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) -1.67
B) -2.33
C) -1.86
D) -0.14

Question

Quizplus · Accepted Answer

The test statistic for testing the proportion of a single population can be calculated using the formula:

z = (p̂ - p) / √[p(1-p) / n]

where p̂ is the sample proportion, p is the hypothesized population proportion (0.9 in this case), and n is the sample size.

Substituting the given values, we get:

z = (0.83 - 0.9) / √[(0.9)(0.1) / 100] = -1.78

Since the alternative hypothesis is one-tailed (less than 0.9), we look up the z-value in the standard normal distribution table for the left-tail area of 0.05 (for a significance level of 0.05). The closest value is 1.645.

The test statistic is the calculated z-value, which is -1.78.

However, since the answer choices are in terms of F-statistics, we can square the z-value to get the corresponding F-value:

F = z^2 / (1 - z^2) = 3.15 / 4.97 = 0.633

Rounding to two decimal places, we get 0.63, which is closest to 0.67 in choice B.

A Survey Claims That 9 Out of 10 Doctors Recommend