Question

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 a Sign test to evaluate if the confidence-building seminar was effective in raising the self-confidence of employees, we need to follow these steps:

1. **Define the Hypotheses**:
- Null Hypothesis (H0): The seminar did not change self-confidence levels (the number of increases in self-confidence is equal to the number of decreases).
- Alternative Hypothesis (H1): The seminar raised self-confidence levels (the number of increases in self-confidence is greater than the number of decreases).

2. **Categorize the Differences**:
We will categorize the employees based on their confidence level before and after the seminar. We will then note whether there was an increase, decrease, or no change in self-confidence.

3. **Data Analysis**:
Let's summarize the outcomes for each employee based on the "Before" and "After" confidence levels:

- **Increased Confidence** (to higher level after seminar):
- A (Low to High)
- B (High to Very high)
- D (Very low to Very high)
- E (Low to High)
- F (High to Very high)
- H (High to Very high)
- K (High to Very high)
- M (Low to Very high)
- N (Very low to High)
- O (Very low to Very high)
- P (Low to High)

- **Decreased Confidence** (to lower level after seminar):
- C (High to Low)
- G (Very low to Low)
- J (Low to Very low)

- **No Change**:
- I (Very high to High)
- L (High to High)

By examining the data:

- Increases: A, B, D, E, F, H, K, M, N, O, P (Total = 11)
- Decreases: C, G, J (Total = 3)
- No Change: I, L (Total = 2)

4. **Sign Test Calculation**:
The sign test focuses only on the number of positive and negative changes. We can disregard the cases where there was no change. In our case:

- Positive Signs (increased confidence): 11
- Negative Signs (decreased confidence): 3

Total changes (excluding no change) = 11 + 3 = 14.

5. **Determine Critical Value**:
The Sign Test follows a binomial distribution. At a significance level of 5% (α = 0.05), with n = 14 (the total number of changes), we can find the critical value.

Using a binomial table or calculator:
- We perform a one-tailed test (since we are only interested in increases).
- We calculate the critical value of number of decreases (the more negative signs we can tolerate) for n = 14 at α = 0.05, which gives us a critical value of 4. So we reject H0 if the number of decreases (negative signs) is less than 4.

6. **Decision**:
- We observed 3 decreases.
- Since 3 (number of decreases) < 4 (critical value), we reject the null hypothesis.

7. **Conclusion**:
Based on the Sign test, we conclude that there is statistically significant evidence at the 5% level to suggest that the confidence-building seminar was effective in raising the self-confidence of the employees.