Answer:

1. (a) A Wilcoxon signed-rank test may not be appropriate because the salary data are highly skewed and contain an extreme outlier, so the distribution of salaries is unlikely to be symmetric.

2. (b) There are 4 salaries above \$32500 and 11 below. For a two-sided sign test, the two-sided p-value is \[ 2P(X \le 4)=2(0.0592)=0.1184. \] Since \[ 0.1184 > 0.10, \] there is insufficient evidence to reject the company's claim that the median salary is \$32500.

Solution:

1. (a) The Wilcoxon signed-rank test uses the sizes of the positive and negative differences from the hypothesised median, and it is suitable when the underlying distribution is reasonably symmetric. Here, the salaries include a very large value, \$125000, compared with the rest of the sample. This suggests the salary distribution is skewed and contains an outlier, so the symmetry assumption is doubtful.

2. (b) Let \(m\) be the population median salary.

   The hypotheses are

   \[ H_0: m = 32500, \qquad H_1: m \ne 32500. \]

   Compare each salary with \$32500. The salaries above \$32500 are \$125000, \$42500, \$52500 and \$33000, so there are 4 salaries above \$32500. The remaining 11 salaries are below \$32500, and there are no ties.

   Under \(H_0\), each salary is equally likely to be above or below the median, so if \(X\) is the number of salaries above \$32500, then

   \[ X \sim \mathrm{B}(15,0.5). \]

   The observed value is \(X=4\). Since the test is two-sided, use the probability of being at least as extreme in either tail:

   \[ P(X \le 4) = \sum_{r=0}^{4} \binom{15}{r}(0.5)^{15} = 0.0592. \]

   So the two-sided p-value is

   \[ 2 \times 0.0592 = 0.1184. \]

   Since

   \[ 0.1184 > 0.10, \]

   the result is not significant at the 10% level. Therefore, there is insufficient evidence to reject the company's claim that the median salary is \$32500.