Question

A survey claims that 9 out of 10 doctors (i.e., 90%) recommend brand Z for their patients who have children. To test this claim against the alternative that the actual proportion of doctors who recommend brand Z is less than 90%, a random sample of doctors was taken. Suppose the test statistic is z= -1.75. Can we conclude that He should be rejected at the a) a = 0.10, b) a = 0.05, and c) a = 0.01 level?

A. a) no; b) no; c) yes
B. a) no; b) no; c) no
c. a) yes; b) yes; c) no
D. a) yes; b) yes; c) yes

Answer 1

The test statistic -1.75 leads to rejection of the null hypothesis at alpha levels 0.10 and 0.05, but not at 0.01. Hence the correct answer is yes for both 0.10 and 0.05 levels, and no for the 0.01 level, which is Option C: a) yes; b) yes; c) no.

Step-by-step explanation:

When conducting a hypothesis test, we compare the calculated test statistic to critical values corresponding to the levels of significance (alpha) to decide whether to reject the null hypothesis. For the given test statistic of z = -1.75, we need to look at the z-table to determine the p-value and compare with the alpha levels for the decision.

• At alpha = 0.10, the critical value for a one-tailed test is -1.28. Our test statistic of -1.75 is more extreme, so we reject the null hypothesis at this level.
• At alpha = 0.05, the critical value is -1.645. Our test statistic of -1.75 is more extreme, so we again reject the null hypothesis at this level.
• At alpha = 0.01, the critical value is -2.33. Our test statistic of -1.75 is not as extreme, so we do not reject the null hypothesis at this level.

Therefore, the correct answer is Option C: a) yes; b) yes; c) no, indicating that at the 10% and 5% levels, there is enough statistical evidence to conclude that the actual proportion recommending brand Z is less than 90%, but not at the 1% level.