A survey claims that 9 out of 10 doctors recommend aspirin for their patients with headaches.To test this claim against the alternative that the actual proportion of doctors who recommend aspirin is less than 0.90,a random sample of 100 doctors results in 83 who indicate that they recommend aspirin.The value of the test statistic in this problem is approximately equal to:
A)- 4.12
B)- 2.33
C)- 1.86
D)- 0.07

Question

Quizplus · Accepted Answer

This is a hypothesis testing problem with the following null and alternative hypotheses:

H0: p = 0.90 (proportion of doctors who recommend aspirin is 0.90)
Ha: p < 0.90 (proportion of doctors who recommend aspirin is less than 0.90)

We can use the z-test for proportions to test this hypothesis:

z = (p̂ - p) / sqrt(p*(1-p)/n)

where p̂ is the sample proportion, p is the hypothesized proportion, and n is the sample size.

In this case, p̂ = 83/100 = 0.83, p = 0.90, and n = 100.

Substituting these values into the z-formula, we get:

z = (0.83 - 0.90) / sqrt(0.90*(1-0.90)/100) = -2.33

The test statistic is -2.33, which is the same as the z-score we would get if we standardize the sample proportion using the hypothesized proportion as the mean and the standard deviation based on the null hypothesis.

Since the alternative hypothesis is one-sided (less than), we can reject the null hypothesis at the 5% significance level if the test statistic is less than the critical value of -1.645 (from the standard normal distribution).

Since -2.33 is less than -1.645, we can reject the null hypothesis and conclude that there is evidence to support the claim that the actual proportion of doctors who recommend aspirin is less than 0.90.

A Survey Claims That 9 Out of 10 Doctors Recommend