Exam chapter 15 2016, questions and answers PDF

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Chapter 15 - Correlation. T/F questions and Short Answer with Solutions...

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Chapter 15: Correlation Chapter Outline 15.1 Introduction The Characteristics of a Relationship (Direction, Form, and Degree) 15.2 The Pearson Correlation The Sum of Products of Deviations Calculation of the Pearson Correlation The Pearson Correlation and z-Scores 15.3 Using and Interpreting the Pearson Correlation Where and Why Correlations are Used Interpreting Correlations Correlation and Causation Correlation and Restricted Range Outliers Correlation and the Strength of the Relationship (r2) 15.4 Hypothesis Tests with the Pearson Correlation The Hypotheses Degrees of Freedom for the Correlation Test The Hypothesis Test In the Literature - Reporting Correlations Partial Correlations 15.5 Alternatives to the Pearson Correlation The Spearman Correlation Ranking Tied Scores Special Formula for the Spearman Correlation Testing the Significance of the Spearman Correlation The Point-Biserial Correlation and Measuring Effect Size with r2 Point-Biserial Correlation, Partial Correlation and Effect Size for the Repeated-Measures t Test The Phi-Coefficient Learning Objectives and Chapter Summary 1. Understand the Pearson correlation as a descriptive statistic that measures and describes the relationship between two variables. The Pearson correlation measures the direction and the degree of linear relationship. The sign of the correlation describes the direction of relationship; a positive correlation indicates that X and Y tend to change in the same direction, and a negative correlation indicates that X and Y tend to change in opposite directions. The numerical value of the correlation measures the strength or consistency of the relationship, and describes how Instructor Notes - Chapter 15 - page 217

closely the data points fit on a straight line. A value of 1.00 indicates a perfect fit and a value of zero indicates no fit at all. 2. Be able to compute the Pearson correlation using either the definitional or the computational formula for SP (the sum of products of deviations). The main new element in the calculation of the Pearson correlation is the sum of products of deviations, SP. However, the concept and the calculation of SP are both very similar to the concept and calculation of SS (the sum of squared deviations). To compute SS you must square X values (X times X). To compute SP you must find products of X times Y. The other “trick” to the correlation formula is to remind students to multiply the SS values in the denominator. (In many other formulas such as pooled variance, the two SS values are added.) 3. Students should understand the concept of a partial correlation and be able to follow the formula to calculate the partial correlation between two variables, holding a third variable constant. Partial correlations are valuable when researchers suspect that the true relationship between two variables may be distorted by the influence of a third variable. The partial correlation reveals the relationship while holding the third variable constant. Partial correlations also help researcher interpret the individual contributions of separate predictor variables in a multiple-regression equation. 4. Students should be able to use a sample correlation to test a hypothesis about the corresponding population correlation. The null hypothesis states that there is no correlation in the population (ρ = 0). A sample correlation (r) that is near zero provides support for the null hypothesis. On the other hand, a sample correlation that is much different than zero will provide evidence to reject H0. The hypothesis test consists of comparing the sample correlation with the critical values in the table. If the sample correlation is greater than or equal to the table value, then it is big enough to reject the null hypothesis and conclude that there is a significant, non-zero correlation in the population. 5. Students should understand the Spearman correlation and how it differs from the Pearson correlation in terms of the data that it uses and the type of relationship that it measures. The Spearman correlation measures the relationship between two variables that are both measured on an ordinal scale (both the X values and the Y values are ranks). The Spearman correlation measures the degree of consistency of direction for the relationship but does not require that the points cluster around a straight line. To compute the Spearman correlation, the Pearson formula is applied to ordinal data. 6. Students should understand how the Pearson correlation formula can be used to compute a point-biserial correlation to measure the relationship between two variable when one variable is dichotomous.

Instructor Notes - Chapter 15 - page 218

A dichotomous variable has exactly two categories (for example, male and female or succeed and fail). A point-biserial correlation measures the relationship between one dichotomous variable and one regular, numerical variable; for example the relationship between gender and reaction time scores. The two categories for the dichotomous variable are usually assigned values of 0 and 1, and then the Pearson formula is used to compute the correlation. The point-biserial correlation is strongly related to the independent-measures t hypothesis test. The value of the correlation (r) can be squared to obtain a measure of effect size (r2) for the independent-measures t. 6. Students should understand how the Pearson correlation formula can be used to compute a phi-coefficient to measure the relationship between two dichotomous variables. The phi-coefficient is used to measure the relationship between two dichotomous variables; for example, the relationship between gender (male/female) and color-blindness (yes/no). The two categories are each variable are assigned values of 0 and 1, and the Pearson formula is then used to compute the coefficient. The phi-coefficient is strongly related to the chisquare test for independence that is discussed in Chapter 18.

Other Lecture Suggestions 1. As a general introduction to correlation, you can combine the concepts of relationship and prediction in one example. Begin by supposing that grade point average is determined exclusively by IQ. The student with the highest IQ has the highest GPA, second highest IQ goes with the second highest GPA, and so on. Sketch a graph showing a perfect, straight-line relationship. Also, note that in this situation a student’s GPA is perfectly predictable (100%) from the student’s IQ score. In the real world however, it is extremely rare for one variable to entirely control another. Grade point average, for example, is partially determined by IQ, but it is also partially determined by other factors such as motivation and health. Thus, the student with the highest IQ may not have much motivation and therefore does not have the highest GPA. Start modifying the graph by moving points off the line. The result is that we now have a degree of relationship, and we find that GPA is only partially predictable from IQ. Correlation provides a method for measuring the degree of relationship and determining how much of one variable can be predicted from another. To demonstrate different degrees of relationship, you can introduce new variables. For example, GPA is also related to family income; students from wealthy families tend to have higher grade point averages than students from poorer families, but this relationship is not nearly as strong as the relationship between IQ and GPA. 2. If students are given the value of a correlation, for example r = –0.70, they should be able to sketch a scatterplot showing how the data would appear. In this case the points are scattered around a line that slopes down to the right and the cluster of points is shaped roughly like a football. (Fatter than a football indicates a correlation closer to zero, and thinner than a football indicates a correlation closer to 1.00.) Also, if students are shown a scatterplot, they should be able to estimate the value of the correlation. Instructor Notes - Chapter 15 - page 219

3. To demonstrate the concept that a sample correlation does not necessarily provide an accurate representation of the population you can propose a theory that people’s heights are related to where they live. To demonstrate your theory, get two student volunteers and ask for each student’s height and the distance between their home and the campus. Sketch a scatter plot showing the two data points. Obviously, you will always get a perfect correlation: either r = 1.00 indicating a perfect positive correlation (the farther away people live, the bigger they are) or r = -1.00 indicating a perfect negative correlation. The demonstration shows that a sample (especially a small sample) can produce a correlation that appears to be strong even when the population correlation is zero. Also, the fact that a sample of n = 2 always produces a correlation of 1.00 is a justification for using df = n  2 for the hypothesis test. 4. The following values produce whole number answers for easy classroom examples. X 3 4 0 2 1

Y 3 1 9 2 5

For these data, SP = -18, SSX = 10 and SSY = 40, producing a correlation of r = -0.90. However, the Y values can be re-ordered to produce different correlations. For example, if the Y values from top to bottom are 5, 9, 1, 2, 3, then SP = +18 and r = +0.90.

X 4 7 3 1 5 4

Y 4 7 7 2 6 4

For these data, SP = 14, SSX = 20 and SSY = 20, producing a correlation of r = 0.70. Again, the Y values can be re-ordered to produce different correlations.

Instructor Notes - Chapter 15 - page 220

Exam Items for Chapter 15 Multiple-Choice Questions 1. What is indicated by a positive value for a correlation? a. increases in X tend to be accompanied by increases in Y b. increases in X tend to be accompanied by decreases in Y c. a much stronger relationship than if the correlation were negative d. a much weaker relationship than if the correlation were negative 2. The numerical value for a correlation _____. a. can never be greater than 1.00 b. can never be less than 1.00 c. can never be greater than 1.00 and can never be less than –1.00 d. can be greater than 1.00 and can be less than –1.00 3. (www) What would the scatterplot show for data that produce a Pearson correlation of r = +.88? a. points clustered close to a line that slopes up to the right b. points clustered close to a line that slopes down to the right c. points widely scattered around a line that slopes up to the right d. points widely scattered around a line that slopes down to the right 4. The scatter plot for a set of X and Y values shows the data points clustered in a nearly perfect circle. For these data, what is the most likely value for the Pearson correlation? a. a positive correlation near 0 b. a negative correlation near 0 c. either positive or negative near 0 d. a value near +1.00 or 1.00 5. Which of the following Pearson correlations shows the greatest strength or consistency of relationship? a. –0.90 b. +0.74 c. +0.85 d. –0.33

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6. For which of the following Pearson correlations would the data points be clustered most closely around a straight line? a. r = –0.10 b. r = +0.40 c. r = –0.70 d. There is no relationship between the correlation and how close the points are to a straight line. 7. A scatter plot shows data points that are widely scattered around a line that slopes down to the right. Which of the following values would be closest to the correlation for these data? a. 0.80 b. 0.40 c. 0.40 d. 0.80 8. A college professor reports that students who finish exams early tend to get better grades than students who hold on to exams until the last possible moment. The correlation between exam score and amount of time spent on the exam is an example of a. a positive correlation. b. a negative correlation. c. a correlation near zero. d. a correlation near one. 9. The results from a research study indicate that adolescents who watch more violent content on television also tend to engage in more violent behavior than their peers. The correlation between amount of TV violent content and amount of violent behavior is an example of a. a positive correlation. b. a negative correlation. c. a correlation near zero. d. a correlation near one. 10. A researcher measures IQ and weight for a group of college students. What kind of correlation is likely to be obtained for these two variables? a. a positive correlation. b. a negative correlation. c. a correlation near zero. d. a correlation near one.

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11. (www) A researcher measures driving distance from college and weekly cost of gas for a group of commuting college students. What kind of correlation is likely to be obtained for these two variables? a. a positive correlation. b. a negative correlation. c. a correlation near zero. d. a correlation near one. 12. A researcher records the odometer reading and price for a group of used Hondas. What kind of correlation is likely to be obtained for these two variables? a. a positive correlation. b. a negative correlation. c. a correlation near zero. d. a correlation near one. 13. What is indicated by a Pearson correlation of r = +1.00 between X and Y? a. each time X increases, there is a perfectly predictable increase in Y b. every change in X causes a change in Y c. every increase in X causes an increase in Y d. All of the other 3 choices occur with a correlation of +1.00. 14. A set of n = 10 pairs of scores has ΣX = 20, ΣY = 30, and ΣXY = 74. What is the value of SP for these data? a. 74 b. 24 c. 14 d. –14 15. A set of n = 5 pairs of X and Y scores has ΣX = 15, ΣY = 5, and ΣXY = 10. For these data, the value of SP is a. –5 b. 5 c. 10 d. 25 16. (www) What is the value of SP for the following set of data? a. 5 X Y b. 5 1 4 c. 15 2 4 d. 15 9 1

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17. (www) What is the value of SP for the following set of data? a. 1 X Y b. 5 4 3 c. 5 1 2 d. None of the other 3 choices is correct. 1 5 2 6 18. A set of n = 15 pairs of scores (X and Y values) has SSX = 4, SSY = 25, and SP = 6. The Pearson correlation for these data is a. 6/100 b. 6/10 c. 6/(100/15) d. 6/(10/√15) 19. Which of the following best describes the Pearson correlation for these data? a. is positive X Y b. is negative 2 5 but it says its positive c. is zero 5 1 d. cannot be determined 3 4 4 2 20. A set of n = 5 pairs of X and Y values has SSX = 16, SSY = 4 and SP = 2. For these data, the Pearson correlation is _____. a. r = 2/64 b. r = 2/8 c. r = 2/8 d. r = 64/64 21. Which of the following sets of correlations is correctly ordered from the highest to the lowest degree of relationship? a. 0.91, +0.83, +0.10, 0.03 b. 0.91, +0.83, 0.03, 0.10 c. +0.83, +0.10, 0.91, 0.03 d. +0.83, +0.10, 0.03, 0.91 22. (www) Suppose the correlation between height and weight for adults is +0.40. What proportion (or percent) of the variability in weight can be explained by the relationship with height? a. 40% b. 60% you square it c. 16% d. 84%

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23. A set of n = 15 pairs of scores (X and Y values) produces a correlation of r = 0.40. If each of the X values is multiplied by 2 and the correlation is computed for the new scores, what value will be obtained for the new correlation? a. r = 0.20 b. r = 0.40 c. r = 0.80 d. cannot be determined without knowing all the X and Y scores 24. (www) For a two-tailed hypothesis test evaluating a Pearson correlation, what is stated by the null hypothesis? a. there is a non-zero correlation for the general population b. the population correlation is zero c. there is a non-zero correlation for the sample d. the sample correlation is zero 25. (www) If the Pearson correlation is calculated for a sample of n = 20 individuals, what value for df should be used to determine whether or not the correlation is significant? a. 18 b. 19 c. 20 d. 21 26. A researcher obtains a Pearson correlation of r = 0.43 for a sample of n = 20 participants. For a two-tailed test, which of the following accurately describes the significance of the correlation? a. The correlation is significant with α = .05 but not with α = .01. b. The correlation is significant with either α = .05 or α = .01. c. The correlation is not significant with either α = .05 or α = .01. d. There is not enough information to evaluate the significance of the correlation. 27. For a sample of n = 16 individuals, how large a Pearson correlation is necessary to be statistically significant for a two-tailed test with α = .05? a. 0.497 b. 0.482 c. 0.468 d. 0.456 28. How large a sample is needed for a correlation of r = 0.550 to be significant using a twotailed test with α = .05? a. n = 14 b. n = 13 c. n = 12 d. n = 11

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29. As the sample size gets larger, what happens to the size of the correlation that is needed for significance? a. It also gets larger. b. It gets smaller. c. It stays constant. d. There is no consistent relationship between sample size and the critical value for a significant correlation. 30. A Pearson correlation is computed for a sample of n = 18 pairs of X and Y values. What correlations are statistically significant with α = .05, two tails. a. correlations between 0.468 and –0.468 b. correlations greater than or equal to 0.468 and correlation less than or equal to –0.468 c. correlations between 0.456 and –0.456 d. correlations greater than or equal to 0.456 and correlation less than or equal to –0.456 31. If the following seven scores are ranked from smallest (#1) to largest, then what rank should be assigned to a score of X = 6? Scores: 1, 1, 3, 6, 6, 6, 9 a. 3 b. 4 c. 5 d. 6 32. (www) If the following seven scores are ranked from smallest (#1) to largest, then what rank should be assigned to a score of X = 1? Scores: 1, 1, 1, 1, 3, 6, 6, 6, 9 a. 1 b. 2 c. 2.5 d. 4 33. (www) The Pearson and the Spearman correlations are both computed for the same set of data. If the Pearson correlation is r = +1.00, then what can you conclude about the Spearman correlation? a. It will be positive. b. It will have a value of 1.00 c. It will be positive and have a value of 1.00 d. There is no predictable relationship between the Pearson and the Spearman correlations.

Instructor Notes - Chapter 15 - page 226

34. The Pearson and the Spearman correlations are both computed for the same set of data. If the Spearman correlation is rS = +1.00, then what can you conclude about the Pearson correlation? a. It will be positive. b. It will have a value of 1.00 c. It will be positive and have a value of 1.00 d. There is no predictable relationship between the Pearson and the Spearman correlations 35. What correlation is obtained when the Pearson correlation is computed for data that have been converted to ranks? a. the Spearman correlation. b. the point-biserial correlation. c. the phi coefficient. d. It is still called a Pearson correlation. 36. Under what circumstances is the phi-coefficient used? a. When one variable consists of ranks and the other is regular, numerical scores. b. When both variables consists of ranks. c. When both X and Y are dichotomous variables d. When one variable is dichotomous and the other is regular, numerical scores. 37. Under what circumstances is the point-biserial correlation used? a. in the same circumstances when a repeated-measures t test would be used b. in the same circumstances when an independent-measures t test would be used c. when both X and Y are dichotomous variables d. when both X and Y are measured on an ordinal scale (ranks) 38. The effect size for the data from an independent-measures t test can be measured by r2 which is the percentage of variance accounted for. Which of the following also produces the value for r 2? a. squaring the Spearman correlation for the same data b. squaring the point-biserial correlation for the same data c. squaring the Pearson correlation for the same data d. None of the other options will produce r2. 39. (www) Which correlation should be used to measure the relationship between gender and grade point average for a group of college students? a. Pearson corr...