Question

This problem is based on information taken from The Merck Manual (a reference manual used in most medical and nursing schools). Hypertension is defined as a blood pressure reading over 140 mm Hg systolic and/or over 90 mm Hg diastolic. Hypertension, if not corrected, can cause long-term health problems. In the college-age population (18-24 years), about 9.2% have hypertension. Suppose that a blood donor program is taking place in a college dormitory this week (final exams week). Before each student gives blood, the nurse takes a blood pressure reading. Of 200 donors, it is found that 28 have hypertension. Do these data indicate that the population proportion of students with hypertension during final exams week is higher than 9.2%? Use a 5% level of significance. (a) What is the level of significance? State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test? H0: p = 0.092; H1: p < 0.092; left-tailed H0: p > 0.092; H1: p = 0.092; right-tailed H0: p = 0.092; H1: p ≠ 0.092; two-tailed H0: p = 0.092; H1: p > 0.092; right-tailed (b) What sampling distribution will you use? Do you think the sample size is sufficiently large? The Student's t, since np > 5 and nq > 5. The standard normal, since np < 5 and nq < 5. The standard normal, since np > 5 and nq > 5. The Student's t, since np < 5 and nq < 5. What is the value of the sample test statistic? (Round your answer to two decimal places.) (c) Find the P-value of the test statistic. (Round your answer to four decimal places.) Sketch the sampling distribution and show the area corresponding to the P-value. Maple Generated Plot Maple Generated Plot Maple Generated Plot Maple Generated Plot (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 α? At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. (e) State your conclusion in the context of the application. There is sufficient evidence at the 0.05 level to conclude that the true proportion of students with hypertension during final exams week is higher than 0.092. There is insufficient evidence at the 0.05 level to conclude that the true proportion of students with hypertension during final exams week is higher than 0.092.

Answer 1

(a)

the level of significance = 0.05

Correct option:

H0: p = 0.092; H1: p > 0.092; right-tailed

(b)

Correct option:

The standard normal, since np > 5 and nq > 5.

n = Sample Size = 200

= Sample Proportion = 28/200 = 0.14

p = Population Proportion = 0.092

So,

q = 1 - p = 0.908

SE =

Test Statistic is given by:
Z = (0.14 - 0.092)/0.0204

= 2.35

So,

the value of the sample test statistic = 2.35

Table of Area Under Standard Normal Curve gives area = 0.4906

So,

P - value = 0.5 - 0.4906 = 0.0094

So,

the P-value of the test statistic = 0.0094

(d)

Correct option:

At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.

(e)

Correct option:

There is sufficient evidence at the 0.05 level to conclude that the true proportion of students with hypertension during final exams week is higher than 0.092.