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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.

What is the value of Tws

1 answer

To conduct the Wilcoxon signed-rank test with the provided data, we first need to calculate the differences between the "After" and "Before" scores for each worker, then compute the ranks of these differences (ignoring signs), and finally sum the ranks for the positive and negative differences.

Step 1: Calculate the differences

Let's set up the data:

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

Now we calculate the differences (After - Before):

  1. \( 5 - 7 = -2 \)
  2. \( 4 - 6 = -2 \)
  3. \( 9 - 5 = 4 \)
  4. \( 7 - 8 = -1 \)
  5. \( 8 - 6 = 2 \)
  6. \( 9 - 9 = 0 \)
  7. \( 11 - 7 = 4 \)
  8. \( 8 - 5 = 3 \)

So the differences are:

  • \( D = [-2, -2, 4, -1, 2, 0, 4, 3] \)

Step 2: Rank the absolute differences

Next, we will ignore the sign and rank the absolute differences:

  1. \( |-2| = 2 \)
  2. \( |-2| = 2 \)
  3. \( |4| = 4 \)
  4. \( |-1| = 1 \)
  5. \( |2| = 2 \)
  6. \( |0| = 0 \)
  7. \( |4| = 4 \)
  8. \( |3| = 3 \)

Now we rank their absolute values:

  • Unique absolute differences: \( [0, 1, 2, 2, 2, 3, 4, 4] \)

Ranks:

  • \( 0 \) receives rank \( 1 \)
  • \( 1 \) receives rank \( 2 \)
  • \( 2 \) receives ranks \( 3, 4, 5 \) (assign average rank, \( (3+4+5)/3 = 4 \))
  • \( 3 \) receives rank \( 6 \)
  • \( 4 \) receives ranks \( 7, 8 \) (assign average rank, \( (7+8)/2 = 7.5 \))

Now, we can associate ranks back with the original differences:

  • For \( -2 \), we assign rank \( 4 \) (twice for the two \( -2 \) entries).
  • For \( -1 \), the rank is \( 2 \).
  • For \( 2 \), the rank is \( 4 \).
  • For \( 4 \), we assign rank \( 7.5 \) (twice for the two \( 4 \) entries).
  • For \( 3 \), the rank is \( 6 \).

So we summarize the signed ranks:

| Difference | Rank | | --- | --- | | -2 | \( 4 \) | | -2 | \( 4 \) | | 4 | \( 7.5 \) | | -1 | \( 2 \) | | 2 | \( 4 \) | | 0 | \( 0 \) | | 4 | \( 7.5 \) | | 3 | \( 6 \) |

Calculation of Signed Ranks

Now, we sum the ranks of the positive and negative differences:

  • Negative ranks: \( 4 + 4 + 2 = 10 \)
  • Positive ranks: \( 7.5 + 6 + 4 = 17.5 \)

The test statistic \( T_w \) is then the smaller of the sums of the ranks:

  • \( T_w = 10 \) (since \( T_w \) considers negative and ignoring zero).

Result

Thus, the value of \( T_w \) (the Wilcoxon signed-rank test statistic) is \( 10 \).

Note: Remember to interpret this result against critical values for the relevant \( n \) (in this case, n is the count of non-zero differences, specifically 7 since there are 8 observations but one had a zero difference and doesn’t contribute to ranks).

For a formal conclusion, you would compare \( T_w \) to the critical value for \( n=7 \) at \( \alpha=0.025 \) to determine if there's significant evidence to reject the null hypothesis (no difference in morale). However, for this answer, we have calculated \( T_w \) as requested.