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:

Question

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.