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The assignment involves analyzing probability distributions related to job satisfaction scores for two groups: senior executives and middle managers. The task requires calculating the expected value, variance, and standard deviation for both groups based on their probability distributions. Additionally, the assignment compares overall job satisfaction levels between the two groups. Furthermore, the analysis extends to the number of persons living in rent-controlled and rent-stabilized housing units, where the expected value, variance, and comparative analysis are required based on the given probability distributions.

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Understanding the dynamics of job satisfaction among different managerial levels has been a significant focus in organizational behavior research. Quantitative methods, particularly probability analysis, allow researchers and practitioners to gauge overall satisfaction levels and their variability. Similarly, analyzing demographic distributions within housing data provides insights into social and economic patterns affecting urban populations.

In this analysis, we first consider the probability distributions of job satisfaction scores among senior executives and middle managers. These scores, ranging from 1 (very dissatisfied) to 5 (very satisfied), are examined to compute several statistical measures. The expected value, also known as the mean, provides the central tendency of satisfaction levels for each group, offering a baseline for comparison. Variance and standard deviation, on the other hand, measure the dispersion or spread of the satisfaction scores, indicating how varied individual responses are within each group.

For senior executives, the probability distribution is given as follows:

  • Score 1: 0.05
  • Score 2: 0.42
  • Score 3: 0.28
  • Score 4: 0.25
  • Score 5: 0.00

Similarly, for middle managers:

  • Score 1: 0.09
  • Score 2: 0.41
  • Score 3: 0.28
  • Score 4: 0.22
  • Score 5: 0.00

Calculating the expected value for each group involves multiplying each score by its probability and summing these products:

Expected value for senior executives (E[X]):

E[X] = (1)(0.05) + (2)(0.42) + (3)(0.28) + (4)(0.25) + (5)(0.00) = 0.05 + 0.84 + 0.84 + 1.00 + 0 = 2.73

Expected value for middle managers (E[X]):

E[X] = (1)(0.09) + (2)(0.41) + (3)(0.28) + (4)(0.22) + (5)(0.00) = 0.09 + 0.82 + 0.84 + 0.88 + 0 = 2.63

These calculations suggest that senior executives have a marginally higher average job satisfaction score compared to middle managers.

To compute the variance, the squared difference of each score from the expected value is multiplied by its probability, and summed:

Variance for senior executives:

Var[X] = Σ (x - μ)^2 * P(x)

Calculating:

  • (1 - 2.73)^2 0.05 = (−1.73)^2 0.05 ≈ 2.99 * 0.05 ≈ 0.149
  • (2 - 2.73)^2 0.42 = (−0.73)^2 0.42 ≈ 0.533 * 0.42 ≈ 0.224
  • (3 - 2.73)^2 0.28 = (0.27)^2 0.28 ≈ 0.073 * 0.28 ≈ 0.020
  • (4 - 2.73)^2 0.25 = (1.27)^2 0.25 ≈ 1.613 * 0.25 ≈ 0.403
  • (5 - 2.73)^2 0.00 = (2.27)^2 0 = 0

Summing these gives: 0.149 + 0.224 + 0.020 + 0.403 + 0 = 0.796

Thus, the variance for senior executives is approximately 0.796.

Variance for middle managers:

  • (1 - 2.63)^2 0.09 ≈ 2.66 0.09 ≈ 0.239
  • (2 - 2.63)^2 0.41 ≈ 0.40 0.41 ≈ 0.164
  • (3 - 2.63)^2 0.28 ≈ 0.13 0.28 ≈ 0.036
  • (4 - 2.63)^2 0.22 ≈ 1.88 0.22 ≈ 0.414
  • (5 - 2.63)^2 * 0.00 = 0

Sum: 0.239 + 0.164 + 0.036 + 0.414 + 0 = 0.853

Therefore, middle managers exhibit a slightly higher variance, indicating marginally greater variability in their job satisfaction scores.

The standard deviation, being the square root of variance, quantifies the average deviation from the mean:

Standard deviation for senior executives: √0.796 ≈ 0.89

Standard deviation for middle managers: √0.853 ≈ 0.92

These measures demonstrate that both groups have comparable variability in job satisfaction, with middle managers showing slightly more dispersion.

The comparison indicates that, on average, senior executives are somewhat more satisfied with their jobs than middle managers, but variability within the groups remains similar. Such insights are valuable for organizational interventions aimed at improving satisfaction and reducing disparities among managerial levels.

Moving to the housing data, the analysis involves the distribution of the number of persons living in rent-controlled and rent-stabilized units. The probability distributions for each type are provided, and similar statistical measures are calculated.

For rent-controlled units:

  • 1 person: 0.61
  • 2 persons: 0.27
  • 3 persons: 0.07
  • 4 persons: 0.04
  • 5 persons: 0.01

For rent-stabilized units:

  • 1 person: 0.42
  • 2 persons: 0.41
  • 3 persons: 0.28
  • 4 persons: 0.22
  • 5 persons: 0.00

Calculating the expected values for each type:

Rent-controlled units:

E[X] = 10.61 + 20.27 + 30.07 + 40.04 + 5*0.01 = 0.61 + 0.54 + 0.21 + 0.16 + 0.05 = 1.56

Rent-stabilized units:

E[X] = 10.42 + 20.41 + 30.28 + 40.22 + 5*0.00 = 0.42 + 0.82 + 0.84 + 0.88 + 0 = 2.96

The expected number of persons per unit indicates higher occupancy levels in rent-stabilized units compared to rent-controlled units.

Variance calculations follow similar steps:

Rent-controlled units:

  • (1 - 1.56)^2 0.61 ≈ 0.313 0.61 ≈ 0.191
  • (2 - 1.56)^2 0.27 ≈ 0.19 0.27 ≈ 0.051
  • (3 - 1.56)^2 0.07 ≈ 2.07 0.07 ≈ 0.145
  • (4 - 1.56)^2 0.04 ≈ 5.94 0.04 ≈ 0.238
  • (5 - 1.56)^2 0.01 ≈ 11.99 0.01 ≈ 0.120

Sum: 0.191 + 0.051 + 0.145 + 0.238 + 0.120 ≈ 0.745

Variance of rent-controlled units: approximately 0.745.

Rent-stabilized units:

  • (1 - 2.96)^2 0.42 ≈ 3.84 0.42 ≈ 1.61
  • (2 - 2.96)^2 0.41 ≈ 0.92 0.41 ≈ 0.38
  • (3 - 2.96)^2 0.28 ≈ 0.0016 0.28 ≈ 0.00045
  • (4 - 2.96)^2 0.22 ≈ 1.08 0.22 ≈ 0.24
  • (5 - 2.96)^2 * 0.00 = 0

Sum: 1.61 + 0.38 + 0.00045 + 0.24 + 0 ≈ 2.23

Variance of rent-stabilized units: approximately 2.23.

The higher variance in rent-stabilized units indicates greater variability in household sizes, aligning with the higher average number of persons. The comparative analysis highlights that rent-stabilized units tend to host larger and more varied household sizes than rent-controlled units, which could reflect differences in demographic, economic, or policy factors.

In conclusion, both the job satisfaction and housing occupancy analyses demonstrate the application of probability distributions to derive meaningful insights. The computations of expected values, variances, and standard deviations provide quantitative measures that facilitate comparison and understanding of the underlying social and organizational dynamics. Policymakers and organizational leaders can leverage such statistical analyses to inform decisions aimed at improving employee satisfaction and optimizing housing policies.

References

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  • New York City Housing and Vacancy Survey. (2004). NYC Department of Housing Preservation and Development.
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  • Hogg, R. V., McKean, J., & Craig, A. T. (2019). Introduction to Mathematical Statistics (8th ed.). Pearson.
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