Unit Set 3

1. Howell and Huessy (1981) used a rating scale to classify children in a second-grade class as showing or not showing behavior commonly associated with attention deficit disorder (ADD). The researchers then classified these same children again when they were in fourth and fifth grades. When the children reached the end of the ninth grade, the researchers examined school records and noted which children were enrolled in remedial English. For those who had never been classified as exhibiting ADD-associated behavior, 22 were enrolled in remedial English and 187 were not. For the children who had been classified at least once as exhibiting ADD-associated behavior, the corresponding figures were 19 and 74. In this study,

1. The odds of being in remedial English, given being classified at least once as ADD, are 0.118.
2. Students are 2.18 times as likely to be enrolled in a remedial English class in the ninth grade if they had ever been classified as ADD.
3. A chi-square goodness-of-fit test is the most appropriate procedure for analyzing these data.
4. A test of H _o : Classification as ADD and enrollment in remedial English are independent vs. H _a : Classification as ADD and enrollment in remedial English are not independent cannot be rejected at α = 0.05.

2. Which of the following is(are) correct ?

1. The mean squares in an ANOVA table will add, that is, MST = MSG + MSE.
2. The ANOVA F statistic tests the null hypothesis that the three sample means are equal.
3. If H _o is true, if multiple t -tests are used for comparing pairs of means, and if the critical value for t with α = 0.05 is used, the probability of rejecting at least one of the 15 H _o ’s is slightly less that 0.05.
4. In a study that examined the effects on consuming three different varieties of onions on blood cholesterol level, five golden retrievers were randomly assigned to each treatment. If α = 0.05 is used, the critical value for the F statistic is 3.74.
5. None of the above are correct.

3. In a sample of 250 undergraduate biomedical engineering students, the average score on a measure designed to measure achievement motivation ( x ) was 20. This measure can range from 0 to 30, with higher scores representing higher levels of achievement motivation. The average grade on the final exam ( y ) was 85; a total of 100 points was possible. The correlation between achievement motivation and grade on the final exam was 0.25. This correlation can be interpreted as follows:

1. Achievement motivation accounts for 25% of the variance in final exam grade.
2. About 6.25% of the variance in final exam grade is accounted for by achievement motivation.
3. Final exam grade changes 0.0625 units for every one-unit increase in achievement motivation.
4. Final exam grade changes 1 unit for every 0.25 unit increase in achievement motivation.
5. None of the above.

4. In a multiple regression involving five predictors and 206 persons, the sum of the squared residuals is 800. What is the value of the estimated standard deviation (also known as the standard error of estimate)?

1. 4.00
2. 3.88
3. 2.00
4. Not enough information is given.

5. Suppose you draw a random sample from the population and obtain a Pearson’s r of 0.50. If you use calculate a regression equation and use this equation to make predictions for participants not in your sample, some error will be involved. The amount of error would be less if:

1. your correlation were higher.
2. your sample were larger.
3. you were measuring the entire population.
4. All of the above are true.

6. Multiple regression is used to model y , using x ₁ and x ₂ . Which of the following is(are) correct?

1. It is possible that the partial regression coefficient of x ₂ could be positive in a simple regression model that only used x ₂ to predict y but negative in the multiple regression equation that uses both x ₁ and x ₂ to predict y .
2. If the F statistic for H _o : β ₁ = β ₂ = 0 has a P -value = 0.001, then we can conclude that both predictors have an effect on annual income.
3. If β ₂ = 0, then y is independent of x ₂ .
4. The multiple correlation between y and ŷ could equal -0.40.

7. For data on y = college GPA, x ₁ = high school GPA, and x ₂ = average of mathematics and verbal college entrance SAT score, we get a multiple regression equation of ŷ = 0.30 + 0.40 x ₁ + 0.003 x ₂ . Which of the following is(are) correct?

1. The correlation between y and x ₁ is moderate in size.
2. A one-unit increase in x ₁ corresponds to a change of 0.40 in the predicted value of y .
3. Controlling for x ₁ , a 100-unit increase in x ₂ corresponds to a predicted increase of 0.30 in college GPA.
4. The multiple correlation for this equation is smaller than that for the equation that simply uses high school GPA to predict college GPA.
   1. The variables y = current salary (measured in thousands of dollars), x ₁ = average course evaluation rating (which ranges from 1 = poor to 5 = excellent), and x ₂ = number of publications for the past academic year are measured for all CUNY faculty. The following prediction equations and correlations were obtained.

y = 10 + 1.0 x ₁ , r = 0.30

y = 14 + 0.4 x ₂ , r = 0.60

The correlation is -0.40 between x ₁ and x ₂ . Which of the following is(are) true?

1. The strongest association is between y and x ₁ .
2. It is possible that the standard deviation of y is 12.0 and the standard deviation of x ₁ is 3.6.
3. A standard deviation increase in average course rating corresponds to a predicted increase of 1 standard deviation in current salary.
4. When x ₁ is the predictor of y , the sum of squared residuals (SSE) is larger than when x ₂ is the predictor of y .
5. Each additional publication corresponds to a $400 increase in salary.
6. None of the above.

9. One-way ANOVA provides relatively more evidence that H _o : µ ₁ = µ ₂ = … = µ _I is false:

1. The smaller the between-groups variation and the larger the within-groups variation.
2. The smaller the between-groups variation and the smaller the within-groups variation.
3. The larger the between-groups variation and the smaller the within-groups variation.
4. The larger the between-groups variation and the larger the within-groups variation.

10. The regression equation for using x to predict y has a prediction equation of ŷ = -2.0 + 5.0 x and a correlation of 0.3. Then, the regression equation for using y to predict x

1. has a correlation of 0.3.
2. could have a negative slope.
3. does less well in predicting x than using x to predict y .
4. has an intercept of -2.0.

11. University of Rochester economist Steven Landsburg surveyed economic studies in England and the United States that showed a positive correlation between height and income. The article stated that in the United States, each one-inch increase in height was worth about $1500 extra earnings a year, on the average (Toronto Globe and Mail, 4/1/2002). The regression equation than links y = annual earnings to x = height (in inches)

1. has a y -intercept = $1500.
2. has slope 1500.
3. has slope 1/1500.
4. has correlation 0.150.

12. Imagine that you have four predictors, none of which are correlated with any of the others. If two have a 0.30 correlation with some response variable ( y ), and the other two have a 0.20 correlation with the same response variable, how much of the variance in y is accounted for by the four predictors together?

1. 0.07
2. 0.26
3. 0.51
4. 1.00
5. Not enough information is provided.

13. The slope of the least squares regression equation and the correlation are similar in the sense that

1. They both must fall between -1 and +1.
2. They both describe the strength of association.
3. They have the same t statistic value for testing H _o : The two variables are independent.
4. They both are unaffected by severe outliers.
5. None of the above

14. A student wonders if tall women tend to date taller men than do short women. She measures herself, her dormitory roommate, and four women in the adjoining dormitory rooms. Then she measures the next man each woman dates. The correlation between the heights of the men and women is 0.57.

1. If all the men were 6 inches shorter than the heights given in the table, the r > 0.57.
2. If the heights were measured in centimeters rather than inches (1 inch = 2.54 centimeters), r < 0.57.
3. If every woman dated a man exactly 3 inches taller than herself, r = 1.0.
4. All of the above are correct.
5. None of the above are correct.
   15. Below are three sets of correlations for x ₁ , x ₂ , and y . If you used x ₁ and x ₂ to predict y in a multiple regression model, which set of correlations would you expect to yield the highest R ² value? Provide the rationale for your choice. ( 2 points )
   
   
   16. For 5 groups that each has a sample size of 50, we plan to calculate multiple comparisons for all pairs of population means. We want the probability to equal at least 0.95 that the entire set of confidence intervals contain the true differences. What is the appropriate t* value that should be used in calculating each interval? ( 2 points )
   17. A researcher is investigating possible explanations for deaths in traffic accidents. He examined data from 1991 for each of the 50 states plus Washington, D.C. The data included information on the following variables:
   Deaths The number of deaths in traffic accidents
   Income The average income per family
   Children The number of children (in multiples of 100,000) between the ages of 1 and 14 in the state
   As part of his investigation he ran the following multiple regression model
   Deaths = β ₀ + β ₁ (Children) + β ₂ (Income) + є _i
   where the deviations \(\varepsilon \) _i were assumed to be independent and normally distributed with mean 0 and standard deviation \(\sigma \). This model was fit to the data using the method of least-squares. The following results were obtained from statistical software.

   Source			Sum of Squares				df
   Model		48362278			2
   Error		3042063			48
   Variable				Coefficient				Standard Error
   Constant	593.829					204.114
   Children	90.629					3.305
   Income	–0.039					0.015

   1. Suppose we wish to test the hypotheses H ₀ : β ₁ = β ₂ = 0 versus H _a : at least one of the βs is not 0, using the ANOVA F test. What is the value of the F statistic and its associated P -value? [ 2 points ]
      
   2. What is a 99% confidence interval for β ₂ , the coefficient of the variable Income? [ 2 points ]
   3. What proportion of the variation in the variable Deaths is explained by the explanatory variables Children and Income? [ 2 points ]
   4. Can you tell from this printout whether the correlation between y and x ₁ is negative or positive? Why or why not? [ 2 points ]
   5. Based on the above results, the researcher tested the hypotheses H ₀ : β ₂ = 0 versus
      H _a : β ₂ ≠ 0. What do we know about the P -value of the test? [ 1 point ]
   6. Based on the above results of the two regression analyses, what can we conclude about the value of using Income and Children to predict Deaths? [ 2 points ]

18. Do piano lessons improve the spatial-temporal reasoning of preschool children? A group of researchers examined this question by comparing the change scores for spatial-temporal reasoning (after treatment score minus before treatment score) of 34 children who took piano lessons with the scores of: 10 preschool children who were given singing lessons; 20 preschool children who received some computer instruction; and 14 preschool school who received no extra lessons. The mean change in spatial reasoning score for each group was: Piano Lessons (3.168); Singing (-0.30); Computer (0.45); and No Lesson (0.786).

1. The ANOVA summary table is presented below. Fill in the blanks for SS , df , MS , F , and p . ( 3 points )

   Source of Variation	SS	df	MS	F	p
   Type of Lesson	207.281	3	69.09	9.239	0.000029
   Error	553.437	74	7.4789
   Total	760.718	77

2. State the null and alternative hypotheses. ( 2 point s )
3. What should the researchers conclude about the null hypothesis? Be sure and provide the reasons underlying your conclusion. ( 3 point s )
4. The researchers in this study actually based their research on a biological theory that argues for a causal link between music and spatial-temporal reasoning. Perform the most appropriate contrast(s) or planned comparison(s) that best examines this contention and summarize the results. ( 4 points )

19. Zuckerman, Hodgins, Zuckerman, and Rosenthal (1993) surveyed more than 500 people and asked a number of questions on statistical issues. In one question, a reviewer warned a researcher that she had a high probability of a Type I error because she had a small sample size. The researcher disagreed. Subjects were asked, "Was the researcher correct?" The proportions of respondents, categorized as either students, assistant professors, associate professors, or full professors, who sided with the researcher and the total number of respondents in each category were as follows:

(a) Would you agree with the reviewer or the researcher? Why? ( 1 point )

(b) What is the error in the logic of the person you disagreed with in (a)? ( 1 point )

(c) Analyze these data, using the appropriate statistical procedure(s). What do these data tell you about whether there were differences among the groups of respondents and if so, the nature of these differences? ( 3 points )

20. I conduct a study examining the relationship between scores for students on a verbal aptitude test x and a mathematics aptitude test y . The mean for the verbal aptitude test is 480, with a standard deviation of 80. The mean for the mathematics aptitude test is 500, with a standard deviation of 120. The correlation between x and y is 0.60. What is the prediction equation for using verbal aptitude scores to predict math aptitude test scores? ( 3 points )

21. In Wyoming, four performance levels are used for reporting student performance on the state assessment: "novice," "partially proficient", "proficient", and "advanced." Below are presented the frequency of 11 ^th -grade students in the state who fall in each performance level on the reading portion of this assessment and on the math portion of this assessment. Data also are given for the results for a single school district (District A). Analyze the school district data to address the question as to whether the school district results depart significantly from the statewide profiles in this particular year. [ 5 points ]

Wyoming Statewide Results on the 11 ^th -Grade

Reading and Mathematics Assessments ( n = 6711)

District A Results on the 11 ^th -Grade

Reading and Mathematics Assessments ( n = 266)

22. The following descriptive statistics come from an experiment with 20 scores in each condition.

Can you conduct an ANOVA F test, or would you need more information? Provide an explanation for your choice. ( 3 points )

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