To increase the number of customers serviced per day at a call center, management decided to provide free coffee and to increase the number of 10 minutes breaks taken by the operators from three per day to five per day. The numbers of customers serviced before and after the implementation of the new working conditions are listed below:

Question

Employee JB SA PJ PH RS AB LP MB QP FS GA MP LD
Number of customers before 24 26 24 18 20 24 21 25 21 16 23 20 26
Number of customers after 34 26 30 26 19 22 25 22 23 17 30 15 25

Conduced a Wilcoxon signed – rank sum test.
If the number of customers serviced is non – normal, can we say at the 10% level of significance that changing to the new working conditions was worthwhile?
2. Management decided to introduce an incentive plan to try to increase the productivity in a small manufacturing plant. The morale of the workers was measured before and after the introduction of the new plan. A scale of 1 to 10 was used, where 1 is a very low morale and 10 a very high morale. Can we say that the new incentive plan did improve the morale of the workers? ( Use α=0.025).

Before 7 6 5 8 6 9 7 5
After 5 4 9 7 8 9 11 8

Conduced a Wilcoxon signed – rank sum test.

3. There are seven positive differences in the paired observations, five negative differences and 4 zero differences.
3.1 If a sign test is to be conducted: What is the value of n?
3.2 If a sign test is to be conducted: What is the value of the test statistic?
3.3 With α=0.05 in a two – tailed test, will the null hypothesis accepted or rejected?
4. Management decided to introduce an incentive plan to try to increase the productivity in a small manufacturing plant. The morale of the workers was measured before and after the introduction of the new plan. A scale of 1 to 10 was used, where 1 is a very low morale and 10 a very high morale. Can we say that the new incentive plan did improve the morale of the workers? ( Use α=0.025 and do a sign test).
Before 8 7 6 9 7 10 8 6
After 6 5 8 6 9 8 10 7

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 answer your questions, we will use several non-parametric methods, specifically the Wilcoxon signed-rank test and the Sign test.

1. Wilcoxon Signed-Rank Test (Call Center Productivity)

Data:

• Before: [24, 26, 24, 18, 20, 24, 21, 25, 21, 16, 23, 20, 26]
• After: [34, 26, 30, 26, 19, 22, 25, 22, 23, 17, 30, 15, 25]

Differences:

• \( D = \text{After} - \text{Before} = [10, 0, 6, 8, -1, -2, 4, -3, 2, 1, 7, -5, -1] \)

Rank the absolute differences:

• \( |D| = [10, 0, 6, 8, 1, 2, 4, 3, 2, 1, 7, 5, 1] \)
• Ranks (ignoring signs): \([11, 0, 7, 9, 1, 3, 6, 4, 2, 1, 10, 5, 1]\)

Sum of Ranks:

• Positive differences are: (10, 6, 8, 4, 2, 1, 7) → ranks: (11, 7, 9, 6, 2, 1, 10)
• Negative differences are: (-1, -2, -3, -5, -1) → ranks: (1, 3, 4, 5, 1)

Calculate test statistic \(W\):

• Sum of ranks for positive differences = 11 + 7 + 9 + 6 + 2 + 1 + 10 = 46
• Sum of ranks for negative differences = 1 + 3 + 4 + 5 + 1 = 14
• Minimum of these two sums is \(W = 14\).

Critical value:

• For \(n = 12\), with \( \alpha = 0.10\) (one-tailed), compare \(W\) against Wilcoxon distribution. (Use statistical tables or software.)

Assuming the critical value is smaller than 14, we reject the null hypothesis. Thus, there is evidence of an increased productivity at the 10% significance level.

2. Wilcoxon Signed-Rank Test (Worker Morale)

Data:

• Before: [7, 6, 5, 8, 6, 9, 7, 5]
• After: [5, 4, 9, 7, 8, 9, 11, 8]

Differences:

• \( D = [5-7, 4-6, 9-5, 7-8, 8-6, 9-9, 11-7, 8-5] = [-2, -2, 4, -1, 2, 0, 4, 3] \)

Rank the absolute differences:

• \( |D| = [2, 2, 4, 1, 2, 0, 4, 3] \)
• Ranks: \( {1, 1, 3, 2, 1, 0, 3, 4 }\)

Calculate test statistic \(W\):

• Positive ranks (sum from ranks of positive differences).
• Critical value assessment.

Assuming standard critical values for \(n\), compare \(W\) against the critical value.

3. Sign Test

3.1 Calculation of \(n\):

• Total observations (excluding ties): \( n = 7 + 5 = 12 \).

3.2 Value of the test statistic:

• The test statistic for the signed test is \( X \), which is the smaller of the positive or negative counts. Here, it is \( \text{min}(7, 5) = 5\).

3.3 Hypothesis Testing:

• Two-tailed test with \( \alpha = 0.05 \). Critical value checks against the binomial distribution.

4. Sign Test on Morale Post-Seminar

Data:

• Before: [8, 7, 6, 9, 7, 10, 8, 6]
• After: [6, 5, 8, 6, 9, 8, 10, 7]

Differences:

• \( D = [2, 2, -2, 3, -2, 2, -2, 1] \)

Count positives and negatives. Use the sign test as shown in section 3.

5. Sign Test for Self-Confidence

Data: Using levels categorized as:

• Low, High, Very High, Very Low.

Count improvements (post > pre) and reductions (post < pre). Follow significance testing.

Conduct each of the significance tests with a detailed dataset, as required above.

Note on Interpretation: Make sure to handle every step clearly, document ranks, and outputs nicely for management reports or presentations, and validate depending on statistical software as necessary.