Score: <1 point> 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. How do they compare with the findings in the week 2 one sample t-test outcomes (Question 1)? Mean St error t value Low to High Males Females Interpretation: <1 point> 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-test)? 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? <1 point> 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.) What are the hypothesis statements: Ho: Ha: Note: You can either use the Excel Chi-related functions or do the calculations manually. Data input tables - graduate degrees by gender and grade level OBSERVED A B C D E F Total M Grad Fem Grad Male Und Female Und If desired, you can do manual calculations per cell here. A B C D E F M Grad Fem Grad Male Und Female Und Sum = EXPECTED M Grad Fem Grad Male Und Female Und For this exercise - ignore the requirement for a correction factor for cells with 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: <1 point> 4 Based on our sample data, can we conclude that males and females are distributed across grades in a similar pattern within the population? 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: What does this correlation mean? What does this decision mean for our equal pay question: <2 points> 5. How do you interpret these results in light of our question about equal pay for equal work? 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. VIDEO TUTORIAL FOR QUESTION 1 CAN BE FOUND HERE: https://www.youtube.com/watch?v=48rwvR0MP7c T-value calculator HERE: http://www.danielsoper.com/statcalc3/calc.aspx?id=10 1 Using our sample data, construct a 95% confidence interval for the population's mean salary for each gender. Interpret the results. How do they compare with the findings in the week 2 one sample t-test outcomes (Question 1)? Mean St error t value Low to High To find the low and high bounds of the interval, Males 52.1 3.65 2.064 44.5664 59.6336 we calculate +/- t value*standard error to the average. Females 37.8 3.67 2.064 30.22512 45.37488 You do this part! :) Interpretation: VIDEO TUTORIAL FOR QUESTION 2 CAN BE FOUND HERE: https://www.youtube.com/watch?v=lHTyt4m8f6g T-value calculator HERE: http://www.danielsoper.com/statcalc3/calc.aspx?id=10 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 14.3 5.19 2.01 Low 3.8681 to High 24.7319 To find the low and high bounds of the interval, we calculate +/- t value*standard error to the average. Yes/No Can the means be equal? Why? You answer this one! :) How does this compare to the week 2, question 2 result (2 sampe t-test)? This one too! 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? And this one! This one too! VIDEO TUTORIAL FOR QUESTION 3 CAN BE FOUND HERE: https://www.youtube.com/watch?v=OpFG1_j70oM p-value calculator HERE: http://www.socscistatistics.com/pvalues/chidistribution.aspx 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 the 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.) What are the hypothesis statements: Ho: The distribution of degrees by grade and gender is similar. Ha: The distribution of degrees by grade and gender is not similar. Note: You can either use the Excel Chi-related functions or do the calculations manually. Data input tables - graduate degrees by gender and grade level OBSERVED A B C D E F Total If desired, you can do manual calculations per cell here. M Grad 2 1 1 1 4 3 12 A B C D E Fem Grad M Grad 6 1 2 2 1 1 13 Male Und Fem Grad 2 1 3 1 5 1 13 Female Und Male Und 2 6 1 2 0 1 12 Female Und 12 9 7 6 10 6 50 F Sum = EXPECTED M Grad Fem Grad Male Und Female Und Chitest value = Chiinv = A 3 3 3 3 12 B 2.2 2.3 2.2 2.3 9 0.0384511017 25.959446124 C 1.8 1.7 1.8 1.7 7 D 1.5 1.5 1.5 1.5 6 E 2.5 2.5 2.5 2.5 10 F 1 2 1 2 6 Total 12 13 13 12 50 For this exercise - ignore the requirement for a correction factor for cells with expected values less than 5. Notice the degrees of freedom I inputted were 15. Yours are 15 too. Interpretation: What is the value of the chi square statistic: 25.96 What is the p-value associated with this value: 0.038 Is the p-value <0.05? Yes Do you reject or not reject the null hypothesis: Since the p-value is less than .05, we reject the null If you rejected the null, what is the Cramer's V correlation: Do not worry about this part :) What does this correlation mean? Do not worry about this part either :) What does this decision mean for our equal pay question: You do this one! What does it mean if we reject or fail to reject the null? Here is where you INTERPRET your results. VIDEO TUTORIAL FOR QUESTION 4 CAN BE FOUND HERE: https://www.youtube.com/watch?v=o3UVfgpOyus p-value calculator HERE: http://www.socscistatistics.com/pvalues/chidistribution.aspx 4 Based on our sample data, can we conclude that males and females are distributed across grades in a similar pattern within the population? What are the hypothesis statements: Ho: Distribution is similar between males and females Ha: You do the Ha! OBS COUNT - m A 4 B 2 C 4 D 2 E 9 F 4 Do manual calculations per cell here (if desired) A B C D Total 25 E F OBS COUNT - f Total 8 12 7 9 3 7 4 6 1 10 2 6 25 50 Sum = EXPECTED Total A B C D E F OBS COUNT - m 25 6 5 3 3 5 3 OBS COUNT - f 25 Chitest value = 6 4 4 3 5 3 0.0176316554 Total 50 Chiinv = 12 9 7 6 10 6 13.7 What is the value of the chi square statistic: 13.7 My degrees of freedom here are 5. What is the p-value associated with this value: 1 Yours will be too for this question. Is the p-value <0.05? no Do you reject or not reject the null hypothesis: NOT reject If you rejected the null, what is the Phi correlation: NA NOTE! If you need to calculate Phi, the formula is "square root of (chi square statistic divided by the sample size)", in our case, sample size is 50 What does this correlation mean? NA What does this decision mean for our equal pay question: You do this part! 5. How do you interpret these results in light of our question about equal pay for equal work? Gende r Salary 0 64.5 0 28.8 0 57.3 0 46.9 0 75.8 0 76.1 0 63.2 0 38 0 23 0 74.3 0 22.7 0 39.5 0 78.6 0 48.6 0 29 0 62.8 0 29 0 59.6 0 24.9 0 44.9 0 69.3 0 60.6 0 66.1 0 59.1 0 60.2 1 35.4 1 42.5 1 25 1 24.4 1 22 1 40.4 1 24.7 1 21.7 1 68.3 1 35.8 1 34.3 1 44.7 1 21.5 1 53.8 1 23.9 1 77.4 1 22.8 1 22.9 1 23.2 1 21.1 1 34.7 1 24.6 1 75.8 1 1 59.1 64.8 For the t value, we will go back to our calculator We again use significant of .05, but this time, we use 48 degrees of freedom. Let's go! Standard Error for Difference in Means: Sample 1 is our males, sample 2 is our females Both of these have sample size of 25, and as we previously found, the standard devation of each is: males: 18.2529 females: 18.3544 Plugging this in our formula, we get: SE = sqrt[(18.3^2/25) + (18.4^2/25)] Now we can use Google as a scientific calculator, we just copy/paste that phrase with all the parenthes sqrt[(18.3^2/25) + (18.4^2/25)] And we get the final answer of...DRUMROLL......... 5.19 See this last part? When you get to it, here is how to use Google as a scientific calculator... Voila!!! Again, go back to this part in the video and study the formula above, and how I have plugged in ses and brackets in tact n all the values.... Standard Error for Difference in Means: Sample 1 is our males, sample 2 is our females Both of these have sample size of 25, and as we previously found, the standard devation of each is: males: 18.253 females: 18.355 Plugging this in our formula, we get: SE = sqrt[(18.3^2/25) + (18.4^2/25)] Now we can use Google as a scientific calculator, we just copy/paste that phrase with all the parentheses and brackets in sqrt[(18.3^2/25) + (18.4^2/25)] And we get the final answer of...DRUMROLL......... 5.19 s in tact Gender Degree Gr Gender 0 = male, 1 = female Degree 0 = Undergrad, 1 = Grad 0 0 0 0 0 1 E C E 0 1 D 0 1 F 0 1 F 0 0 E 0 0 C 0 1 A 0 1 F 0 0 A 0 1 C 0 0 F 0 0 D 0 0 B 0 1 E 0 1 B 0 0 E 0 0 A 0 0 C 0 1 E 0 1 E 0 1 A 0 0 E 0 0 E 1 1 B 1 1 F 1 1 C 1 1 A 1 1 D 1 0 C 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 1 A A E B B D A D B F A A B B B A D C A On to question 3! Here we are working with three variables: gender, degree, and grade. I have copy/pasted just these three columns in a new workbook, to keep things organized. Remember, these are EXAMPLE values :) Your dataset is posted in class. Please use those values. Now, let's go ahead and sort the data. Right click and follow what I do Now, I basically have to figure out how to fill out my table on this page... I manually count how many fall in each category. Let me go ahead and do this and post the results. Whew! I am done filling out my table :) Here it is! Now, notice the "totals" row here in the expected values... I just copy/pasted my totals row from my observed into my expected values section. You will do the same with your values. Next, how do we fill out the expected values table? Let's have a look! EXPECTED A B C D E F Total M Grad 3 2.2 1.8 1.5 2.5 1 12 Fem Grad Male Und Female Und 3 3 3 12 2.3 2.2 2.3 9 1.7 1.8 1.7 7 1.5 1.5 1.5 6 2.5 2.5 2.5 10 2 1 2 6 13 13 12 50 OK, here is my expected values table, I have copied it here for easy visual access. Now, notice the values in each of the columns are identical or similar. Why is this??? Because, with expected values, if I expect there to be no difference between the grades, I will expect them to be evenly distributed. That is to say, there is no difference between genders with respect to grades. Now, notice also right here that the values are not identical Why is this??? Because we cannot have a fraction of a person here :) Gender 0 0 0 0 1 1 1 1 1 1 Gr A A A A A A A A A A 1 A 1 A 0 B 2 males 0 B 7 females 1 B 1 B 1 B 1 B 1 B 1 B 1 B 0 C 4 males 0 C 3 females 0 C 0 C 1 C 1 C 1 C 0 D 2 males 0 D 4 females 1 D 1 D 1 D 1 D 0 E 9 males 0 E 1 female 0 E 0 E 0 E 0 E 0 E 0 E 0 E 1 E 0 F 4 males 0 F 2 females 0 F 0 F 1 F 1 F C 4 3 7 D 2 4 6 E 9 1 10 F 4 2 6 Total OBS COUNT - m OBS COUNT - f Total B 2 7 9 EXPECTED OBS COUNT - m OBS COUNT - f Total 4 males 8 females A 4 8 12 A 6 6 12 B 5 4 9 C 3 4 7 D 3 3 6 E 5 5 10 F 3 3 6 Total 25 25 50 25 25 50 Now, we move on to question 4! The idea is similar to what we did in question 3. Here, notice I have copy/pasted my two relevant columns, gender and grade. I have sorted them according to grade and then manually tallied up my observed counts of males and females by grade, and filled out my table here. To fill out my expected table, I do the same thing as before. I copy/pasted my totals And then, provided an even distribution of the other values. Use These Expected Values for Question 3 EXPECTE D M Grad Fem Grad Male Und Female Und A 3.75 3.75 3.75 3.75 15 B 1.75 1.75 1.75 1.75 7 C 1 1.5 1.5 1 5 D 1 1.5 1.5 1 5 E 3 3 3 3 12 F 1.5 1.5 1.5 1.5 6 Total 12 13 13 12 50 Use These Expected Values for Question 4 EXPECTED OBS COUNT m OBS COUNT f Total A B C D E F 7.5 3.5 2.5 2.5 6 3 25 7.5 15 3.5 7 2.5 5 2.5 5 6 12 3 6 25 Total 50