclude that the true proportion of patients having headaches is less than 12%. There is insufficient evidence at the 0.01 level to conclude that the true proportion of patients having headaches is less than 12%.

Answer 1

Solution: Sample size = n = 211 Sample number of patients having headaches = x = 19 Sample proportion of patients having headaches: x 19 P = = 241 = 0.0900 Part-A: Level of significance =a= 1% = 0.01 Ho:p = 0.12 H:p < 0.12 Part-B: np = 211 x 0.12 = 25.3 nq = n(1-p) = 211 x (1 – 0.12) = 185.7 The standard normal distribution will be used, since np > 5 and nq > 5 Standard error of the sampling distribution: p(1 - p) 0.12 x (1 – 0.12) L= 0.0224 п 211 Op = 20 = Test Statistic: php Op 0.0900 -0.12 → 20 = -= -1.34 0.0224 Part-C: From normal distribution table; area under the normal curve to the left of (z = -1.34), is given by: 0.0901 Hence, p-value = 0.0901

Part-D: Since p-value > a, we fail to reject the null hypothesis. At the a = 0.01 level, we fail to reject the null hypothesis and conclude that the data are not statistically significant. Part-E: There is insufficient evidence at the 0.01 level to conclude that the true proportion of patients having headaches is less than 12%.