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statistics for engineers and scientists
Statistics For The Behavioral And Social Sciences A Brief Course 6th Edition Arthur Aron Elliot J Blows Elaine N Aron - Solutions
3. An education researcher has been studying test anxiety using a particular measure, which she administers to students prior to midterm exams. On this measure, she has found that the distribution follows a normal curve.Using a normal curve table, what percentage of students have Z scores(a) below
2. Using the information in problem 1 and the 50%–34%–14% figures, what is the minimum score a person has to have to be in the top (a) 2%,(b) 16%, (c) 50%, (d) 84%, and (e) 98%?
1. Suppose the people living in a particular city have a mean score of 40 and a standard deviation of 5 on a measure of concern about the environment. Assume that these concern scores are normally distributed.Using the 50%–34%–14% figures, approximately what percentage of people in this city
4. Briefly describe the steps required to figure a percentage from a Z score or vice versa (as required by the question). Be sure to draw a diagram of the normal curve with appropriate numbers and shaded areas marked on it from the relevant question (e.g., the mean, 1 and 2 standard deviations
3. Describe the link between the normal curve and the percentage of scores between the mean and any Z score. Be sure to include a description of the normal curve table and explain how it is used.
2. If necessary (that is, if required by the question), explain the mean and standard deviation (using the points in the essay outlined in Chapter 2).
1. Note that the normal curve is a mathematical (or theoretical)distribution, describe its shape (be sure to include a diagram of the normal curve), and mention that many variables in nature and in the behavioral and social sciences approximately follow a normal curve.
1. 4.5 Describe how normal curves, samples and populations, and probability are typically presented in behavioral and social science research articles
4. What is meant by p
3. Suppose people’s scores on a particular measure are normally distributed with a mean of 50 and a standard deviation of 10. If you were to pick a person completely at random, what is the probability you would pick someone with a score on this measure higher than 60?
2. Suppose you have 400 coins in a jar and 40 of them are more than 12 years old. You then mix up the coins and pull one out hoping to get one that is more than 12 years old. (a) What is the number of possible successful outcomes? (b) What is the number of all possible outcomes?(c) What is the
1. The probability of an event is defined as the expected relative frequency of a particular outcome. Explain what is meant by (a) relative frequency and (b) outcome.
3. Divide the number of possible successful outcomes (Step ) by the number of all possible outcomes (Step ): 3/6 =.5.
2. Determine the number of possible outcomes. There are six possible outcomes in the throw of a die: 1, 2, 3, 4, 5, or 6.
1. Determine the number of possible successful outcomes. There are three outcomes of 3 or lower: 1, 2, or 3.
3. Divide the number of possible successful outcomes (Step ) by the number of all possible outcomes (Step ).Let’s apply these three steps to the probability of getting a number 3 or lower on a throw of a die:
2. Determine the number of all possible outcomes.
1. Determine the number of possible successful outcomes.
4. Explain the difference between a population parameter and a sample statistic.
3. Explain the difference between random sampling and haphazard sampling.
2. Why do behavioral and social scientists usually study samples and not populations?
1. Explain the difference between the population and a sample for a research study.
6. Using the normal curve table, what Z score would you have if (a) 20%are above you, and (b) 80% are below you?
5. Using the normal curve table, what percentage of scores are (a) between the mean and a Z score of 2.14, (b) above 2.14, and (c) below 2.14?
4. Without using a normal curve table, about what Z score would a person have who is at the start of the top (a) 50%, (b) 16%, (c) 84%, and (d)2%?
3. Without using a normal curve table, about what percentage of scores on a normal curve are (a) between the mean and 2 SD above the mean, (b)below 1 SD above the mean, and (c) above 2 SD below the mean?
2. Without using a normal curve table, about what percentage of scores on a normal curve are (a) above the mean, (b) between the mean and 1 SD above the mean, (c) between 1 and 2 SD above the mean, (d) below the mean, (e) between the mean and 1 SD below the mean, and (f) between 1 and 2 SD below
1. Why is the normal curve (or at least a curve that is symmetrical and unimodal) so common in nature?
5. If you want to find a raw score, change it from the Z score. For the high end, using the usual formula, X=(1.96)(16)+100=131.36. For the low end, X=(−1.96)(16)+100=68.64. In sum, the middle 95% of IQ scores run from 68.64 to 131.36.I. How are you doing?Answers can be found at the end of this
4. Check that your exact Z score is within the range of your rough estimate from Step . As we estimated, +1.96 is between +1 and +2 and is very close to +2, and −1.96 is between −1 and −2 and very close to −2.
3. Find the exact Z score using the normal curve table (subtracting 50%from your percentage if necessary before looking up the Z score). Being in the top 2.5% means that 2.5% of the IQ scores are in the upper tail. In the normal curve table, the closest percentage to 2.5% in the “% in
2. Make a rough estimate of the Z score where the shaded area stops. You can see from the picture that the Z score for where the shaded area stops above the mean is just below +2. Similarly, the Z score for where the shaded area stops below the mean is just above −2.
1. Draw a picture of the normal curve, and shade in the approximate area for your percentage using the 50%–34%–14% percentages. Let’s start where the top 2.5% begins. This point has to be higher than 1 SD (16%of scores are higher than 1 SD). However, it cannot start above 2 SD because there
5. If you want to find a raw score, change it from the Z score. Use the usual formula, X=(Z)(SD)+M.
4. Check that your exact Z score is within the range of your rough estimate from Step .
3. Find the exact Z score using the normal curve table (subtracting 50%from your percentage if necessary before looking up the Z score).Looking at your picture, figure out either the percentage in the shaded tail or the percentage between the mean and where the shading stops.For example, if your
2. Make a rough estimate of the Z score where the shaded area stops.
1. Draw a picture of the normal curve and shade in the approximate area for your percentage using the 50%–34%–14% percentages.
3. Make a rough estimate of the shaded area’s percentage based on the 50%–34%–14% percentages. If the shaded area started at a Z score of 1, it would have 16% above it. If it started at a Z score of 2, it would have only 2% above it. So, with a Z score of 1.56, it has to be somewhere between
2. Draw a picture of the normal curve, decide where the Z score falls on it, and shade in the area for which you are finding the percentage. This is shown in Figure 4–5 (along with the exact percentages figured later).
1. If you are beginning with a raw score, first change it to a Z score. Using the usual formula, Z=(X−M)/SD,Z=(125−100)/16=+1.56.
23. ADVANCED TOPIC: Hahlweg, Fiegenbaum, Frank, Schroeder, and von Witzleben (2001) carried out a study of a treatment method for agoraphobia, a condition that affects about 4% of the population and involves unpredictable panic attacks in public spaces such as shopping malls, buses, or movie
22. Ask five other students of the same gender as yourself (each from different families) to give you their own height and also their mother’s height. Based on the numbers these five people give you, (a) figure the correlation coefficient, and (b) determine the Z-score prediction model for
21. A researcher studying adjustment to the job of new employees found a correlation of .30 between amount of employees’ education and rating by job supervisors 2 months later. The researcher now plans to use amount of education to predict supervisors’ later ratings of employees.Indicate the
20. Arbitrarily select eight people from a news website or newspaper. Do each of the following: (a) make a scatter diagram for the relation between the number of letters in each person’s first and last name, (b)figure the correlation coefficient for the relation between the number of letters in
19. Gable and Lutz (2000) studied 65 children, 3–10 years old, and their parents. One of their results was: “Parental control of child eating showed a negative association with children’s participation in extracurricular activities (r=.34; p
18. As part of a larger study, Speed and Gangestad (1997) collected ratings and nominations on a number of characteristics for 66 fraternity men from their fellow fraternity members. The following paragraph is taken from their “Results” section:… men’s romantic popularity significantly
17. Five university students were asked about how important a goal it is to them to have a family and about how important a goal it is for them to be highly successful in their work. Each variable was measured on a scale from 0 “Not at all important goal” to 10 “Very important goal.”(For
16. Four research participants take a test of manual dexterity (high scores mean better dexterity) and an anxiety test (high scores mean more anxiety). (For part (f), assume that dexterity is the predictor variable.)The scores are as follows:Person Dexterity Anxiety A 1 10 B 1 8 C 2 4 D 4 −2
15. In a study of people first getting acquainted with each other, researchers reasoned the amount of self-disclosure of one’s partner (on a scale from 1 to 30) and one’s liking for one’s partner (on a scale from 1 to 10). The Z scores for the variables are given to save you some figuring.
14. A researcher studying people in their 80s was interested in the relation between number of very close friends and overall health (on a scale from 0 to 100). The scores for six research participants are shown below.Research Participant Number of Friends Overall Health A 2 41 B 4 72 C 0 37 D 3 84
13. Make up a scatter diagram with 10 dots for each of the following situations: (a) perfect positive linear correlation, (b) strong but not perfect positive linear correlation, (c) weak positive linear correlation,(d) strong but not perfect negative linear correlation, (e) no correlation,(f) clear
10. ADVANCED TOPIC: Buboltz, Johnson, and Woller (2003) conducted a study of the relationship between various aspects of college students’family relationships and students’ level of “psychological reactance.”“Reactance” in this study referred to a tendency to have an extreme reaction
9. A researcher working with hockey players found that knowledge of fitness training principles correlates .40 with number of injuries received over the subsequent year. The researcher now plans to test all new athletes for their knowledge of fitness training principles and use this information to
4. In multiple regression, why are the standardized regression coefficients for each predictor variable often smaller than the ordinary correlation coefficient of that predictor variable with the criterion variable?
3. In a multiple regression model, the standardized regression coefficient for the first predictor variable is .40 and for the second predictor variable is .70. What is the predicted criterion variable Z score for (a) a person with a Z score of +1 on the first predictor variable and a Z score of +2
2. Write the multiple regression prediction model with two predictors, and define each of the symbols.
1. What is multiple regression?
4. For a variable X, the mean is 20 and the standard deviation is 5. For a variable Y, the mean is 6 and the standard deviation is 2. The correlation of X and Y is .80. (a) Predict the score on Y for a person who has a score on X of 20. (b) Predict the score on Y for a person who has a score on X
3. For a variable X, the mean is 10 and the standard deviation is 3. For a variable Y, the mean is 100 and the standard deviation is 10. The correlation of X and Y is .60. (a) Predict the score on Y for a person who has a score on X of 16. (b) Predict the score on Y for a person who has a score on
2. List the steps of making predictions using raw scores.
1. Explain the principle behind prediction using raw scores.
3. Write the formula for the prediction model using Z scores, and define each of the symbols.
2. Why does the standardized regression coefficient have this name? That is, explain the meaning of each of the three words that make up the term:standardized, regression, and coefficient.
1. In words, what is the prediction model using Z scores?
2. Multiply the standardized regression coefficient (β) by the person’s Z score on the predictor variable. Your Z score on the predictor variable is–1.84. Multiplying .85 by –1.84 gives a predicted Z score on happy mood of –1.56.
1. Determine the standardized regression coefficient (β). Because the correlation coefficient is .85, the standardized regression coefficient (β)is also .85.
2. Multiply the standardized regression coefficient (β) by the person’s Z score on the predictor variable. In the example, the person’s Z score on the predictor variable is 2. Multiplying .30 by 2 gives .60. Thus, .60 is the person’s predicted Z score on the criterion variable (college GPA).
1. Determine the standardized regression coefficient (β). In the example, it was .30.
2. Multiply the standardized regression coefficient (β) by the person’s Z score on the predictor variable.We can illustrate the steps using the same example as above for predicting college GPA of a person at your school with an entering SAT 2 standard deviations above the mean.
1. Determine the standardized regression coefficient (β).
4. How much stronger is a correlation of r=.60 compared to a correlation of r=.30?
3. What does it mean to say that a particular correlation coefficient is statistically significant?
2. A researcher randomly assigns participants to eat zero, two, or four cookies and then asks them how full they feel. The number of cookies eaten and feeling full are highly correlated. What directions of causality can and cannot be ruled out?
1. If anxiety and depression are correlated, what are three possible directions of causality that might explain this correlation?
4. Figure the correlation coefficient for the Z scores shown below for three people who were each tested on two variables, X and Y.Person ZX ZY K .5 −.7 L −1.4 −.8 M .9 1.5
3. Write the formula for the correlation coefficient and define each of the symbols.
2. When figuring the correlation coefficient, why do you divide the sum of cross-products of Z scores by the number of people in the study?
1. Give two reasons why we use Z scores for figuring the exact linear correlation between two variables, thinking of correlation as how much high scores go with high scores and lows go with lows (or vice versa for negative correlations).
3. Mark a dot for each pair of scores. The completed scatter diagram is shown in Figure 3–12.Figure 3–
2. Determine the range of values to use for each variable and mark them on the axes. We will assume that the achievement test scores go from 0 to 100. Class size has to be at least 1 (thus close to zero so we can use the standard of starting our axes at zero) and in this example we guessed that it
1. Draw the axes and decide which variable goes on which axis. Because it seems more reasonable to think of class size as affecting achievement test scores rather than the other way around, we will draw the axes with class size along the bottom.
3. What is the difference between a positive and negative linear
2. What does it mean to say that two variables have no correlation?
1. What is the difference between a linear and curvilinear correlation in terms of how they appear in a scatter diagram?
1. 3.2 Identify the pattern of correlation for a given scatter diagram There are many different patterns of correlation. We focus here on the most common types of correlation
3. Make a scatter diagram for the following scores for four people who were each tested on two variables, X and Y. X is the variable we are predicting from; it can have scores ranging from 0 to 6. Y is the variable being predicted; it can have scores from 0 to 7.Person X Y A 3 4 B 6 7 C 1 2 D 4 6
2. (a) For a study in which one variable can be thought of as predicting another variable, which variable goes on the horizontal axis? (b) Which variable goes on the vertical axis?
1. What does a scatter diagram show, and what does it consist of?
3. Mark a dot for each pair of scores. For the first student, the number of hours slept last night was 5. Move across to 5 on the horizontal axis. The happy mood rating for the first student was 2, so move up to the point across from the 2 on the vertical axis. Place a dot at this point, as shown
2. Determine the range of values to use for each variable and mark them on the axes. For the horizontal axis, we start at 0 as usual. We do not know the maximum possible, but let us assume that students rarely sleep more than 12 hours. The vertical axis goes from 0 to 8, the lowest and highest
24. Selwyn (2007) conducted a study of gender-related perceptions of information and communication technologies (such as games machines, DVD players, and cell phones). The researchers asked 406 university students in Wales to rate eight technologies in terms of their level of masculinity or
23. A study involves measuring the number of days absent from work for 216 employees of a large company during the preceding year. As part of the results, the researcher reports, “The number of days absent during the preceding year (M=9.21; SD=7.34) … .” Explain the material in parentheses to
22. A person scores 81 on a test of verbal ability and 6.4 on a test of math ability. For the verbal ability test, the mean for people in general is 50 and the standard deviation is 20. For the math ability test, the mean for people in general is 0 and the standard deviation is 5. Which is this
21. On a standard measure of peer influence among adolescents, the mean is 300 and the standard deviation is 20. Give the Z scores for adolescents who score (a) 340, (b) 310, and (c) 260. Give the raw scores for adolescents whose Z scores on this measure are (d) 2.4, (e) 1.5, (f) 0, and(g) −4.5.
20. On a measure of artistic ability, the mean for university students in New Zealand is 150 and the standard deviation is 25. Give the Z scores for New Zealand university students who score (a) 100, (b) 120, (c) 140, and (d) 160. Give the raw scores for students whose Z scores on this test are (e)
19. A developmental specialist studies the number of words seven infants have learned at a particular age. The numbers are 10, 12, 8, 0, 3, 40, and 18. Figure the (a) mean, (b) median, and (c) standard deviation for the number of words learned by these seven infants. (d) Explain what you have done
18. A researcher interested in political behavior measured the square footage of the desks in the official office of four U.S. governors and of four chief executive officers (CEOs) of major U.S. corporations. The figures for the governors were 44, 36, 52, and 40 square feet. The figures for the
17. Make up three sets of scores: (a) one with the mean greater than the median, (b) one with the median and the mean the same, and (c) one with the mode greater than the median. (Each made-up set of scores should include at least five scores.)
15. For the following scores, find the (a) mean, (b) median, (c) sum of squared deviations, (d) variance, and (e) standard deviation:3.0, 3.4, 2.6, 3.3, 3.5, 3.2.16. For the following scores, find the (a) mean, (b) median, (c) sum of squared deviations, (d) variance, and (e) standard deviation:8,
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