ID Salary Compa Midpoint Age Performance Service Gender Rating Raise Degree Gender 1 Gr Students: Copy the Student Data file data values into this sheet to assist in doing your weekly assignments. The ongoing question that the weekly assignments will focus on is: Are males and fe Note: to simplfy the analysis, we will assume that jobs within each grade comprise eq The column labels in the table mean: ID - Employee sample number Salary - Salary in thousands Age - Age in years Performance Rating - Appraisal rating (e Service - Years of service (rounded) Gender - 0 = male, 1 = female Midpoint - salary grade midpoint Raise - percent of last raise Grade - job/pay grade Degree (0= BS\\BA 1 = MS) Gender1 (Male or Female) Compa - salary divided by midpoint emales paid the same for equal work (under the Equal Pay Act)? qual work. employee evaluation score) Week 1. Measurement and Description - chapters 1 and 2 1 The goal this week is to gain an understanding of our data set - what kind of data we are looking at, some descriptive measurse, and a look at how the data is distributed (shape). Measurement issues. Data, even numerically coded variables, can be one of 4 levels nominal, ordinal, interval, or ratio. It is important to identify which level a variable is, as this impact the kind of analysis we can do with the data. For example, descriptive statistics such as means can only be done on interval or ratio level data. Please list under each label, the variables in our data set that belong in each group. Nominal Ordinal Interval Ratio b. For each variable that you did not call ratio, why did you make that decision? 2 The first step in analyzing data sets is to find some summary descriptive statistics for key variables. For salary, compa, age, performance rating, and service; find the mean, standard deviation, and range for 3 groups: overall sample, Females, and Males. You can use either the Data Analysis Descriptive Statistics tool or the Fx =average and =stdev functions. (the range must be found using the difference between the =max and =min functions with Fx) functions. Note: Place data to the right, if you use Descriptive statistics, place that to the right as well. Some of the values are completed for you - please finish the table. Salary Compa Age Perf. Rat. Service Overall Mean 35.7 85.9 9.0 Standard Deviation 8.2513 11.4147 5.7177 Note - data is a sample from the larger company population Range 30 45 21 Female Mean 32.5 84.2 7.9 Standard Deviation 6.9 13.6 4.9 Range 26.0 45.0 18.0 Male Mean 38.9 87.6 10.0 Standard Deviation 8.4 8.7 6.4 Range 28.0 30.0 21.0 3 What is the probability for a: Probability a. Randomly selected person being a male in grade E? b. Randomly selected male being in grade E? Note part b is the same as given a male, what is probabilty of being in grade E? c. Why are the results different? 4 5. A key issue in comparing data sets is to see if they are distributed/shaped the same. We can do this by looking at some measures of where some selected values are within each data set - that is how many values are above and below a comparable value. For each group (overall, females, and males) find: Overall Female Male A The value that cuts off the top 1/3 salary value in each group "=large" function i The z score for this value within each group? Excel's standize function ii The normal curve probability of exceeding this score: 1-normsdist function iii What is the empirical probability of being at or exceeding this salary value? B The value that cuts off the top 1/3 compa value in each group. i The z score for this value within each group? ii The normal curve probability of exceeding this score: iii What is the empirical probability of being at or exceeding this compa value? C How do you interpret the relationship between the data sets? What do they mean about our equal pay for equal work question? What conclusions can you make about the issue of male and female pay equality? Are all of the results consistent? What is the difference between the sal and compa measures of pay? Conclusions from looking at salary results: Conclusions from looking at compa results: Do both salary measures show the same results? Can we make any conclusions about equal pay for equal work yet? Week 2 1 Testing means - T-tests In questions 2, 3, and 4 be sure to include the null and alternate hypotheses you will be testing. In the first 4 questions use alpha = 0.05 in making your decisions on rejecting or not rejecting the null hypothesis. Below are 2 one-sample t-tests comparing male and female average salaries to the overall sample mean. (Note: a one-sample t-test in Excel can be performed by selecting the 2-sample unequal variance t-test and making the second variable = Ho value - a Note: These values are not the same as the data the assignment uses. The purpose is to analyze the results of t-tests rather than directly answer our eq Based on these results, how do you interpret the results and what do these results suggest about the population means for male and female average sal Males Ho: Mean salary = Ha: Mean salary =/= 45.00 45.00 Females Ho: Mean salary = Ha: Mean salary =/= 45.00 45.00 Note: While the results both below are actually from Excel's t-Test: Two-Sample Assuming Unequal Variances, having no variance in the Ho variable makes the calculations default to the one-sample t-test outcome - we are tricking Excel into doing a one sample Male Ho Female Ho Mean 52 45 Mean 38 45 Variance 316 0 Variance 334.66667 0 Observations 25 25 Observations 25 25 Hypothesized Mean D 0 Hypothesized Mean D 0 df 24 df 24 t Stat 1.9689038266 t Stat -1.913206 P(T<=t) one-tail 0.0303078503 P(T<=t) one-tail 0.0338621 t Critical one-tail 1.7108820799 t Critical one-tail 1.7108821 P(T<=t) two-tail 0.0606157006 P(T<=t) two-tail 0.0677242 t Critical two-tail 2.0638985616 Conclusion: Do not reject Ho; mean equals 45 t Critical two-tail 2.0638986 Conclusion: Do not reject Ho; mean equals 45 Note: the Female results are done for you, please complete the male results. Is this a 1 or 2 tail test? Is this a 1 or 2 tail test? 2 tail - why? - why? Ho contains = P-value is: Is P-value < 0.05 (one tail test) or 0.25 (two tail test)? Why do we not reject the null hypothesis? P-value is: 0.0677242 Is P-value < 0.05 (one tail test) or 0.25 (two tail test)? No Why do we not reject the null hypothesis? P-value greater than (>) rejection alpha Interpretation of test outcomes: 2 Based on our sample data set, perform a 2-sample t-test to see if the population male and female average salaries could be equal to each other. (Since we have not yet covered testing for variance equality, assume the data sets have statistically equal variances.) Ho: Ha: Test to use: P-value is: Is P-value < 0.05 (one tail test) or 0.25 (two tail test)? Reject or do not reject Ho: If the null hypothesis was rejected, calculate the effect size value: If calculated, what is the meaning of effect size measure: Interpretation: Male salary mean = Female salary mean Male salary mean =/= Female salary mean t-Test: Two-Sample Assuming Equal Variances b. Is the one or two sample t-test the proper/correct apporach to comparing salary equality? Why? 3 Based on our sample data set, can the male and female compas in the population be equal to each other? (Another 2-sample t-test.) Again, please assume equal variances for these groups. Ho: Ha: Statistical test to use: What is the p-value: Is P-value < 0.05 (one tail test) or 0.25 (two tail test)? Reject or do not reject Ho: If the null hypothesis was rejected, calculate the effect size value: If calculated, what is the meaning of effect size measure: Interpretation: 4 Since performance is often a factor in pay levels, is the average Performance Rating the same for both genders? NOTE: do NOT assume variances are equal in this situation. Ho: Ha: Test to ust-Test: Two-Sample Assuming Unequal Variances What is the p-value: Is P-value < 0.05 (one tail test) or 0.25 (two tail test)? Do we REJ or Not reject the null? If the null hypothesis was rejected, calculate the effect size value: If calculated, what is the meaning of effect size measure: Interpretation: 5 If the salary and compa mean tests in questions 2 and 3 provide different results about male and female salary equality, which would be more appropriate to use in answering the question about salary equity? Why? What are your conclusions about equal pay at this point? lue - a constant.) our equal pay question. age salaries? sample test for us. Week 3 Paired T-test and ANOVA For this week's work, again be sure to state the null and alternate hypotheses and use alpha = 0.05 for our decision value in the reject or do not reject decision on the null hypothesis. 1 Many companies consider the grade midpoint to be the "market rate" - the salary needed to hire a new employee. Does the company, on average, pay its existing employees at or above the market rate? Use the data columns at the right to set up the paired data set for the analysis. Salary Midpoint Diff Null Hypothesis: Alt. Hypothesis: Statistical test to use: What is the p-value: Is P-value < 0.05 (one tail test) or 0.25 (two tail test)? What else needs to be checked on a 1-tail test in order to reject the null? Do we REJ orrejected, what is the If the null hypothesis was Not reject the null? effect effect size meaning of size value: measure: Interpretation of test results: Let's look at some other factors that might influence pay - education(degree) and performance ratings. 2 Last week, we found that average performance ratings do not differ between males and females in the population. Now we need to see if they differ among the grades. Is the average performace rating the same for all grades? (Assume variances are equal across the grades for this ANOVA.) Here are the data values sorted by grade level. The rating values sorted by grade have been placed in columns I - N for you. A B C D E 90 80 100 90 85 Null Hypothesis: Ho: means equal for all grades 80 75 100 65 100 Alt. Hypothesis: Ha: at least one mean is unequal 100 80 90 75 95 Place B17 in Outcome range box. 90 70 80 90 55 80 95 80 95 90 85 80 95 65 90 90 70 75 95 95 60 90 90 95 75 80 95 90 100 Interpretation of test results: What is the p-value: Is P-value < 0.05? Do we REJ or Not reject the null? 0.57 If the ANVOA was done correctly, this is the p-value shown. F 70 100 95 95 95 95 If the null hypothesis was rejected, what is the effect size value (eta squared): Meaning of effect size measure: What does that decision mean in terms of our equal pay question: 3 While it appears that average salaries per each grade differ, we need to test this assumption. Is the average salary the same for each of the grade levels? Use the input table to the right to list salaries under each grade level. (Assume equal variance, and use the analysis toolpak function ANOVA.) Null Hypothesis: If desired, place salaries per grade in these columns A B C D E Alt. Hypothesis: F Place B51 in Outcome range box. Note: Sometimes we see a p-value in the format of 3.4E-5; this means move the decimal point left 5 places. In this example, the p-value is 0.000034 What is the p-value: Is P-value < 0.05? Do we REJ or Not reject the null? If the null hypothesis was rejected, calculate the effect size value (eta squared): If calculated, what is the meaning of effect size measure: Interpretation: 4 The table and analysis below demonstrate a 2-way ANOVA with replication. Please interpret the results. Note: These values are not the same as the data the assignment uses. The purpose of this question is to analyze the result of a 2-way ANOVA test rather than directly ans BA MA Ho: Average compas by gender are equal Male 1.017 1.157 Ha: Average compas by gender are not equal 0.870 0.979 Ho: Average compas are equal for each degree 1.052 1.134 Ha: Average compas are not equal for each degree 1.175 1.149 Ho: Interaction is not significant 1.043 1.043 Ha: Interaction is significant 1.074 1.134 1.020 1.000 Perform analysis: 0.903 1.122 0.982 0.903 Anova: Two-Factor With Replication 1.086 1.052 1.075 1.140 SUMMARY BA MA Total 1.052 1.087 Male 1.096 1.050 Female Count 12 12 24 1.025 1.161 Sum 12.349 12.9 25.249 1.000 1.096 Average 1.029083333 1.075 1.0520417 0.956 1.000 Variance 0.006686447 0.006519818 0.006866 1.000 1.041 1.043 1.043 Female 1.043 1.119 Count 12 12 24 1.210 1.043 Sum 12.791 12.787 25.578 1.187 1.000 Average 1.065916667 1.065583333 1.06575 1.043 0.956 Variance 0.006102447 0.004212811 0.0049334 1.043 1.129 1.145 1.149 Total Count Sum 24 24 25.14 25.687 Average 1.0475 1.070291667 Variance 0.006470348 0.005156129 ANOVA Source of Variation SS Sample 0.002255021 Columns 0.006233521 (This is the row variable or gend 1 0.0062335 1.060054 0.3088296 4.0617065 (This is the column variable or D Interaction 0.006417188 1 0.0064172 1.0912878 0.3018915 4.0617065 Within Total 0.25873675 0.273642479 df MS F P-value F crit 1 0.002255 0.3834821 0.538939 4.0617065 44 0.0058804 47 Interpretation: For Ho: Average compas by gender are equal Ha: Average compas by gender are not equal What is the p-value: Is P-value < 0.05? Do you reject or not reject the null hypothesis: If the null hypothesis was rejected, what is the effect size value (eta squared): Meaning of effect size measure: For Ho: Average compas are equal for all degrees Ha: Average compas are not equal for all grades What is the p-value: Is P-value < 0.05? Do you reject or not reject the null hypothesis: If the null hypothesis was rejected, what is the effect size value (eta squared): Meaning of effect size measure: For: Ho: Interaction is not significant Ha: Interaction is significant What is the p-value: Is P-value < 0.05? Do you reject or not reject the null hypothesis: If the null hypothesis was rejected, what is the effect size value (eta squared): Meaning of effect size measure: What do these three decisions mean in terms of our equal pay question: Place data values in these columns 5. Using the results up thru this week, what are your conclusions about gender equal pay for equal work at this point? Dif swer our equal pay question. der.) Degree.) ns Week 4 Confidence Intervals and Chi Square (Chs 11 - 12) For questions 3 and 4 below, be sure to list the null and alternate hypothesis statements. Use .05 for your significance level in making your decisions. For full credit, you need to also show the statistical outcomes - either the Excel test result or the calculations you performed. 1 Using our sample data, construct a 95% confidence interval for the population's mean salary for each gender. Interpret the results. Mean St error t value Low to High Males Females Interpretation: 2 Using our sample data, construct a 95% confidence interval for the mean salary difference between the genders in the population. How does this compare to the findings in week 2, question 2? Difference St Err. T value Low to High Yes/No Can the means be equal? Why? How does this compare to the week 2, question 2 result (2 sampe t-testResults are the same - means are not equal. a. Why is using a two sample tool (t-test, confidence interval) a better choice than using 2 one-sample techniques when comparing two samples? 3 We found last week that the degree values within the population do not impact compa rates. This does not mean that degrees are distributed evenly across the grades and genders. Do males and females have athe same distribution of degrees by grade? (Note: while technically the sample size might not be large enough to perform this test, ignore this limitation for this exercise.) Ignore any cell size limitations. What are the hypothesis statements: Ho: Ha: Note: You can either use the Excel Chi-related functions or do the calculations manually. Data InTables The Observed Table is completed for you. OBSERVED A B C D E F Total If desired, you can do manual calculations per cell here. M Grad 1 1 1 1 5 3 12 A B C D E F Fem Grad M Grad 5 3 1 1 1 2 13 Male Und Fem Grad 2 2 2 1 5 1 13 Female Und Male Und 7 1 1 2 1 0 12 Female Und 15 7 5 5 12 6 50 Sum = EXPECTED M Grad Fem Grad Male Und Female Und For this exercise - ignore the requirement for a correction for expected values less than 5. Interpretation: What is the value of the chi square statistic: What is the p-value associated with this value: Is the p-value <0.05? Do you reject or not reject the null hypothesis: If you rejected the null, what is the Cramer's V correlation: What does this correlation mean? What does this decision mean for our equal pay question: 4 Based on our sample data, can we conclude that males and females are distributed across grades in a similar pattern within the population? Again, ignore any cell size limitations. What are the hypothesis statements: Ho: Ha: A OBS COUNT - m OBS COUNT - f B C D E Do manual calculations per cell here (if desired) A B C D E F M F F Sum = EXPECTED What is the value of the chi square statistic: What is the p-value associated with this value: Is the p-value <0.05? Do you reject or not reject the null hypothesis: If you rejected the null, what is the Phi correlation: If calculated, what is the meaning of effect size measure: What does this decision mean for our equal pay question: 5. How do you interpret these results in light of our question about equal pay for equal work? Week 5 Correlation and Regression 1. Create a correlation table for the variables in our data set. (Use analysis ToolPak or StatPlus:mac LE function Correlation.) a. Reviewing the data levels from week 1, what variables can be used in a Pearson's Correlation table (which is what Excel produces)? b. Place table here (C8): c. d. Looking at the above correlations - both significant or not - are there any surprises -by that I mean any relationships you expected to be meaningful and are not and vice-versa? e. 2 Using r = approximately .28 as the signicant r value (at p = 0.05) for a correlation between 50 values, what variables are significantly related to Salary? To compa? Does this help us answer our equal pay for equal work question? Below is a regression analysis for salary being predicted/explained by the other variables in our sample (Midpoint, age, performance rating, service, gender, and degree variables. (Note: since salary and compa are different ways of expressing an employee's salary, we do not want to have both used in the same regression.) Plase interpret the findings. Note: These values are not the same as the data the assignment uses. The purpose is to analyze the result of a regression test rather than directly answer our equal pay question. Ho: The regression equation is not significant. Ha: The regression equation is significant. Ho: The regression coefficient for each variable is not significant Note: technically we have one for each input variable. Ha: The regression coefficient for each variable is significant Listing it this way to save space. Sal SUMMARY OUTPUT Regression Statistics Multiple R 0.99155907 R Square 0.9831894 Adjusted R Square 0.98084373 Standard Error 2.65759257 Observations 50 ANOVA df Regression Residual Total SS MS F Significance F 6 17762.3 2960.383 419.15161 1.812E-036 43 303.70033 7.062798 49 18066 Standard Coefficients Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept -1.74962121 3.6183677 -0.483539 0.6311665 -9.046755 5.54751262 -9.0467550427 5.547512618 Midpoint 1.21670105 0.0319024 38.13829 8.66E-035 1.15236383 1.28103827 1.1523638283 1.2810382727 Age -0.00462801 0.0651972 -0.070985 0.943739 -0.1361107 0.1268547 -0.1361107191 0.1268546987 Performace Rating -0.05659644 0.0344951 -1.640711 0.1081532 -0.1261624 0.01296949 -0.1261623747 0.0129694936 Service -0.04250036 0.084337 -0.503935 0.6168794 -0.2125821 0.12758138 -0.2125820912 0.1275813765 Gender 2.420337212 0.8608443 2.811585 0.0073966 0.68427919 4.15639523 0.684279192 4.156395232 Degree 0.27553341 0.7998023 0.344502 0.7321481 -1.3374217 1.88848848 -1.3374216547 1.8884884833 Note: since Gender and Degree are expressed as 0 and 1, they are considered dummy variables and can be used in a multiple regression equation. Interpretation: For the Regression as a whole: What is the value of the F statistic: What is the p-value associated with this value: Is the p-value <0.05? Do you reject or not reject the null hypothesis: What does this decision mean for our equal pay question: For each of the coefficients: Intercept What is the coefficient's p-value for each of the variables: NA Is the p-value < 0.05? NA Do you reject or not reject each null hypothesis: NA What are the coefficients for the significant variables? Midpoint Age Perf. Rat. Service Gender Degree Note: These Using the intercept coefficient and only the significant variables, what is the equation? Is gender a significant factor in salary: If so, who gets paid more with all other things being equal? How do we know? 3 Salary = Perform a regression analysis using compa as the dependent variable and the same independent variables as used in question 2. Show the result, and interpret your findings by answering the same questions. Note: be sure to include the appropriate hypothesis statements. Regression hypotheses Ho: Ha: Coefficient hyhpotheses (one to stand for all the separate variables) Ho: Ha: Place c94 in output box. Interpretation: For the Regression as a whole: What is the value of the F statistic: What is the p-value associated with this value: Is the p-value < 0.05? Do you reject or not reject the null hypothesis: What does this decision mean for our equal pay question: For each of the coefficients: Intercept What is the coefficient's p-value for each of the variables: NA Is the p-value < 0.05? NA Do you reject or not reject each null hypothesis: NA What are the coefficients for the significant variables? Midpoint Age Perf. Rat. Service Gender Degree Using the intercept coefficient and only the significant variables, what is the equation? Compa = Is gender a significant factor in compa: Regardless of statistical significance, who gets paid more with all other things being equal? How do we know? 4 Based on all of your results to date, Do we have an answer to the question of are males and females paid equally for equal work? Does the company pay employees equally for for equal work? How do we know? Which is the best variable to use in analyzing pay practices - salary or compa? Why? What is most interesting or surprising about the results we got doing the analysis during the last 5 weeks? 5 Why did the single factor tests and analysis (such as t and single factor ANOVA tests on salary equality) not provide a complete answer to our salary equality question? What outcomes in your life or work might benefit from a multiple regression examination rather than a simpler one variable test? se values are not the same as in the data the assignment uses. The purpose is to analyze the result of a 2-way ANOVA test rather than directly answer our equal pay