Examine the distribution of salaries for only faculty members. Because upper-level administrators may have been included in this sample, Dr. Smith also wants analyses of annual salaries for only faculty members in the sample. She is particularly interested in the probability that faculty members have salaries in various ranges. Prior to your involvement in the project, another analyst found the following descriptive statistics for this subgroup: mean = 41.579, standard deviation = 7.39, and n = 316. Your predecessor estimated that approximately 26.8% of sampled faculty members had salaries below $37,000 (based on a Z-score of -.619). She also estimated that a third of faculty members had salaries below $38,327 and a third had salaries above $44,831. (You are encouraged to test your ability to use the Z-score formulas by obtaining these numbers yourself.) Now Dr. Smith wants estimates for different benchmarks: (a) the percentage of faculty members had salaries above $52,000 and (b) the top and bottom 20% of salaries. Be sure to show your work for any CALCULATIONS to at least the third decimal point. Part A: Obtain the following output and CALCULATE the following numbers (8 points): CALCULATE the appropriate Z-score using at least 3 decimal points for estimating the probability that a faculty member has a salary over $52,000. Find the appropriate probability using a normal probability table (e.g., Levine Table E.2). Use a normal probability table to find the appropriate Z-scores for the bottom and top 20% of salaries for faculty members. CALCULATE the cut-point for the upper boundary of the bottom 20% and the bottom boundary of the top 20%. Part B: Based on Part A results, answer the following questions (10 points): What does the Z-score from A.II above indicate about how many standard deviations a salary of $52,000 is above or below the mean salary? What is the probability that a faculty member had a salary less than $52,000? Briefly explain. If the districts specify no minimum salary, what is the lowest possible salary if another sample of faculty members were drawn? If you wanted to make similar estimates about the distribution of years of service, what Z-scores would be used estimate the upper and lower 20% of years worked in VSCCS? Briefly explain. Part C: (8 points) Write a brief (1-2 paragraphs) description of conclusions that might be drawn from analyses in Part A.I-.II above concerning the distributions of salaries among this sample of VSCCS faculty members. Make this text suitable for inclusion in the first interim report to Dr. Smith by describing the analysis in grammatically correct everyday language, include substantive interpretations of appropriate statistics (also giving the numbers in the text) obtained for this question, and discuss the basis of your judgements concerning the importance of the results for the union negotiations.
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