Let's go through the problem step by step.

(a) Hypothesis to be Tested

To perform the sign test, the null and alternative hypotheses are:

• Null Hypothesis ($H_0$): The median of the differences in job satisfaction scores after promotion minus before promotion is zero. This implies that there is no change in job satisfaction scores after the promotion.

  $H_0: \text{The median of differences} = 0$

• Alternative Hypothesis ($H_1$): The median of the differences in job satisfaction scores after promotion minus before promotion is not zero. This implies that there is a change in job satisfaction scores after the promotion.

  $H_1: \text{The median of differences} \neq 0$

(b) Number of Values Above and Below the Hypothesized Value, and Test Statistic

The sign test compares the number of positive and negative differences from the hypothesized median (which is 0). We only consider non-zero differences.

Given differences:

• Negative Differences ($d < 0$):

  • Employees: 1, 3, 8, 16, 17, 18
  • Differences: $-2, -13, -15, -2, -13, -7$
  • Count: 6

• Positive Differences ($d > 0$):

  • Employees: 2, 4, 6, 7, 10, 11, 12, 13, 14
  • Differences: $3, 7, 7, 7, 6, 14, 1, 14, 7$
  • Count: 9

• Zero Differences ($d = 0$):

  • Employees: 5, 9, 15
  • Differences: $0, 0, 0$
  • Count: 3 (these do not count towards the test statistic)

The sign test statistic is the smaller of the number of positive or negative differences.

• Test Statistic: The value of the test statistic is $\min(9, 6) = 6$.

Summary:

• Number of Positive Differences: 9
• Number of Negative Differences: 6
• Test Statistic: 6

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Would you like to explore more about how this result might be interpreted or the next steps in the analysis? Here are some related questions:

1. How do you determine the critical value for the sign test?
2. What would be the p-value associated with this test statistic?
3. How does the sign test differ from the Wilcoxon signed-rank test?
4. What conclusions can be drawn if the p-value is significant?
5. How would the results change if there were more tied (zero) values?

Tip: When dealing with small samples or non-parametric data, the sign test is a straightforward method to check for differences without needing to assume normality.