Question

5. A lack of self – confidence often influences the quality of work of employees who need to work with the public. A group of employees were sent to a confidence-building seminar. Employees were asked to classify their confidence before and after the seminar as extremely high, high, low and extremely low. The before and after levels of self-confidence were measured for the employees and are shown in the table below:
Employee Before After Difference Employee Before After Difference
A Low High I Very high High
B High Very high J Low Very low
C High Low K High Very high
D Very low Very high L High High
E Low High M Low Very high
F High Very high N Very low High
G Very low Low O Very low Very high
H High Very high P Low High

Use Sign test to see if the confidence-building seminar was effective in raising the self-confidence of the employees. (Use the 5% significant level.)

Answer 1

To perform the sign test, we will assess whether the confidence-building seminar led to a significant increase in self-confidence by comparing the "Before" and "After" classifications for each employee.

### Steps to perform the Sign Test:

1. **Assign a rank to each level of confidence** (for simplicity, we can use the following numeric assignments):
- Extremely low: 1
- Low: 2
- High: 3
- Extremely high: 4

Based on this scale, the rank can be assigned as follows:
- Extremely low = 1
- Low = 2
- High = 3
- Extremely high = 4

2. **Calculate the differences** between "Before" and "After" measures for each employee and count the number of positive, negative, and ties:
- **Count the number of increases (positive signs)**
- **Count the number of decreases (negative signs)**
- **Count the number of ties (no change)**

### Data Analysis

Now, let's list out the "Before" and "After" confidence levels, assign numeric values, and find the differences:

| Employee | Before (Value) | After (Value) | Difference (After - Before) |
|----------|----------------|---------------|------------------------------|
| A | Low (2) | High (3) | 3 - 2 = 1 (Increase) |
| B | High (3) | Very high (4) | 4 - 3 = 1 (Increase) |
| C | High (3) | Low (2) | 2 - 3 = -1 (Decrease) |
| D | Very low (1) | Very high (4) | 4 - 1 = 3 (Increase) |
| E | Low (2) | High (3) | 3 - 2 = 1 (Increase) |
| F | High (3) | Very high (4) | 4 - 3 = 1 (Increase) |
| G | Very low (1) | Low (2) | 2 - 1 = 1 (Increase) |
| H | High (3) | Very high (4) | 4 - 3 = 1 (Increase) |
| I | Very high (4) | High (3) | 3 - 4 = -1 (Decrease) |
| J | Low (2) | Very low (1) | 1 - 2 = -1 (Decrease) |
| K | High (3) | Very high (4) | 4 - 3 = 1 (Increase) |
| L | High (3) | High (3) | 3 - 3 = 0 (Tie) |
| M | Low (2) | Very high (4) | 4 - 2 = 2 (Increase) |
| N | Very low (1) | High (3) | 3 - 1 = 2 (Increase) |
| O | Very low (1) | Very high (4) | 4 - 1 = 3 (Increase) |
| P | Low (2) | High (3) | 3 - 2 = 1 (Increase) |

### Counting Signs:
- **Increases (Positive)**: A, B, D, E, F, G, H, K, M, N, O, P (Total = 12)
- **Decreases (Negative)**: C, I, J (Total = 3)
- **Ties**: L (Total = 1)

### Total Count:
- Total = Increases + Decreases = 12 + 3 = 15 (ties excluded)

### Total Positive Signs = 12
### Total Negative Signs = 3

### Test Statistic:
The smaller of the two counts (positive signs or negative signs) will be our test statistic:
- Test Statistic = 3 (the number of decreases)

### Critical Value:
Since we are using a two-tailed test with α = 0.05 and n = 15 (15 non-tied observations):
- The critical value from the sign test table for n=15 at α = 0.05 is 2.

### Conclusion:
- Since the test statistic (3) is greater than the critical value (2), we do not reject the null hypothesis.

### Interpretation:
We conclude that there is not enough statistical evidence at the 5% significance level to suggest that the confidence-building seminar was effective in raising the self-confidence of the employees.