Chapter 16, Problem 1SR

a.

To determine

State whether the test is one-tailed or two-tailed test of hypothesis.

Answer to Problem 1SR

Two-tailed test.

Explanation of Solution

In this context, it is known that the sample size is 12 and significance level is 0.10.

The test hypothesis is given as follows:

Denote $\pi$ as the proportion of successes.

Null hypothesis:

$H_0:\pi =0.5$

That is, the proportion of successes is 0.5.

Alternative hypothesis:

$H_1:\pi >0.5$

That is, the proportion of successes is greater than 0.5.

For sign test, the test statistic follows binomial distribution. In this context, It is appropriate to use binomial distribution. Because, the sign test satisfies all the binomial assumptions.

Here, the test is two-tailed test of hypothesis.

b.

To determine

State the decision rule.

Explanation of Solution

However, the test is two-tailed. The significance level for one-tail of the curve is 0.05.

For the significance level of 0.05, the cumulative probability should fall nearest to the level of significance, but it should not be greater than the level of significance.

In Appendix A, Table B.1 Binomial Probability Distribution, at 0.05, n=12, the cumulative probability for the observation ≤ 2  is nearest to 0.05 (=0.000+0.003+0.016) and the sum of cumulative probability for the observation ≥ 10 is next nearest to 0.05(=0.016+0.003+0.000).

Decision rule:

• If the number of successes in the samples is ≤ 2  or ≥ 10, than reject $H_0$.
• Otherwise, fail to reject $H_0$.

c.

To determine

State the decision that has been made regarding the null hypothesis.

Answer to Problem 1SR

The null hypothesis is rejected.

Explanation of Solution

For sign test, the test statistic follows binomial distribution. In this context, It is appropriate to use binomial distribution. Because, the sign test satisfies all the binomial assumptions.

It is given that the number of customers who preferred coffee is 2 (number of success).

Conclusion:

Here, the number of success is equal to  2.

By the rejection rule, reject the null hypothesis.

Thus, there is a preference for a decaffeinated coffee.