1997315. 6.6.any. STATS UNITS 4, 5, &6 11958 CORRELATION Fill in the blanks: 8.1. The magnitude of the correlation is indicated by the correlation COEFICIENT, which can range from -1.00 to +1.00. 8.2. The most common and efficient way to present the correlations of several variables with each other is by using the ANOVA summary table and the Equation variable table. 8.3. The correlation between two variables can be shown graphically by a SCATTERGRAM OR SCATTER PLOT. 8.4. The null hypothesis predicts that the correlation coefficient is equal to 0. 8.5. The Spearman rank order correlation is used when the variables to be correlated are measured on an ORDINAL scale. Circle the correct answer: 8.6. The hypothesis that states that r=0 is an example of NULL hypothesis. 8.7. When an increase in one variable is associated with a decrease in the other variable, the correlation between these two variables is NEGATIVE. 8.8. In order to use the Pearson product-moment correlation, the variables to be correlated should be measured on an INTERVAL scale. 8.9. When the points on a scattergram go from the bottom left to the top right they represent a POSITIVE correlation. 8.10. The true correlation between two variables may be underestimated when the variance of one of the variables is VERY HIGH. 8.11. When the null hypothesis is rejected at p<.001, it means that the chance that r=0 is VERY SMALL. 8.12. The null hypothesis is rejected when the obtained correlation coefficient is LOWER than the critical value. Answer/compute the following questions: 8.13 Which correlation coefficient (a or b) shows a stronger relationship between the two variables being correlated? b. X2&Y2: r = -.94 8.14. Following are two scattergrams (in Figure A and in Figure B). Four different correlation coefficients are listed under each scattergram. Choose the coefficient that best matches each scattergram. Y Y • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • X X Figure A Figure B A1. r= .50 and B3. r= .38 8.15 Following is a scattergram showing the scores of 8 statistics students on two tests, X and Y. Each of the first 7 students is represented by a dot and their scores are listed in the table that follows. Use the scattergram to find the scores of student #8 on test X and test Y. The location of this student on the scattergram is represented by a large dot (•) next to number 8. Y 1 4 4 • • 5 2 3 • • 6 8 2 • • 7 3 1 • • X 1 2 3 4 5 Student # X Y 1 2 4 2 3 3 3 2 1 4 5 4 5 2 3 6 2 2 7 1 1 8 1 4 8.16 What do these two scattergrams have in common? Y Y • • • •• • •• • • • • • •• • •• • • • • • •• • •• • • •• • • • • • • • • • • • • • • • • •• • •• • • •• • • • •• • •• • • • • • •• • •• • • •• • • •• • •• • • •• Answer: they are both positively correlated 8.17 Estimate (do not calculate!) the correlation between the advertising spending and sales that were obtained over a 5 year span. Indicate whether the correlation is positive or negative, and whether it is high or low. Explain your answer. Year Ad Spending Sales 1 $21,000 $83 2 15,000 70 3 17,000 68 4 25,000 90 5 19,000 74 Answer: The correlation is positive, it is high as, generally, an increase or decrease in AD spending proportionally corresponds with increase or decrease in sales. 8.18 Estimate (do not calculate!) which of the two sets of consumer research studies (A&B or X&Y) has a higher correlation. Explain your answer. Set 1 Set 2 Study # A B Study # X Y 1 41 50 1 66 47 2 41 47 2 53 36 3 38 43 3 50 45 4 30 39 4 48 38 5 28 37 5 45 39 Answer: STUDY A and B has a higher correlation because the decrease/increase in values is more vividly represented 8.19 Match the correlation coefficient and the diagram illustrating this correlation. a. r = .91 b. r = .28 c. r = .15 8.20. Study the following intercorrelation table. Two surveys measure consumer attitudes and two surveys measure marketers’ promotional spending. Knowing that Test 1 measures consumer attitudes, speculate which is the other consumer attitudes survey (2, 3, or 4) and which two surveys measure marketers’ promotional spending. (A hint: the two consumer attitudes surveys should correlate higher with each other than with the two marketers’ promotional spending surveys, and the two marketers’ promotional spending surveys should correlate higher with each other than with the two consumer attitudes surveys.) 2 3 4 1 .35 .89 .23 2 .39 .92 3 .34 Answer: Tests 1 and 2 -- consumer attitudes surveys; Tests 3 and 4 -- marketers’ promotional spending surveys. 8.21. Following are results from a study correlating gross domestic product (GDP) and the Consumer Price Index (CPI) from a group of 16 Asian countries, a group of 15 European countries, and the two groups combined. Group Correlation (r)/ Group Size (n)/ Significance (p value) Asia r =.59 n = 16 p =.02 Europe r =.52 n = 15 p =.05 Combined r =.53 n = 31 p =.01 a. Which correlation coefficient is the highest? ASIA, r =.59 b. Which correlation coefficient has the highest statistical significance? Combined, r =.53 c. How can a correlation of r=.53 (from the combined group) be more statistically significant (better p value) than a correlation of r=.59 (from the Asian countries)? Answer: In this case, the significance level indicates how possibly it is that the correlations shown may be due to chance (thus untrue) in the form of random sampling error. Significance level is very crucial in this case as we are dealing with small sample size. Chapter 9 PREDICTION AND REGRESSION Fill in the blanks: 9.1. In simple regression, there is one predictor. 9.2. The regression line is also called the regression line of Y on X 9.3. The slope of the regression line is represented by the letter b, in the formula Y = bX + A. 9.4. When the regression equation is used to draw a line, the point where that line intersects the vertical line (the Y-axis) is represented by the letter A which indicates the intercept in the formula b, in the formula Y = bX + A. 9.5. When the correlation between two variables is perfect and positive, and we use one of these variables to predict the other one, the standard error of estimate (SE) is 0. 9.6. The difference between an actual Y score and its corresponding predicted Y score (Y’) is called the Total Distance (that if we consider Y’ to be the grand mean of the distribution). 9.7. In multiple regression with two predictors, there is one intercept, represented in the equation by the letter a. It shows the points at which the regression plane crosses the y-axis in the case when values of the predictor scores are 0. Circle the correct answer: 9.8. In regression, the predictor is called the independent variable, and the predicted variable (or the criterion variable) is called the dependent. 9.9. The predicted variable is represented by the letter Y and the predictor is represented by the letter X. 9.10. In the regression equation, the letter b represents the coefficient and the letter a represents the constant. 9.11. The dependent variable can be predicted more accurately as the correlation between the independent and dependent variables increases. 9.12. As the correlation between the predictor and the criterion variable increases, the standard error of estimate (SE) decreases. 9.13. The predicted Y scores are expected to be around the regression line. 9.14. The criterion is predicted more accurately when SE is smaller. Answer/compute the following questions: 9.15. Study the following graph. If a company is able to maintain a 25% gross margin (the predictor X), what is the company’s predicted Return on Investment (ROI) (the criterion Y)? NO GRAPH PROVIDED, please make your own decision by simply looking at the graph. 9.16. Compute SE (the standard error of estimate) when the standard deviation of the Y-variable is 5 (SDY=5) and the correlation is 0.00 (r=0.00). Use the following equation to compute the SE: What is the relationship between the SE (the standard error of estimate) and the SD of the dependent variable Y (SDY=5) when the correlation is zero (r=.00)? Explain. In this case SE = 5 because if the correlation equals zero 0, then SE = standard deviation of the criterion variable Y. 9.17. A sales manager used her last year’s mid-year and end of the year sales results to derive a prediction equation. This year her sales force has the same 6 salespeople as last year and the market conditions are very similar to last year. In the middle of this year the manager wants to predict the year-end sales results for each of her 6 salespeople. Following are the mid-year sales results of the 6 salespeople, and the prediction equation. a. Calculate the salespeople’s' predicted end of the year sales results (Y’ scores).
(Note: Use the prediction equation to compute the students’ Y’ scores.) Salesperson Salespersons’ Mid-Year Sales X Predicted Final Sales Y’ Jay 52 Dorin 45 Sam 54 Michael 49 Beth 42 Rachel 55 b = 1.5 a = 13.2 Y’ = 13.2 + 1.5 * 52 + 45 * 1.5 + 54 * 1.5 + 49 * 1.5 + 42 * 1.5 + 55 * 1.5 = 458.7 Individual results can be derived by using one predictor in the formula (e.g. Jay: 13.2 + 52 * 1.5 = 91.2) 9.18. Study the following graph, and determine what is the value of the slope of the regression line (i.e., b). Explain your answer. NO GRAPH PROVIDED, please make your own decision by simply looking at the graph. 9.19. Draw a regression line in the following graph, when the intercept is 10 (a=10), the mean of the X-variable is 20 (X¯=20), and the mean of the Y-variable is 30 (Y¯=30). This regression line intercepts Y-axes at 10. The regression line runs in the middle of the points that are structured on the graphs as follows: The regression line is linked through the point of averages, generally its slope is equal to Standard Deviation of Y divided by the Standard Deviation of X if correlation is greater than or equals zero (this value is negative if correlation coefficient is negative). 9.20. Figure A shows three predictors, X1, X2, and X3, and their correlations with the criterion variable Y1. Figure B shows three predictors, Z1, Z2, and Z3, and their correlations with a criterion variable Y2. a. Which predictor variables, those depicted in Figure A or those depicted in Figures B, correlate higher with each other? b. Which set of three predictors, those shown in Figure A (X1, X2, and X3) or those shown in Figure B (Z1, Z2, and Z3) is likely to predict the criterion variables (Y1 or Y2) more accurately? Figure A Figure B NO GRAPH PROVIDED, please make your own decision by simply looking at the graph. Unit 5 Chapter 10 t TEST Fill in the blanks: 10.1. In a t test for a single sample, the sample's mean is compared to the population Mean. 10.2. When we use a paired-samples t test to compare the pretest and posttest scores for a group of 45 people, the degrees of freedom (df) are (n-1) = 44 . 10.3. If we conduct a t test for independent samples, and n1 = 32 and n2 = 35, the degrees of freedom (df) = n1 + n2 – 2; = 65 10.4. A researcher wants to study the effect of college education on people’s earning by comparing the annual salaries of a randomly-selected group of 100 college graduates to the annual salaries of 100 randomly-selected group of people whose highest level of education is high school. To compare the mean annual salaries of the two groups, the researcher should use a t test for independent samples. 10.5. A training coordinator wants to determine the effectiveness of a program that makes extensive use of educational technology when training new employees. She compares the scores of her new employees who completed the training on a nationally-normed test to the mean score of all those in the country who took the same test. The appropriate statistical test the training coordinator should use for her analysis is the t test for independent samples. 10.6. As part of the process to develop two parallel forms of a questionnaire, the persons creating the questionnaire may administer both forms to a group of students, and then use a t test for paired samples to compare the mean scores on the two forms. Circle the correct answer: 10.7. A difference of 4 points between two homogeneous groups is likely to be more statistically significant than the same difference (of 4 points) between two heterogeneous groups, when all four groups are taking completing the same survey and have approximately the same number of subjects. 10.8. A difference of 3 points on a 100-item test taken by two groups is likely to be more statistically significant than a difference of 3 points on a 30-item test taken by the same two groups. 10.9. When a t test for paired samples is used to compare the pretest and the posttest means, the number of pretest scores is the same as than the number of posttest scores. 10.10. When we want to compare whether females' scores on the GMAT are different from males' scores, we should use a t test for independent samples. 10.11. In studies where the alternative (research) hypothesis is directional, the critical values for a one-tailed test should be used to determine the level of significance (i.e., the p value). 10.12. When the alternative hypothesis is: HA: µ1 = µ2, the critical values for two-tailed test should be used to determine the level of statistical significance. Answer/compute the following questions: 10.13. In a study conducted to compare the test scores of experimental and control groups, a 50-item test is administered to both groups at the end of the study. The mean of the experimental group on the test is 1 point higher than the mean of the control group. The researchers conduct a t test for independent samples to compare the two means. The obtained t value is 1.89, and the p-value is .05. Can we conclude that the experimental treatment was clearly effective because the t value is statistically significant? Explain. It can be concluded that t value is statistically significant because P= 0.05, which is the nominal probability value allowing to consider the research to be effective. 10.14. Study the following formula for a t test for independent samples. What measures (e.g., the mean of Group 1) are needed in order to calculate the t value? (Respond in words, not symbols). According to Statistical theory, the two P values have to be equal. The t statistics has to be equal in effect. The signs are the same if the t statistic is derievd by subtracting the mean of group 0 from the mean of group 1. 10.15. Identify each of the following as a null hypothesis, a directional hypothesis, or a nondirectional hypothesis. a. µ1 ≠ µ2 is a NON-DIRECTIONAL hypothesis b. µ1 = µ2 is a NULL hypothesis c. µ1 > µ2 is a DIRECTIONAL hypothesis d. µ1 - µ2 = 0 is a NULL hypothesis 10.16. A company psychologist wants to compare the scores of a group of 35 employees on two different IQ tests: one is a group IQ test and one is an individually-administered IQ test. The psychologist compares the mean scores of the employees on the two tests. Which t test should the psychologist use to determine whether there is a significant difference between the two sets of IQ scores? Explain. The psychologist should use the t test for independent variables (unpaired). We use a simple check to see if the paired t test can apply - if one sample can contain a dissimilar number of data points from the other, then we can’t use the paired t test. In our case, the data points in two tests may be different. 10.17. The CEO of ABC Company wishes to determine whether there are differences between union members and managers in their attitude toward the company. The CEO asks 30 randomly selected union members and 28 managers to complete a 40-item questionnaire designed to measure attitudes toward the company. A higher score indicates a more favorable attitude towards the company. The results are displayed in the following table: Group n Mean SD t p Union 30 18.30 9.85 1.92 .03 Managers 28 23.07 9.07 a. Which t test should the CEO use to compare the responses of the union members and the managers? Explain. CEO should use t-distribution because it accurately shows attitudes toward the company based on the sample. b. What are the degrees of freedom (df)? The degrees of freedom for an estimate equals the number of observations (values) minus the number of additional parameters estimated for that calculation. c. What are the conclusions of the CEO based on the results in the table? Explain. CEO needs to take additional sampling to assure accurate results. 10.18. Based on the results of the study described in the previous question, the CEO and the HR manager decide to implement several programs to get the union more actively involved in making decisions at ABC. After implementing the programs for a year, the CEO asks the same group of union members (n=30) to about their attitudes towards the company again, using the same questionnaire as the one used the year before. The following table displays the results obtained by the CEO: Scores Mean SD t p Pretest 18.30 9.85 7.13 .0001 Posttest 22.10 8.88 a. Which t test should be used to analyze the data? Explain. Paired samples t test should be used. This type of test should be used when the number of points in each data set is the same and is organized in pairs, in which there is a specific relationship between each pair of numbers. b. What are the degrees of freedom (df)? (df) = n-1 = 29. c. Based on the results in the table, can the CEO conclude that the programs worked? Explain. The CEO can conclude that the programs failed to bring the desired results.
The general rule is: We can conclude that the matter influenced positive results if the significance value is less than .05. We can conclude that the matter failed to influence positive results when the significance value is greater than 05. 10.19. A plant manager randomly divides her employees into two groups. One group (Group A) includes 32 employees and the second one (Group B) includes 30 employees. After dividing the employees, the manager wants to confirm that the two groups are indeed similar in performance. He hypothesizes that there is no statistically significant difference between the two groups. To compare the two groups, the plant manger use ratings given by the employees’ front-line supervisors at the end of the previous year. The rating scale ranges from 5 (excellent employee) to 1 (on the brink of termination). Using these ratings, the manager conducts a t test to determine whether the two groups are similar. The results are as follows: Group n Mean SD t A 32 3.66 1.31 2.008 B 30 3.00 1.26 tcrit(.05,df)=2.000; tcrit(.02,df)=2.390; tcrit(.01,df)=2.660 a. Which t test was used and why? Independent samples t test was used. The general rule states that when samples are taken from two dissimilar populations or from randomly chosen individuals from the same population at diverse times, we have to use the independent samples t test. b. What were the degrees of freedom (df)? (df) = n1+n2-2 = 60 c. What are the manager’s conclusions? Explain. There is a difference between two groups. The point at which degrees of freedom intersect is the critical value of t. Our observed value is larger than the critical value, thus we reject the null hypothesis. 10.20. Joe, a new manager suspects that the 23 new employees assigned to his department have lower levels of social skills than the new employees hired into the all the other departments in the rest of the company. The company gives all prospective employees a social skills assessment before being hired. The mean score obtained by Joe’s new employees is 635.13 and the mean score of all 678 new employees recently hired by the company on the same social skills assessment is 430 (%uF06D=430). A t test is used to compare the social skills assessment scores of Joe’s new employees to all the new employees in the company. The results of the t test are: X¯ = 635.13 S2 = 71.53 t value = 13.75 p = .0001 a. Which t test is used and why? Independent samples t test was used (2-tailed). In our case the samples are taken from randomly chosen individuals from the same population at diverse times, thus we have to use the independent samples t test. b. What are the degrees of freedom (df)? (df) = n1+n2-2 = 699 c. What are Joe’s conclusions? Explain. Two groups’ skills differ from each other. In our case, the mean for group 1 is significantly greater than that for group 2, thus we reject the null hypothesis. Chapter 11 ANOVA Fill in the blanks: 11.1. While a t test is used to compare two means, the one-way ANOVA can be used to simultaneously compare more than 2 groups. 11.2. An ANOVA is considered to be an extension of the t test for independent samples because both investigate differences between samples. 11.3. By conducting a one-way ANOVA test to compare multiple (more than 2) group means simultaneously instead of conducting a series of t tests to compare these means, the potential level of error is reduced. 11.4. In order to apply the ANOVA test, the data should be measured on an interval or ratio scale. 11.5. The one-way ANOVA is used when there is one independent variable. 11.6. With 3 groups, the null hypothesis (Ho) in ANOVA is more than 1: 11.7. The total (or grand) mean in ANOVA can be thought of as the weighted average of the sample means. 11.8. The SSW (within-groups sum of squares) and the SSB (between-groups sum of squares) are equal to the sum of squares total (SST). 11.9. To find the MSB, we divide the SSB by dfB. 11.10. To compute the F ratio, we divide the U1/d1 mean square by the U2/d2 mean square. 11.11. Factorial ANOVA is commonly used when there are at least 2 independent variables. Circle the correct answer: 11.12. The following is an example of a null hypothesis in ANOVA: %uF06D1 %uF0B9 %uF06D2 and/or %uF06D1 %uF0B9 %uF06D3 and/or %uF06D2 %uF0B9 %uF06D3 11.13. Post hoc comparisons should be conducted in cases where the F ratio IS statistically significant. 11.14. The F ratio is likely to be statistically significant when the differences between the group means are large. 11.15. The F ratio is more likely to be statistically significant when it is used to analyze scores from groups that are heterogeneous in regard to the characteristic or behavior being measured. Answer the following questions: 11.16. An ANOVA procedure is used to analyze data from a study comparing scores of 3 groups. Following are the obtained mean squares and the appropriate critical values for the F ratio at p=.05 and p=.01. MSB = 26.50 MSW = 3.23 Fcrit (.05,2,20) = 3.49 AND Fcrit (.01,2,20) = 5.85 a. Compute the obtained F ratio. F = MSB / MSW = 8.2 b. Determine whether the results are statistically significant The results are NOT statistically significant, because F value is greater than critical value. c. Report your conclusions. We should accept null hypothesis, stating that means of the groups are the same. 11.17. Three different age groups of consumers (ages 18-25, 26-35, 36-45) in two different regions of the country (Midwest, South) were surveyed about their likelihood of buying a new product. Following are the means and standard deviations obtained by the 3 age groups in each of the two regions: Means Standard Deviations Region Ages 18-25 Ages 26-35 Ages 36-45 Ages 18-25 Ages 26-35 Ages 36-45 Midwest 50.2 52.8 53.3 2.5 3.1 2.7 South 41.0 48.5 55.9 2.7 3.2 2.8 Two separate one-way ANOVA procedures are conducted to test whether the differences between the three means of the three age groups in each of the two regions are statistically significant. Estimate which F ratio would be larger: The one resulting from analyzing the survey results obtained from the three age groups in the Midwest or the one from analyzing the survey results obtained by the three age groups in the South. Explain your answer. The F ratio resulting from analyzing the survey results obtained by the three age groups in the South will be larger because the means’ difference within these groups is larger the one in Midwest. 11.18. Three statistics classes at University A took the same test as did 3 other statistics classes at College B. Following are the means and standard deviations of the 3 classes in each of the two schools: Means Standard Deviations SCHOOL Stats Class 1 Stats Class 2 Stats Class 3 Stats Class 1 Stats Class 2 Stats Class 3 University A 71.4 78.8 90.2 3.2 4.3 4.8 College B 72.1 78.6 89.3 9.3 6.6 8.7 Two separate one-way ANOVA procedures are used to test whether the differences between the means in each of the two schools are statistically significant. Estimate which F ratio would be larger: the one resulting from analyzing the test scores obtained by the three groups at University A or the one resulting from analyzing the test scores of the three groups at College B. Explain your answer. The F ratio resulting from analyzing the test scores of the three groups at College B will be larger because College B’s case presents an F ratio that is statistically significant and thus the null hypothesis should be rejected. 11.19. Each of the two figures below (Figure A and Figure B) depicts a set of 3 distributions. Two separate one-way ANOVA analyses are performed to test whether there are statistically significant differences between the three means in each set and two F ratios are computed. Estimate which of the two F ratios is likely to be higher and explain your answer. Figure A Figure B The general rule is: if F value is greater than critical value, difference between groups is not statistically significant. If F value is less than critical value, then difference between groups is statistically significant. 11.20. Match the interaction shown in the following graph with one of the three F ratios (a, b, or c) that was calculated for the interaction. Explain your answer. ANSWER: a. F=3.23 (p=.07) NOT statistically significant b. F=5.86 (p=.002) statistically significant c. F=2.90 (p=.08) NOT statistically significant 11.21. In a study comparing means of 4 groups, the F ratio was significant at the p<.05 level. The 4 means are Mean 1=13.12; Mean 2=9.31; Mean 3=13.65; and Mean 4=11.34. Tukey’s post hoc comparison is used to test which means statistically differ from each other. The obtained HSD value at the p=.
05 level is 3.74. Which means are statistically significantly different from each other? Explain. Mean 2 and Mean 3 statistically differ from each other. Since our F score is statistically significant, we compare sets of two various means at a time in order to decide where the significance difference is situated. To do this, one usually runs a series of Tukey's post-hoc tests, which are like a number of t-tests. 11.22. A pilot-test marketing research study comparing two new models of widgets was conducted in two companies (Company A and Company B). In each of the two companies, one manufacturing plant used one new widget and the other plant used the other new widget. At the end of the year, the employees using the new widgets in their jobs completed a questionnaire assessing their satisfaction with the widgets. Following is a table listing the mean scores of the two plants questionnaire scores for each of the two companies (those that used new widget 1 and those that used new widget 2). Study the data in the table. (Note: Do not attempt to compute the F ratios or the exact level of significance in order to answer the questions below.) Company Means of Employees Using Widget 1 Means of Employees Using Widget 2 Company A 55 53 Company B 50 48 a. Are there differences in questionnaire scores as a result of the using the two widgets? Explain. Yes, there are differences in questionnaire scores as a result of the using the two widgets. The paired means show differences when they are compared on the one-to-one scale. b. Are there differences in questionnaire scores between the two companies? Explain. Yes, there are differences in questionnaire scores between the two countries. Varying psychological and social factors influence this difference between the countries. c. Graph the interaction. Is there an interaction effect? Explain. Chapter 12 CHI SQUARE Fill in the blanks: 12.1. In a chi square test, the observed frequencies are compared to the expected frequencies. 12.2. The null hypothesis for the chi square test states that there is no statistically significant difference between the expected frequencies and the observed frequencies. 12.3. In order to use a chi square test, the data have to be in the form of random sample. 12.4. In order to create categories for a chi square test, observations that are measured on an interval scale should first be divided into categories in a raw frequency way. 12.5. The degrees of freedom (df) for a 3x4 chi square table are 6. 12.6. In a goodness of fit chi square test with 4 cells, the degrees of freedom (df) are 3. Circle the correct answer: 12.7. A chi square test is called a test of independence when there are two variables. 12.8. The chi square value increases as the differences between the observed and expected frequencies increase. 12.9. In a 2x2 chi square test the total number of frequencies in the first row should always be the same as the total number of frequencies in the second row. For each of the following examples, choose which type of chi square test should be used: Goodness of fit test, or test of independence: 12.10. To test whether a four-sided spinning top has an equal probability of landing on any of its four sides the chi square test of independence should be used. 12.11. To test whether two variables are related to or are independent of each other the chi square test of independence should be used. 12.12. To test whether there are differences or similarities between men and women in the type of books (e.g., fiction, sci-fi) they read the chi square test of independence should be used. 12.13. To test whether the number of right- and left-handed subjects in a given population is higher than the national proportions the chi square Goodness of fit test should be used. Answer/compute the following questions: 12.14. A random sample of 100 men and 100 women were asked whether they would be willing to work as unpaid volunteers. The results indicate that 54% of the men and 64% of the women responded YES to this question, while 46% of the men and 36% of the women responded NO to the question. The chi square test was used to determine whether the men and women in the study differ in their willingness to work as unpaid volunteers. The obtained chi square value was 2.06 (%uF0632(obt)=2.06). The appropriate critical value at p=.05 is 3.841 (%uF0632crit (.05)=3.841). a. Which chi square test was used to analyze the data and determine whether there are gender differences between the responses of the men and women who participated in the survey? Explain. Chi-square test of independence was used. In our case, we test whether men and women differ or not in their volunteering attitude, which is whether there is any dependence between being a man or a woman and being more open to working as a volunteer. b. What was the null hypothesis for the study? The null hypothesis states the proportion of the population in the selected category. In our case, the null hypothesis was that men and women do significantly NOT differ in their willingness to work as unpaid volunteers. c. Was there a gender difference between the responses of the male and female participants? Explain. Because the obtained chi-square value is smaller than the critical value, we accept the null hypothesis: there is no significant difference between men and women’s willingness to work as unpaid volunteers. 12.15. Undergraduate students in a large state university are required to take an introduction to psychology course. Three sections of this course are offered at the same time and are taught by three different instructors. The chi square test is used to determine if any of these sections has significantly more students, which may indicate that the instructor of that section is more popular than the other two instructors. The enrollment figures for the three sections are presented in the following table. The chi square value is 6.29 (%uF0632=6.29) significant at the p<.05 level. INSTRUCTOR NUMBER OF STUDENTS ENROLLED Dr. Smith 114 Dr. Brown 128 Dr. Johnson 91 a. Which chi square test should be used to determine whether any of the sections has a statistically significantly higher number of students? Explain. Chi-square test of independence should be used. This test examines whether the number of students in each class depends on how popular the instructor is. b. What are the expected frequencies? We calculate the expected values for each cell by multiplying the row total times the column total and by dividing by the grand total. 114*3/333= 1.03 128*3/333= 1.15 91*3/333= 0.82 c. What are the degrees of freedom (df)? (df) = N-1 = 2 d. What are your conclusions? Explain. Since chi-square value is significant at p<0.5, we reject the null hypothesis and thus, the instructor of one section is more popular than the other two. 12.16. Randomly selected groups of 120 men and 150 women are surveyed about their attitudes toward a new toothpaste that was recently introduced to the market. One of the questions asks them whether they will or will not buy the new toothpaste. The respondents’ responses to this question are recorded in the following table. The data were analyzed using a chi square test. The obtained chi square value is 5.65, significant at the .02 level (p=.02). GROUP WILL BUY WILL NOT BUY Men 75 45 Women 72 78 a. Which chi square test should be used to analyze the data and answer the research questions? Explain. Chi-square test of independence was used in order to test whether there was a ny dependence between gender and toothpaste buying habits. b. Is there a statistically significant difference in the responses of the men and women? Explain. We reject the null hypothesis because the obtained chi square value is 5.65, significant at the p<0.5 and thus, there is a statistically significant difference in the responses of the men and women. 12.17 . In a recent national poll, people were asked the following question: "In your opinion, how important is it for the Federal government to regulate retail gas prices?" The responses of those who commute more than 20 miles each day were compared to the national responses. A chi square test was used to analyze the data in order to determine whether there is a difference in responses between those who commute more than 20 miles and the national responses. The results of the study are displayed in the following table. The analysis revealed a chi square value of 4.32, significant at p=.36. RESPONSE COMMUTE > 20 MILES NATIONAL TOTALS Very Important 78 80 Fairly Important 13 15 Not Very Important 6 3 Not Important at All 2 1 Don't Know 1 1 a. Which chi square test was used to analyze the data? Explain. Goodness of fit test should be used. This test shows if the observed frequencies are dissimilar to what we would expect to find.
b. What was the null hypothesis? The null hypothesis was that there was NO significant difference in responses between those who commute more than 20 miles and the national responses. c. What are the conclusions of the study? Explain. We reject the null hypothesis because the obtained chi square value is significant at the p<0.5 and thus, there is significant difference in responses between those who commute more than 20 miles and the national responses. 12.18. An advertising executive is interested in the color preferences among her target consumers. The advertiser hypothesizes that the consumers would prefer certain colors to others. For the purpose of the study, only five primary colors are included: yellow, red, blue, green, and black. A group of 50 consumers are brought to a marketing research firm and are asked to select one ball from a box full of balls. In the box there are 50 yellow balls, 50 red, 50 blue, 50 green, and 50 black. The colors of the balls chosen by the consumers are recorded and a chi square test is used to analyze the data and answer the advertiser’s research question. The obtained chi square value is 8.800 (%uF0632obt = 8.800) and the appropriate critical value is %uF0632crit (.05,4) = 9.488. Following are the colors chosen by the consumers: Color No. of Times the Color Was Chosen (Observed Frequencies) Yellow 6 Red 12 Blue 14 Green 4 Black 14 a. Which chi square test was used to analyze the data? Explain. Chi-square test of independence should be used. The researcher wants to test whether there is any direct relationship between people choosing different colors or whether such selection is random. b. What are the expected frequencies? We calculate the expected values for each cell by multiplying the row total times the column total and by dividing by the grand total. Thus, 6*5/50= 0.6 12*5/50= 1.2 14*5/50= 1.16 4*5/50=0.4 14*5/50= 1.16 c. What are the study’s conclusions? Explain. Because the obtained chi-square value is smaller than the critical value, we accept the null hypothesis: there is no real relation between colors and customers choosing them as some particular ones. 12.19. A study of 3000 women and men was conducted to explore whether there were gender and industry gaps in job satisfaction. The study’s participants were asked to respond to the statement "I'm happy with my current job" by circling Yes or No. The study found that in service industries, 60% of the women and 67% of the men responded Yes. When the same statement was posed to employees in manufacturing industries, 29% of the women and 48% of the men responded Yes. To answer the research questions, two chi square tests were conducted. The first one compared the responses of women and men working in the service industry, and the second one compared the responses of women and men employed in the manufacturing industry. The results of the analyses are summarized in the following table. Is there a difference in the responses of women and men? Explain. GROUP PERCENT RESPONDING "YES" χ2 p Service Women Men 60 67 0.30 .53 Manufacturing Women Men 29 48 4.69 .03 There is NO significant difference in responses of women and men in service industry because we accept the null hypothesis in this case: obtained chi-square value is less than the critical value and p>0.5. In the second case, observed frequencies are greater than expected frequencies There is a significant difference in responses of women and men in manufacturing industry because we reject the null hypothesis in this case: obtained chi-square value is greater than the critical value and p<0.5. Unit 6 Chapter 13 RELIABILITY Fill in the blanks: 13.1. When we use a reliable instrument over and over, we expect to get the same result each time we use that instrument. 13.2. The consistency of scores obtained for the same group of people upon repeated measures is an indication of the instrument's reliability. 13.3. To assess the test-retest reliability of an instrument, the statistical test of difference between two means is used. 13.4. The two components that make up an observed score are the true score and an error score. 13.5. One way to assess a test’s reliability is to compare two alternate forms of the test with each other. 13.6. The Spearman Brown Prophecy formula is used to calculate the reliability of a full-length test by first calculating the difference or time-length between the two split halves. 13.7. When a person has a score of 85 and the test has a standard error of estimate (SEM) of 5, it means that 68% of the time the person's true score is expected to be between the scores of 75 and 88. Circle the correct answer: 13.8. Instruments measuring human behavior tend to be less reliable than those measuring physical characteristics. 13.9. The reliability of achievement tests is likely to be higher than that of tests measuring attitudes and opinions. 13.10. The test's reliability is likely to increase when the test's error component is decreased. 13.11. One way to increase the test's reliability is to increase the number of items in the test. 13.12. An instrument is administered twice to the same group of people and the correlation between the two set of scores is used to assess the reliability of the test. The reliability would be higher when the time interval between the two testing sessions is shorter. 13.13. The correlation between two split halves of a test is likely to be lower than the correlation between two alternate forms of the same test. 13.14. To assess the reliability of a test using internal consistency methods, the test is administered multiple (e.g. for test-rest) times. 13.15. To assess the inter-scorer reliability of an essay, the degree of agreement between people who score the same essay is commonly used. 13.16. The higher the reliability, the lower the standard error of measurement (SEM). 13.17. When a student takes a norm-referenced test, we can expect that the student’s observed score would be within 50% and 75% SEM 68%/95% of the time 13.18. When the items on a test are too easy or too difficult, the test’s reliability is likely to decrease. 13.19. Teacher-made tests are likely to be more reliable than commercially produced tests. 13.20. The reliability of tests used for decisions about individual students should be higher than the reliability of tests used for group decisions. Chapter 14 VALIDITY Circle the best answer: 14.1. Correlation may be used to assess the content validity of a test. a. face b. content c. criterion-related 14.2. A high correlation of a newly developed instrument with another well-established instrument measuring the same thing indicates a high concurrent validity. a. content b. concurrent c. face 14.3. The type of validity that is most important for achievement tests is the construct validity. a. face b. content c. construct 14.4. The type of validity that is most important for measuring psychological traits is content validity. a. construct b. content c. predictive Circle the correct answer: 14.5. Most tests are valid for a single purpose. 14.6. When there is a poor match between course content and a test that is used to assess students in the course, the test is likely to have a low content validity. 14.7. Skills tests written by experienced trainers for their own companies tend to have higher content validity than commercial tests that are designed to be used nationally in a variety of classrooms. 14.8. Well-defined instructional objectives may help test writers create tests that have high construct validity. 14.9. Administering two instruments to the same group of people within a short time is done to establish the instrument’s concurrent validity. Fill in the blanks: 14.10. In studies conducted to assess how well the GMAT (Graduate Management Admission Test) predicts graduate school GPA (grade point average), the GMAT is considered the quantitative variable and the GPA is the qualitative variable. 14.11. To establish the concurrent validity of a newly-developed instrument, we can compare it with a well-established instrument which measures the same thing. 14.12. Correlating the scores from a newly-developed short version of a personality inventory with a similar full-length personality inventory may be used to establish the convergent validity of the short-version inventory. 14.13. When a test simply appears to measure what it is intended to measure, we conclude that the test has a high curricular validity. 14.14. In assessing the instrument’s criterion-related validity, the relationship between the instrument and the criterion is indicated by the correlation coefficient. 14.15. When a test systematically discriminates against a group of test-takers, the test is considered invalid.