Chapter 15, Problem 62SE

To determine

Check whether the data provide sufficient evidence to indicate that one of gourmets tends to give higher ratings than the other.

Answer to Problem 62SE

The conclusion is that the data provide sufficient evidence to indicate that one of gourmets tends to give higher ratings than the other.

Explanation of Solution

The hypotheses are given below:

Null hypothesis:

$H_0:p=0.5$

Alternative hypothesis:

$H_a:p\ne 0.5$

Consider, the p-value is the probability of gourmet A’s rating exceeds gourmet B’s rating for a given meal.

From the given table, the number of gourmet A’s rating exceeds gourmet B’s rating for a given meal is 4. Hence, the test statistic (M) is 4.

The p-value is obtained below:

$\begin{array}{c}p\text{-value}=2P\left(M\le 4\right)\\ =2\left[\left(\begin{array}{l}17\\ 0\end{array}\right){0.5}^0{\left(1-0.5\right)}^{17-0}+\cdot \cdot \cdot +\left(\begin{array}{l}17\\ 4\end{array}\right){0.5}^4{\left(1-0.5\right)}^{17-4}\right]\\ =2\left(0.02452\right)\\ =0.04904\end{array}$

Thus, the p-value is 0.04904.

Decision rule:

If $p\text{-value}\le \alpha$, then reject the null hypothesis.

Conclusion:

Here, the p-value is almost equal to the significance level 0.05.

From the rejection rule, the null hypothesis is rejected.

Hence, it can be concluded that the data provide sufficient evidence to indicate that one of gourmets tends to give higher ratings than the other.