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Statistical Analysis of Gender Differences in Employment Competency Scores

A solved assignment analyzing employment competency scores by gender using statistical methods.

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    1 Statistical Analysis of Gender Differences in Employment Competency Scores Question 1 Data on 200 men and 200 women were obtained from representative random samples of young men and women who applied for employment at a large U.S. corporation in 2012. During the application process, each applicant was given a Reading comprehension test, a Mathematics reasoning test, along with several other exercises which were used to generate an overall competency score. A review committee used these scores along with other factors when considering the applicants for employment. All of the applicants were between the ages of 18 and 25. The data file is posted as exam1_employment_competency.csv attached. You may read the file into JMP, or any other software package of your choice, to answer the following questions. The data file contains information on the following five variables  Subject : subject identification number  Gender : coded 1 for females and 2 for males.  Reading: score on a reading comprehension test  Mathematics: score on a mathematics reasoning test  Competency: an overall assessment of employment competency obtained from a complex combination of the scores on the reading and mathematics tests and the scores on several other tests of reasoning, communication, organization, and social skills. There is one line of data for each of the 400 individuals in the sample. To help you check if you correctly entered the data into JMP, the first five lines of the data file are shown below. Subject Gender Reading Mathematics Competency 1 1 11 20 60.4 2 1 13 21 84.7 3 1 11 12 44.5 4 1 6 4 4.0 5 1 6 7 11.8 Use these data to answer the following questions. (a) Do these data provide evidence that women perform better at Reading tasks than men ? Set up appropriate null and alternate hypotheses to answer this question, report a formula and a value for your test statistic, and clearly indicate how you reached your conclusion. Use an α = .05 significance level, and state your conclusion in the context of this study. Here we want to test that women perform better at Reading tasks than men thus the null and alternative hypotheses are, 𝐻 0 : μ ௐ − μ ெ ≤ 0 𝑎𝑛𝑑 𝐻 𝑎 : μ ௐ − μ ெ > 0 It can be seen that, this is a two sample test for mean, the test is one tailed (right tailed). So here the t - test for two sample mean would be the most appropriate test.

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    2 The test statistic can be calculated using the formula, T = ௫ ̅ 1 − ௫ ̅ మ ඨ ೄ 1 మ 𝑛 1 ା ೄ మ మ 𝑛 మ Using Minitab the obtained output is given below, Two - Sample T - Test and CI: Reading, Gender Two - sample T for Reading Gender N Mean StDev SE Mean 1 200 10.63 3.47 0.25 2 200 9.03 3.83 0.27 Difference = μ (1) - μ (2) Estimate for difference: 1.600 95% lower bound for difference: 0.997 T - Test of difference = 0 (vs >): T - Value = 4.38 P - Value = 0.000 DF = 394 The above output indicates that the test statistic value is 4.38 with associated p - value 0.000. As the p - value is smaller than the significance level of 0.05 thus we are rejecting the null hypothesis indicating that, the data gives enough evidence that women perform better at Reading tasks than men . (b) Construct a 99 percent confidence interval for the difference between mean Mathematics performance for women and mean Mathematics performance for men . Interpret your confidence interval in the context of the employment competence tests . Using an α = .01 significance level, would you conclude that there is a significant difference between mean Mathematics performance for women and mean Mathematics performance for men ? Justify you answer. The obtained MINITAB output is given below, Two - Sample T - Test and CI: Mathematics, Gender Two - sample T for Mathematics Gender N Mean StDev SE Mean 1 200 12.00 5.91 0.42 2 200 10.99 6.18 0.44 Difference = μ (1) - μ (2) Estimate for difference: 1.005 99% CI for difference: ( - 0.560, 2.570) T - Test of difference = 0 (vs ≠ ): T - Value = 1.66 P - Value = 0.097 DF = 397

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    3 The output gives us the 99% confidence interval for the difference between mean Mathematics performance for women and mean Mathematics performance for men as ( - 0.560, 2.570) . The interval indicates that, we are 95% confident that the population difference between mean Mathematics performance for women and mean Mathematics performance for men falls in the interval ( - 0.560, 2.570) . As the confidence interval contains the value 0 so at 0.01 significance level we can conclude that there is no significant difference between the mean Mathematics performance between the two genders. (c) Do these data provide evidence that men perform better on Mathematics related tasks than they perform on Reading related tasks? Use a significance level of α = .05, and justify your answer. Here we want to test that men perform better on Mathematics related tasks than they perform on Reading related tasks thus the null and alternative hypotheses are (here the sample is dependent, d = mathematics score performance – reading performance) , 𝐻 0 : μ 𝑑 ≤ 0 𝑎𝑛𝑑 𝐻 𝑎 : μ 𝑑 > 0 It can be seen that, this is a dependent sample test for mean, the test is one tailed (right tailed). So here the paired t - test would be the most appropriate test. The test statistic can be calculated using the formula, T = ௫ ̅ 𝑑 ೄ 𝑑 √ 𝑛 Using Minitab the obtained output is given below, Paired T - Test and CI: Mathematics, Reading Paired T for Mathematics - Reading N Mean StDev SE Mean Mathematics 200 10.995 6.182 0.437 Reading 200 9.030 3.830 0.271 Difference 200 1.965 4.641 0.328 95% lower bound for mean difference: 1.423 T - Test of mean difference = 0 (vs > 0): T - Value = 5.99 P - Value = 0.000 The above output indicates that the test statistic value is 5.99 with associated p - value 0.000. As the p - value is smaller than the significance level of 0.05 thus we are rejecting the null hypothesis indicating that, the data gives enough evidence that men perform better on Mathematics related tasks than they perform on Reading related tasks .

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