Get questions and answers for Essentials of Statistics

Use these “enjoyment” data to perform the following:

a. Calculate the cell and marginal means.

b. Draw a bar graph.

c. Calculate the five different degrees of freedom, and indicate the critical F value based on each set of degrees of freedom, assuming the p level is 0.01.

d. Calculate the total sum of squares.

e. Calculate the between-groups sum of squares for the independent variable gender.

f. Calculate the between-groups sum of squares for the independent variable sporting event.

g. Calculate the within-groups sum of squares.

h. Calculate the sum of squares for the interaction.

i. Create a source table.

Use these data—incidents of reports of underage drinking—to perform the following:

“Dry” campus, state school: 47, 52, 27, 50

“Dry” campus, private school: 25, 33, 31

“Wet” campus, state school: 77, 61, 55, 48

“Wet” campus, private school: 52, 68, 60

a. Calculate the cell and marginal means. Notice the unequal Ns.

b. Draw a bar graph.

c. Calculate the five different degrees of freedom, and indicate the critical F value based on each set of degrees of freedom, assuming the p level is 0.05.

d. Calculate the total sum of squares.

e. Calculate the between-groups sum of squares for the independent variable campus.

f. Calculate the between-groups sum of squares for the independent variable school.

g. Calculate the within-groups sum of squares.

h. Calculate the sum of squares for the interaction.

i. Create a source table. 

Using what you know about the expanded source table, fill in the missing values in the table shown here:

Using the information in the source table provided here, compute R²values for each effect. Using Cohen’s conventions, explain what these values mean.

Using the information in the source table provided here, compute R²values for each effect. Using Cohen’s conventions, explain what these values mean.

A researcher wondered about the degree to which age was a factor for those posting personal ads on Match.com. He randomly selected 200 ads and examined data about the posters (the people who posted the ads). Specifically, for each ad, he calculated the difference between the poster’s age and the oldest acceptable age in a romantic prospect. So, if someone were 23 years old and would date someone as old as 30, his or her score would be 7; if someone were 25 and would date someone as old as 23, his or her score would be −2. The researcher then categorized the scores into male versus female and seeking a same-sex date versus seeking an opposite-sex date.

a. List any independent variables, along with the levels.

b. What is the dependent variable?

c. What kind of ANOVA would he use?

d. Now name the ANOVA using the more specific language that enumerates the numbers of levels.

e. Use your answer to part (d) to calculate the number of cells. Explain how you made this calculation.

f. Draw a table that depicts the cells of this ANOVA. 

In a study of racism, Nail, Harton, and Decker (2003) had participants read a scenario in which a police officer assaulted a motorist. Half the participants read about an African American officer who assaulted a European American motorist, and half read about a European American officer who assaulted an African American motorist. Participants were categorized based on political orientation: liberal, moderate, or conservative. Participants were told that the officer was acquitted of assault charges in state court but was found guilty of violating the motorist’s rights in federal court. Double jeopardy occurs when an individual is tried twice for the same crime. Participants were asked to rate, on a scale of 1–7, the degree to which the officer had been placed in double jeopardy by the second trial. The researchers reported the interaction as F(2, 58) = 10.93, p < 0.0001. The means for the liberal participants were 3.18 for those who read about the African American officer and 1.91 for those who read about the European American officer. The means for the moderate participants were 3.50 for those who read about the African American officer and 3.33 for those who read about the European American officer. The means for the conservative participants were 1.25 for those who read about the African American officer and 4.62 for those who read about the European American officer.

a. Draw a table of cell means that includes the actual means for this study.

b. Do the reported statistics indicate that there is a significant interaction? If yes, describe the interaction in your own words.

c. Draw a bar graph that depicts the interaction. Include lines that connect the tops of the bars and show the pattern of the interaction.

d. Is this a quantitative or qualitative interaction? Explain.

e. Change the cell mean for the conservative participants who read about an African American officer so that this is now a quantitative interaction.

f. Draw a bar graph that depicts the pattern that includes the new cell means.

g. Change the cell means for the moderate and conservative participants who read about an African American officer so that there is now no interaction.

h. Draw a bar graph that depicts the pattern that includes the new cell means.

Ratner and Miller (2001) wondered whether people are uncomfortable when they act in a way that’s not obviously in their own self-interest. They randomly assigned 33 women and 32 men to read a fictional passage saying that federal funding would soon be cut for research into a gastrointestinal illness that mostly affected either (1) women or (2) men. They were then asked to rate, on a 1–7 scale, how comfortable they would be “attending a meeting of concerned citizens who share your position” on this cause (p. 11). A higher rating indicates a greater degree of comfort. The journal article reported the statistics for the interaction as F(1, 58) = 9.83, p < 0.01. Women who read about women had a mean of 4.88, whereas those who read about men had a mean of 3.56. Men who read about women had a mean of 3.29, whereas those who read about men had a mean of 4.67.

a. What are the independent variables and their levels? What is the dependent variable?

b. What kind of ANOVA did the researchers conduct?

c. Do the reported statistics indicate that there is a significant interaction? Explain your answer.

d. Draw a table that includes the cells of the study. Include the cell means.

e. Draw a bar graph that depicts these findings.

f. Describe the pattern of the interaction in words. Is this a qualitative or a quantitative interaction? Explain your answer.

g. Draw a new table of cells, but change the means for male participants reading about women so that there is now a quantitative, rather than a qualitative, interaction.

h. Draw a bar graph of the means in part (g).

i. Draw a new table of cells, but change the means for male participants reading about women so that there is no interaction.

Eleanor Barkhorn (2012) reported in the Atlantic about differences in women’s and men’s negotiating styles. She first explained that researchers did not find a significant difference in how likely women and men are to negotiate salaries. But this did not tell the whole story. Barkhorn wrote: “Women are more likely to negotiate when an employer explicitly says that wages are negotiable. Men, on the other hand, are more likely to negotiate when the employer does not directly state that they can negotiate.” For each of the following, state whether the finding is a result of examining a main effect or examining an interaction. Explain your answer.

a. The finding that women and men do not significantly differ, on average, in their likelihood of negotiating.

b. The finding of a gender difference in the circumstances under which one will negotiate.

Hugenberg, Miller, and Claypool (2007) conducted a study to better understand the cross-race effect, in which people have a difficult time recognizing members of different racial groups—colloquially known as the “they all look the same to me” effect. In a variation on this study, white participants viewed either 20 black faces or 20 white faces for 3 seconds each. Half the participants were told to pay particular attention to distinguishing features of the faces. Later, participants were shown 40 black faces or 40 white faces (the same race that they were shown in the prior stage of the experiment), 20 of which were new. Each participant received a score that measured his or her recognition accuracy. The researchers reported two effects, one for the race of the people in the pictures, F(1, 136) = 23.06, p < 0.001, such that white faces were more easily recognized, on average, than black faces. There also was a significant interaction of the race of the people in the pictures and the instructions, F(1, 136) = 5.27, p < 0.05. When given no instructions, the mean recognition scores were 1.46 for white faces and 1.04 for black faces. When given instructions to pay attention to distinguishing features, the mean recognition scores were 1.38 for white faces and 1.23 for black faces.

a. What are the independent variables and their levels? What is the dependent variable?

b. What kind of ANOVA did the researchers conduct?

c. Do the reported statistics indicate that there is a significant main effect? If yes, describe it.

d. Why is the main effect not sufficient in this situation to understand the findings? Be specific about why the main effect is misleading by itself.

e. Do the reported statistics indicate that there is a significant interaction? Explain your answer.

f. Draw a table that includes the cells of the study and the cell means.

g. Draw a bar graph that depicts these findings.

h. Describe the pattern of the interaction in words. Is this a qualitative or a quantitative interaction? Explain your answer.

A sample of students from our statistics classes reported their GPAs, indicated their genders, and stated whether they were in the university’s Greek system (i.e., in a fraternity or sorority). Following are the GPAs for the different groups of students:

Men in a fraternity: 2.6, 2.4, 2.9, 3.0

Men not in a fraternity: 3.0, 2.9, 3.4, 3.7, 3.0

Women in a sorority: 3.1, 3.0, 3.2, 2.9

Women not in a sorority: 3.4, 3.0, 3.1, 3.1

a. What are the independent variables and their levels? What is the dependent variable?

b. Draw a table that lists the cells of the study design. Include the cell means.

c. Conduct all six steps of hypothesis testing.

d. Draw a bar graph for all statistically significant effects.

e. Is there a significant interaction? If yes, describe it in words and indicate whether it is a qualitative or a quantitative interaction. Explain.

f. Compute the effect sizes, R², for the main effects and interaction. Using Cohen’s conventions, interpret the effect-size values. 

The data below were from the same 25-yearold participants described in How It Works 12.1, but now the scores represent the oldest age that would be acceptable in a dating partner.

25-year-old women seeking men: 40, 35, 29, 35, 35

25-year-old men seeking women: 26, 26, 28, 28, 28

25-year-old women seeking women: 35, 35, 30, 35, 45

25-year-old men seeking men: 33, 35, 35, 36, 38

a. What are the independent variables and their levels? What is the dependent variable?

b. Draw a table that lists the cells of the study design. Include the cell means.

c. Conduct all six steps of hypothesis testing.

d. Is there a significant interaction? If yes, describe it in words, indicate whether it is a quantitative or a qualitative interaction, and draw a bar graph.

e. Compute the effect sizes, R², for the main effects and interaction. Using Cohen’s conventions, interpret the effect-size values.

Heyman and Ariely (2004) were interested in whether effort and willingness to help were affected by the form and amount of payment offered in return for effort. They predicted that when money was used as payment, in what is called a money market, effort would increase as a function of payment level. On the other hand, if effort were performed out of altruism, in what is called a social market, the level of effort would be consistently high and unaffected by level of payment. In one of their studies, college students were asked to estimate another student’s willingness to help load a sofa into a van in return for a cash payment or candy of equivalent value. Willingness to help was assessed using an 11-point scale ranging from “Not at all likely to help” to “Will help for sure.” Data are presented here to re-create some of their findings.

Cash payment, low amount of $0.50: 4, 5, 6, 4

Cash payment, moderate amount of $5.00: 7, 8, 8, 7

Candy payment, low amount valued at $0.50: 6, 5, 7, 7

Candy payment, moderate amount valued at $5.00: 8, 6, 5, 5

a. What are the independent variables and their levels?

b. What is the dependent variable?

c. Draw a table that lists the cells of the study design. Include the cell and marginal means.

d. Create a bar graph.

e. Using this graph and the table of cell means, describe what effects you see in the pattern of the data.

f. Write the null and research hypotheses.

g. Complete all of the calculations and construct a full source table for these data.

h. Determine the critical value for each effect at a p level of 0.05.

i. Make your decisions. Is there a significant interaction? If yes, describe it in words and indicate whether it is a qualitative or a quantitative interaction. Explain.

j. Compute the effect sizes, R², for the main effects and interaction. Using Cohen’s conventions, interpret the effect-size values. 

German psychologist David Loschelder and his colleagues conducted an experiment on negotiations (2014). They cited tennis player Andy Roddick’s agent who thought it was always detrimental to make an initial offer, saying “The first offer gives you an insight into their [the other party’s] thought process.” The researchers wondered if this was always true. So, they conducted an experiment with two independent variables. One independent variable was the person’s role in the negotiations—either the person starting the negotiation (the sender) or the person being targeted (the receiver). The second independent variable was the type of information in the initial offer— different or the same. That is, the sender was either asking for something that is different from what the other partner wants or asking for something that the other person also wants. For example, if you are negotiating with a new employer, you might ask for five weeks of vacation and a higher salary than you think you can get. And maybe the employer was already prepared to give you five weeks vacation. So, the researchers thought the type of information that matches what the other person wants (like the information about vacation time) might give the receiver a bargaining chip. Knowing what the sender really wants might let you lowball on other aspects of the negotiation. So, the employer can then grant the vacation time, and perhaps not have to offer the higher salary. The graph here depicts the results of this experiment, in which success in the negotiation was measured in the percentage of a pool of money that could be earned.

a. Based on this graph, what type of ANOVA did the researchers conduct?

b. Does it seem as if there’s a main effect of role in the negotiations (sender or receiver)? If yes, explain the effect in your own words. If not, explain your answer.

c. Does it seem as if there’s a main effect of type of information provided? If yes, explain the effect in your own words? If not, explain your answer.

d. Describe the interaction in your own words. Is this

a quantitative interaction or a qualitative interaction? Explain your answer.

e. Based on what you learned about graphing in Chapter 3, explain an important problem with the y-axis

A study on motivated skepticism examined whether participants were more likely to be skeptical when it served their self-interest (Ditto & Lopez, 1992). Ninety-three participants completed a fictitious medical test that told them they had high levels of a certain enzyme, TAA. Participants were randomly assigned to be told either that high levels of TAA had potentially unhealthy consequences or potentially healthy consequences. They were also randomly assigned to complete a dependent measure before or after the TAA test. The dependent measure assessed their perception of the accuracy of the TAA test on a scale of 1 (very inaccurate) to 9 (very accurate). Ditto and Lopez found the following means for those who completed the dependent measure before taking the TAA test: unhealthy result, 6.6; healthy result, 6.9. They found the following means for those who completed the dependent measure after taking the TAA test: unhealthy result, 5.6; healthy result, 7.3. From their ANOVA, they reported statistics for two findings. For the main effect of test outcome, they reported the following statistic: F(1,73) = 7.74, p < 0.01. For the interaction of test outcome and timing of the dependent measure, they reported the following statistic: F(1, 73) = 4.01, p < 0.05.

a. State the independent variables and their levels. State the dependent variable.

b. What kind of ANOVA would be used to analyze these data? State the name using the original language as well as the more specific language.

c. Use the more specific language of ANOVA to calculate the number of cells in this research design.

d. Draw a table of cell means, marginal means, and the grand mean. Assume that equal numbers of participants were assigned to each cell (even though this was not the case in the actual study).

e. Describe the significant main effect in your own words.

f. Draw a bar graph that depicts the main effect.

g. Why is the main effect misleading by itself?

h. Is the main effect qualified by a statistically significant interaction? Explain. Describe the interaction in your own words.

i. Draw a bar graph that depicts the interaction. Include lines that connect the tops of the bars and show the pattern of the interaction.

j. Is this a quantitative or qualitative interaction? Explain.

k. Change the cell mean for the participants who had a healthy test outcome and completed the dependent measure before the TAA test so that this is now a qualitative interaction.

l. Draw a bar graph depicting the pattern that includes the new cell mean.

m. Change the cell mean for the participants who had a healthy test outcome and completed the dependent measure before the TAA test so that there is now no interaction.

n. Draw a bar graph that depicts the pattern that includes the new cell mean. 

In 2013, Forbes reported the 10 top-earning comedians, and all 10 were men—Louis C.K., Kevin Hart, and Larry the Cable Guy among them. A number of online journalists wanted to know why there were no women on the list. Erin Gloria Ryan of Jezebel, for example, wondered where women like Ellen DeGeneres and Amy Poehler were. Under a “methodology” heading, Forbes explained the process of gathering these data. “To compile our earnings numbers, which consist of pretax gross income, we talked to agents, lawyers and other industry insiders to come up with an estimate for what each comedian earned.” Forbes added a caveat: “In order for comics to make the cut, their primary source of income had to come from concert ticket sales.” In response to that caveat, Ryan wrote: “Okay, guys? That’s a super weird definition of what constitutes a comedian.”

a. Explain how Forbes is operationalizing the earnings of comedians.

b. Explain why Jezebel’s Ryan might have a problem with this definition.

c. Ryan wrote: “If Dr. Dre doesn’t have to record an album or perform a concert to be considered a real ‘hip hop artist,’ then why does Mindy Kaling need to hold a mic in front of a brick wall to be a real ‘comedian’?” Based on Ryan’s critique, offer at least one different way of operationalizing the earnings of comedians.

Noting marked increases in weight across the population, researchers, nutritionists, and physicians have struggled to find ways to stem the tide of obesity in many Western countries. They have advocated a number of exercise programs, and there has been a flurry of research to determine the effectiveness of these programs. Pretend that you are in charge of a research study to examine the effects of an exercise program on weight loss in comparison with a program that does not involve exercise.

a. Describe how you could study the exercise program using a between-groups research design.

b. Describe how you could study the exercise program using a within-groups design.

c. What is a potential confound of a within-groups design?

For decades, researchers, politicians, and tobacco company executives debated the relation between smoking and health problems such as cancer.

a. Why was this research necessarily correlational in nature?

b. What confounding variables might make it difficult to isolate the health effects of smoking tobacco?

c. How might the nature of this research and these confounds buy time for the tobacco industry with regard to acknowledging the hazardous effects of smoking?

d. All ethics aside, how could you study the relation between smoking and health problems using a between-groups experiment?

A researcher interested in the cultural values of individualistic and collectivist societies collects data on the rate of relationship conflict experienced by 32 people who test high for individualism and 37 people who test high for collectivism.

a. Is this research experimental or correlational? Explain.

b. What is the sample?

c. Write a possible hypothesis for this researcher.

d. How might we operationalize relationship conflict?

The following statements are wrong but can be corrected by substituting one word or phrase. (See the instructions in Exercise 1.13.) Identify the incorrect word or phrase in each of the following statements and supply the correct word.

a. A researcher examined the effect of the ordinal variable “gender” on the scale variable “hours of

reality television watched per week.”

b. A psychologist used a between-groups design to study the effects of an independent variable (a workout video) on the dependent variable (the weight) of a group of undergraduate students before and after viewing the video.

c. In a study on the effects of the confounding variable of noise level on the dependent variable of memory, researchers were concerned that the memory measure was not valid.

d. A researcher studied a population of 20 rats to determine whether changes in exposure to light would lead to changes in the dependent variable of amount of sleep. 

In the fall of 2008, the U.S. stock market plummeted several times, with grave consequences for the world economy. A researcher might assess the economic effect this situation had by seeing how much money people saved in 2009. That amount could be compared to the amount people saved in more economically stable years. How might you operationalize the economic implications at a national level? 

What is the relation between an independent variable and a dependent variable?

What is the difference between reliability and validity, and how are the two concepts related? 

Report the following numbers to two decimal places: 0.0391; 198.2219; and 17.886. 

A population has a mean of 250 and a standard deviation of 47. Calculate z scores for each of the following raw scores:

a. 391

b. 273

c. 199

d. 160 

What is the difference between a null hypothesis and a research hypothesis? 

Create a histogram for these three sets of scores. Each set of scores represents a sample taken from the same population.

a. 6 4 11 7 7

b. 6 4 11 7 7 2 10 7 8 6 6 7 5 8

c.

d. What do you observe happening across these three distributions?

For a population with a mean of 250 and a standard deviation of 47, convert each of the following z scores to raw scores.

a. 0.54

b. −2.66

c. −1.00

d. 1.79 

For a population with a mean of 1179 and a standard deviation of 164, convert each of the following z scores to raw scores.

a. −0.23

b. 1.41

c. 2.06

d. 0.03 

A study of the Consideration of Future Consequences (CFC) scale found a mean score of 3.20, with a standard deviation of 0.70, for the 800 students in the sample (Adams, 2012). (Treat this sample as the entire population of interest.)

a. If the CFC score is 4.2, what is the z score? Use symbolic notation and the formula. Explain why this answer makes sense.

b. If the CFC score is 3.0, what is the z score? Use symbolic notation and the formula. Explain why this answer makes sense.

c. If the z score is 0, what is the CFC score? Explain. 

Using the instructions in Example 6.8, compare the following “apples and oranges”: a score of 45 when the population mean is 51 and the standard deviation is 4, and a score of 732 when the population mean is 765 and the standard deviation is 23.

a. Convert these scores to standardized scores.

b. Using the standardized scores, what can you say about how these two scores compare to each other?

Compare the following scores:

a. A score of 811 when μ = 800 and σ = 29 against a score of 4524 when μ = 3127 and σ = 951

b. A score of 17 when μ = 30 and σ = 12 against a score of 67 when μ = 88 and σ = 16 

Assume a normal distribution when answering the following questions.

a. What percentage of scores falls below the mean?

b. What percentage of scores falls between 1 standard deviation below the mean and 2 standard deviations above the mean?

c. What percentage of scores lies beyond 2 standard deviations away from the mean (on both sides)?

d. What percentage of scores is between the mean and 2 standard deviations above the mean?

e. What percentage of scores falls under the normal curve? 

A population has a mean of 55 and a standard deviation of 8. Compute μ_M and σ_M for each of the following sample sizes:

a. 30

b. 300

c. 3000

Compute a z statistic for each of the following, assuming the population has a mean of 100 and a standard deviation of 20:

a. A sample of 43 scores has a mean of 101.

b. A sample of 60 scores has a mean of 96.

c. A sample of 29 scores has a mean of 100.

A sample of 100 people had a mean depression score of 85; the population mean for this depression measure is 80, with a standard deviation of 20. A different sample of 100 people had a mean score of 17 on a different depression measure; the population mean for this measure is 15, with a standard deviation of 5.

a. Convert these means to z statistics.

b. Using the z statistics, what can you say about how these two means compare to each other? 

See the description of the MMPI-2 in Exercise 6.47. The mean T score is always 50, and the standard deviation is always 10. Imagine that you administer the MMPI-2 to 50 respondents who do not use Instagram or any other social media; you wonder whether their scores on the social introversion scale will be, on average, higher than the norms. You find a mean score on the social introversion scale of 60 in your sample.

a. Using symbolic notation, report the mean and standard deviation of the population.

b. Using symbolic notation and formulas (where appropriate), report the mean and standard error for the distribution of means to which your sample will be compared.

c. In your own words, explain why it makes sense that the standard error is smaller than the standard deviation.

The General Social Survey (GSS) is a survey of approximately 2000 adults conducted each year since 1972, for a total of more than 38,000 participants. During several years of the GSS, participants were asked how many close friends they have. The mean for this variable is 7.44 friends, with a standard deviation of 10.98. The median is 5.00 and the mode is 4.00.

a. Are these data for a distribution of scores or a distribution of means? Explain.

b. What do the mean and standard deviation suggest about the shape of the distribution? 

c. What do the three measures of central tendency suggest about the shape of the distribution?

d. Let’s say that these data represent the entire population. Pretend that you randomly selected a person from this population and asked how many close friends she or he had. Would you compare this person to a distribution of scores or to a distribution of means? Explain your answer.

e. Now pretend that you randomly selected a sample of 80 people from this population. Would you compare this sample to a distribution of scores or to a distribution of means? Explain your answer.

f. Using symbolic notation, calculate the mean and standard error of the distribution of means. 

g. What is the likely shape of the distribution of means? Explain your answer. 

What is a percentile?

What is the difference between parametric tests and non-parametric tests? 

For each of the following examples, identify whether the research has expressed a directional or a nondirectional hypothesis:

a. A researcher is interested in studying the relation between the use of antibacterial products and the dryness of people’s skin. He thinks these products might alter the moisture in skin differently from other products that are not antibacterial.

b. A student wonders if grades in a class are in any way related to where a student sits in the classroom. In particular, do students who sit in the front row get better grades, on average, than the general population of students? 

c. Cell phones are everywhere, and we are now available by phone almost all of the time. Does this translate into a change in the closeness of our longdistance relationships? 

For each of the following examples (the same as those in Exercise 7.41), state the null hypothesis and the research hypothesis, in both words and symbolic notation:

a. A researcher is interested in studying the relation between the use of antibacterial products and the dryness of people’s skin. He thinks these products might alter the moisture in skin differently from other products that are not antibacterial.

b. A student wonders if grades in a class are in any way related to where a student sits in the classroom. In particular, do students who sit in the front row get better grades, on average, than the general population of students?

c. Cell phones are everywhere, and we are now available by phone almost all of the time. Does this translate into a change in the closeness of our long distance relationships? 

Hurricane Katrina hit New Orleans on August 29, 2005. The National Weather Service Forecast Office maintains online archives of climate data for all U.S. cities and areas. These archives allow us to find out, for example, how the rainfall in New Orleans that August compared to that in the other months of 2005. The table below shows the National Weather Service data (rainfall in inches) for New Orleans in 2005.

a. Calculate the z score for August, the month in which Hurricane Katrina hit.

b. What is the percentile for the rainfall in August? Does this surprise you? Explain.

c. When results surprise us, it is worthwhile to examine individual data points more closely or even to go beyond the data. The daily climate data as listed by this source for August 2005 shows the code “M” next to August 29, 30, and 31 for all climate statistics. The code says: “[REMARKS] ALL DATA MISSING AUGUST 29, 30, AND 31 DUE TO HURRICANE KATRINA.” Pretend you were hired as a consultant to determine the percentile for that August. Write a brief paragraph for your report, explaining why the data you generated are likely to be inaccurate.

d. What raw scores mark the cutoff for the top and bottom 10% for these data? Based on these scores, which months had extreme data for 2005? Why should we not trust these data? 

Researchers often use z tests to compare their samples to known population norms. The Graded Naming Test (GNT) asks respondents to name objects in a set of 30 black-and-white drawings. The test, often used to detect brain damage, starts with easy words like kangaroo and gets progressively more difficult, ending with words like sextant. The GNT population norm for adults in England is 20.4. Roberts (2003) wondered whether a sample of Canadian adults had different scores than adults in England. If they were different, the English norms would not be valid for use in Canada. The mean for 30 Canadian adults was 17.5. For the purposes of this exercise, assume that the standard deviation of the adults in England is 3.2.

a. Conduct all six steps of a z test. Be sure to label all six steps.

b. Some words on the GNT are more commonly used in England. For example, a mitre, the headpiece worn by bishops, is worn by the archbishop of Canterbury in public ceremonies in England. No Canadian participant correctly responded to this item, whereas 55% of English adults correctly responded. Explain why we should be cautious about applying norms to people different from those on whom the test was normed.

c. When we conduct a one-tailed test instead of a two-tailed test, there are small changes in steps 2 and 4 of hypothesis testing. (Note: For this example, assume that those from populations other than the one on which it was normed will score lower, on average. That is, hypothesize that the Canadians will have a lower mean.) Conduct steps 2, 4, and 6 of hypothesis testing for a one-tailed test.

d. Under which circumstance—a one-tailed or a twotailed test—is it easier to reject the null hypothesis? Explain.

e. If it becomes easier to reject the null hypothesis under one type of test (one-tailed versus two tailed), does this mean that there is a bigger difference between the groups with a one tailed test than with a two-tailed test? Explain.

f. When we change the p level that we use as a cutoff, there is a small change in step 4 of hypothesis testing. Although 0.05 is the most commonly used p level, other values, such as 0.01, are often used. For this example, conduct steps 4 and 6 of hypothesis testing for a two-tailed test and p level of 0.01, determining the cutoff and drawing the curve.

g. With which p level—0.05 or 0.01—is it easiest to reject the null hypothesis? Explain.

h. If it is easier to reject the null hypothesis with certain p levels, does this mean that there is a bigger difference between the samples with one p level versus the other p level? Explain. 

In your own words, define the word effect—first as you would use it in everyday conversation and then as a statistician would use it. 

In an exercise in Chapter 7, we asked you to conduct a z test to ascertain whether the Graded Naming Test (GNT) scores for Canadian participants differed from the GNT norms based on adults in England. We also used these data in the How It Works section of this chapter. The mean for a sample of 30 adults in Canada was 17.5. The normative mean for adults in England is 20.4, and we assumed a population standard deviation of 3.2. With 30 participants, the z statistic was −4.97, and we were able to reject the null hypothesis.

a. Calculate the test statistic for 3 participants. How does the test statistic change compared to when N of 30 was used? Conduct step 6 of hypothesis testing. Does your conclusion change? If so, does this mean that the actual difference between groups changed? Explain.

b. Conduct steps 3, 5, and 6 for 100 participants. How does the test statistic change?

c. Conduct steps 3, 5, and 6 for 20,000 participants. How does the test statistic change?

d. What is the effect of sample size on the test statistic?

e. As the test statistic changes, has the underlying difference between groups changed? Why might this present a problem for hypothesis testing? 

Unsavory researchers know that one can cheat with hypothesis testing. That is, they know that a researcher can stack the deck in her or his favor, making it easier to reject the null hypothesis.

a. If you wanted to make it easier to reject the null hypothesis, what are three specific things you could do?

b. Would it change the actual difference between the samples? Why is this a potential problem with hypothesis testing?

Identify the critical t values for each of the following tests:

a. A single-sample t test examining scores for 26 participants to see if there is any difference compared to the population, using a p level of 0.05

b. A one-tailed, single-sample t test performed on scores on the Marital Satisfaction Inventory for 18 people who went through marriage counseling, as compared to the population of people who had not been through marital counseling, using a p level of 0.01

c. A two-tailed, single-sample t test, using a p level of 0.05, with 34 degrees of freedom

Researchers explored whether there were mean differences between students who were randomly assigned to take notes longhand and students who were randomly assigned to take notes on their laptops (Mueller & Oppenheimer, 2014). They had observed that students who took notes by hand performed better, on average, on conceptual questions— those that involved thinking beyond just recalling facts—than students who took notes on their laptops. To explore reasons for this difference, they examined the students’ notes. The researchers “found that laptop notes contained an average of 14.6% verbatim overlap with the lecture (SD = 7.3%), whereas longhand notes averaged only 8.8% (SD = 4.8%), t(63) = −3.77, p < .001, d = 0.94” (p. 3). They concluded that when people took notes longhand, they were more likely to put ideas in their own words, which likely led to deeper processing and better learning of information.

a. What kind of t test did the researchers use? Explain your answer.

b. How do we know this finding is statistically significant?

c. How many participants were in this experiment?

d. Identify the means of the two groups.

e. What is the effect size for this finding? Interpret what that means in terms of Cohen’s conventions. 

For the following data, assuming a between-groups design, determine:

a. df_between

b. df_within

c. df_total

d. The critical value, assuming a p value of 0.05

e. The mean for each group and the grand mean

f. The total sum of squares

g. The within-groups sum of squares

h. The between-groups sum of squares

i. The rest of the ANOVA source table for these data

j. The effect size and an indication of its size 

Chang, Sandhofer, and Brown (2011) wondered whether mothers used number words more, on average, with their preschool sons than with their preschool daughters. Each participating family included one mother and one child—either female or male. They speculated that early exposure to more number words might predispose children to like mathematics. They reported the following: “An independent- samples t test revealed statistically significant differences in the percentages of overall numeric speech used when interacting with boys compared with girls, t(30) = 2.40, p < .05, d = .88. That is, mothers used number terms with boys an average of 9.49% of utterances (SD = 6.78%) compared with 4.64% of utterances with girls (SD = 4.43%)” (pp. 444–445).

a. Is this a between-groups or within-groups design? Explain your answer.

b. What is the independent variable? What is the dependent variable?

c. How many children were in the total sample? Explain how you determined this.

d. Is the sample likely randomly selected? Is it likely that the researchers used random assignment?

e. Were the researchers able to reject the null hypothesis? Explain.

f. What can you say about the size of the effect?

g. Describe how you could design an experiment to test whether exposure to more number words in preschool leads children to like mathematics more when they enter school.

Alice Waters, owner of the Berkeley, California, restaurant Chez Panisse, has long been an advocate for the use of simple, fresh, organic ingredients in home and restaurant cooking. She has also turned her considerable expertise to school cafeterias. Waters (2006) praised changes in school lunch menus that have expanded nutritious offerings, but she hypothesizes that students are likely to circumvent healthy lunches by avoiding vegetables and smuggling in banned junk food unless they receive accompanying nutrition education and hands-on involvement in their meals. She has spearheaded an Edible Schoolyard program in Berkeley, which involves public school students in the cultivation and preparation of fresh foods, and states that such interactive education is necessary to combat growing levels of childhood obesity. “Nothing less,” Waters writes, “will change their behavior.”

a. In your own words, what is Waters predicting? Citing the confirmation bias, explain why Waters’ program, although intuitively appealing, should not be instituted nationwide without further study.

b. Describe a simple between-groups experiment with a nominal independent variable with two levels and a scale-dependent variable to test Waters’ hypothesis. Specifically identify the independent variable, its levels, and the dependent variable. State how you will operationalize the dependent variable.

c. Which hypothesis test would be used to analyze this experiment? Explain your answer.

d. Conduct step 1 of hypothesis testing.

e. Conduct step 2 of hypothesis testing.

f. State at least one other way you could operationalize the dependent variable.

g. Let’s say, hypothetically, that Waters discounted the need for the research you propose by citing her own data that the Berkeley school in which she instituted the program has lower rates of obesity than other California schools. Describe the flaw in this argument by discussing the importance of random selection and random assignment.

Researchers at the Cornell University Food and Brand Lab conducted an experiment at a fitness camp for adolescents (Wansink & van Ittersum, 2003). Campers were given either a 22-ounce glass that was tall and thin or a 22-ounce glass that was short and wide. Campers with the short glasses tended to pour more soda, milk, or juice than campers with the tall glasses.

a. Is it likely that the researchers used random selection? Explain.

b. Is it likely that the researchers used random assignment? Explain.

c. What is the independent variable, and what are its levels?

d. What is the dependent variable?

e. Which hypothesis test would the researchers use? Explain.

f. Conduct step 1 of hypothesis testing.

g. Conduct step 2 of hypothesis testing.

h. How could the researchers redesign this study so that they could use a paired-samples t test?

Find the error in the statistics language in each of the following statements about z, t, or F distributions or their related tests. Explain why it is incorrect and provide the correct word.

a. The professor reported the mean and standard error for the final exam in the statistics class.

b. Before we can calculate a t statistic, we must know the population mean and the population standard deviation.

c. The researcher calculated the parameters for her three samples so that she could calculate an F statistic and conduct an ANOVA.

d. For her honors project, Evelyn calculated a z statistic so that she could compare the mean video game scores of a sample of students who had ingested caffeine with a sample of students who had not ingested caffeine.

Find the incorrectly used symbol or symbols in each of the following statements or formulas. For each statement or formula, (i) state which symbol(s) is/are used incorrectly, (ii) explain why the symbol(s) in the original statement is/are incorrect, and (iii) state which symbol(s) should be used.

a. When calculating an F statistic, the numerator includes the estimate for the between-groups variance, s.

b. SS_between = (X − GM )²

c. SS_within = (X − M )

d. F = √t

What are the four assumptions for a within-groups ANOVA?

What are order effects?

Explain the source of variability called “subjects.”

What is the advantage of the design of the within-groups ANOVA over that of the between-groups ANOVA? 

What is counterbalancing?

Why is it appropriate to counterbalance when using a within-groups design? 

For the following data, assuming a between-groups design, determine:

a. df_between

b. df_within

c. df_total

d. The critical value, assuming a p value of 0.05

e. The mean for each group and the grand mean

f. The total sum of squares

g. The within-groups sum of squares

h. The between-groups sum of squares

i. The rest of the ANOVA source table for these data

j. Tukey HSD values 

Calculate the F statistic, writing the ratio accurately, for each of the following cases:

a. Between-groups variance is 29.4 and within-groups variance is 19.1

b. Within-groups variance is 0.27 and between groups variance is 1.56

c. Between-groups variance is 4595 and within-groups variance is 3972 

In a classic prisoner’s dilemma game with money for prizes, players who cooperate with each other both earn good prizes. If, however, your opposing player cooperates but you do not (the term used is defect), you receive an even bigger payout and your opponent receives nothing. If you cooperate but your opposing player defects, he or she receives that bigger payout and you receive nothing. If you both defect, you each get a small prize. Because of this, most players of such games choose to defect, knowing that if they cooperate but their partners don’t, they won’t win anything. The strategies of U.S. and Chinese students were compared. The researchers hypothesized that those from the market economy (United States) would cooperate less (i.e., would defect more often) than would those from the nonmarket economy (China).

a. How many variables are there in this study? What are the levels of any variables you identified?

b. Which hypothesis test would be used to analyze these data? Justify your answer.

c. Conduct the six steps of hypothesis testing for this example, using the above data.

d. Calculate the appropriate measure of effect size. According to Cohen’s conventions, what size effect is this?

e. Report the statistics as you would in a journal article.

f. Draw a table that includes the conditional proportions for participants from China and from the United States.

g. Create a graph with bars showing the proportions for all four conditions.

h. Create a graph with two bars showing just the proportions for the defections for each country.

i. Calculate the relative risk (or relative likelihood) of defecting, given that one is from China versus the United States. Show your calculations.

j. Explain what we learn from this relative risk.

k. Now calculate the relative risk of defecting, given that one is from the United States versus China. Show your calculations.

l. Explain what we learn from this relative risk.

m. Explain how the calculations in parts (i) and (k) provide us with the same information in two different ways. 

The following figures display data that depict the relation between students’ monthly cell phone bills and the number of hours they report that they study per week.

a. What does the accompanying scatterplot suggest about the shape of the distribution for hours studied per week? What does it suggest about the shape of the distribution for a monthly cell phone bill?

b. What does the accompanying grouped frequency histogram suggest about the shape of the distribution for a monthly cell phone bill?

c. Is it a good idea to use a parametric hypothesis test for these data? Explain. 

Using the following data:

a. Create a scatterplot.

b. Calculate deviation scores and products of the deviations for each individual, and then sum all products. This is the numerator of the correlation coefficient equation.

c. Calculate the sum of squares for each variable. Then compute the square root of the product of the sums of squares. This is the denominator of the correlation coefficient equation.

d. Divide the numerator by the denominator to compute the coefficient, r.

e. Calculate degrees of freedom.

f. Determine the critical values, or cutoffs, assuming a two-tailed test with a p level of 0.05.

Explain how the correlation coefficient can be used as a descriptive or an inferential statistic. 

When we have a straight-line relation between two variables, we use a Pearson correlation coefficient. What does this coefficient describe? 

What magnitude of a correlation coefficient is large enough to be considered important, or worth talking about? 

Stacey Finkelstein and Ayelet Fishbach (2012) examined the impact of feedback in the learning process. The following is an excerpt from their abstract: “This article explores what feedback people seek and respond to. We predict and find a shift from positive to negative feedback as people gain expertise. We document this shift in a variety of domains, including feedback on language acquisition, pursuit of environmental causes, and use of consumer products. Across these domains, novices sought and responded to positive feedback, and experts sought and responded to negative feedback” (p. 22).

a. Based on the abstract, what are the independent variables and what are their levels?

b. What are possible dependent variables, based on the description in the abstract?

c. The researchers conducted several experiments, one of which examined students in beginning and advanced French classes. Here is the result of one analysis: “The analysis also yielded the predicted expertise × feedback interaction (F(1,79) = 7.31, p < .01). Is this interaction statistically significant? Explain your answer.

d. What important statistic is missing from their report? Why would it be helpful to include this statistic?

e. The results in part (c) are represented by the graph here. We would, of course, have to conduct additional analyses to know exactly which bars are significantly different from each other. That said, what does the overall pattern seem to indicate for this analysis?

f. How would you redesign this graph in line with what you learned in Chapter 3? Give at least two specific suggestions.

Focusing on coverage of the U.S. presidential election, Julia R. Fox, a telecommunications professor at Indiana University, wondered whether The Daily Show, despite its comedy format, was a valid source of news. She coded a number of half-hour episodes of The Daily Show as well as a number of half-hour episodes of the network news (Indiana University Media Relations, 2006). Fox reported that the average amounts of “video and audio substance” were not statistically significantly different between the two types of shows. Her analyses are described as “second by second,” so, for this exercise, assume that all outcome variables are measures of time.

a. As the study is described, what are the independent and dependent variables? For nominal variables, state the levels.

b. As the study is described, what type of hypothesis test would Fox use?

c. Now imagine that Fox added a third category, a cable news channel such as CNN. Based on this new information, state the independent variable or variables and the levels of any nominal independent variables. What hypothesis test would she use?

For each of the following situations, state whether the distribution of interest is a z distribution, a t distribution, or an F distribution. Explain your answer.

a. A city employee locates a U.S. Census report that includes the mean and standard deviation for income in the state of Wyoming and then takes a random sample of 100 residents of the city of Cheyenne. He wonders whether residents of Cheyenne earn more, on average, than Wyoming residents as a whole.

b. A researcher studies the effect of different contexts on work interruptions. Using discreet video cameras, she observes employees working in enclosed offices in the workplace, in open cubicles in the workplace, and in home offices.

c. An honors student wondered whether an education in statistics reduces the tendency to believe advertising that cites data. He compares social science majors who had taken statistics and social science majors who had not taken statistics with respect to their responses to an interactive advertising assessment. 

Catherine Ruby, a doctoral student at New York University, conducted an online survey to ascertain the reasons that international students chose to attend graduate school in the United States. One of several dependent variables that she considered was reputation; students were asked to rate the importance in their decision of factors such as the reputation of the institution, the institution and program’s academic accreditations, and the reputation of the faculty. Students rated factors on a 1–5 scale, and then all reputation ratings were averaged to form a summary score for each respondent. For each of the following scenarios, state the independent variable with its levels (the dependent variable is reputation in all cases). Then state what kind of an ANOVA she would use.

a. Ruby compared the importance of reputation among graduate students in different types of programs: arts and sciences, education, law, and business.

b. Imagine that Ruby followed these graduate students for 3 years and assessed their rating of reputation once a year.

c. Ruby compared international students working toward a master’s, a doctoral, or a professional degree (e.g., MBA) on reputation.

d. Imagine that Ruby followed international students from their master’s program to their doctoral program to their postdoctoral fellowship, assessing their ratings of reputation once at each level of their training.

Pilots’ mental efforts and a one-way withingroups ANOVA: Researchers examined the amount of mental effort that participants felt they were expending on a cognitively complex task, piloting an unmanned air vehicle (UAV) (Ayaz, Shewokis, Bunce, Izzetoglu, Willems, & Onaral, 2012). The researchers used the Task Load Index (TLX), a measure that assesses participants’ perception of their mental effort following a series of approach and landing tasks in simulated UAV tasks. They wondered whether expertise would have an effect on perceptions of mental effort. In the results section, the researchers reported the results of their analyses, a series of one-way repeated measures ANOVA. “The results indicated a significant main effect of practice level (beginner/intermediate/ advanced conditions) for mental demand (F (2, 8) = 17.87, p < 0.01, η² = 0.817), effort (F (2, 8) = 16.32, p < 0.01, η² = 0.803), and frustration (F (2, 8) = 8.60, p < 0.01, η² = 0.682).” They went on to explain that mental demand, effort, and frustration all tended to decrease with expertise.

a. What is the independent variable in this study?

b. What are the dependent variables in this study?

c. Explain why the researchers were able to use a oneway within-groups ANOVA in this situation.

d. η² is roughly equivalent to R². How large are each of these effects, based on Cohen’s conventions?

e. The researchers drew a specific conclusion beyond that there was some difference, on average, in the dependent variables, depending on the particular levels of the independent variable. What additional test were they likely to have conducted? Explain your answer.

In Chapter 10, we introduced a study by Steele and Pinto (2006) that examined whether people’s level of trust in their direct supervisor was related to their level of agreement with a policy supported by that leader. Steele and Pinto found that the extent to which subordinates agreed with their supervisor was related to trust and showed no relation to gender, age, time on the job, or length of time working with the supervisor. Let’s assume we used a scale that sorted employees into three groups: low trust, moderate trust, and high trust in supervisors. Below are fictional data regarding level of agreement with a leader’s decision for these three groups. The scores presented are the level of agreement with a decision made by a leader, from 1, the least agreement, to 40, the highest level of agreement. Note: These fictional data are different from those presented in Chapter 10. Employees with low trust in their leader: 9, 14, 11, 18 Employees with moderate trust in their leader: 14, 35, 23 Employees with high trust in their leader: 27, 33, 21, 34

a. What is the independent variable? What are its levels?

b. What is the dependent variable?

c. Conduct all six steps of hypothesis testing for a one-way between-groups ANOVA.

d. How would you report the statistics in a journal article?

e. Conduct a Tukey HSD test. What did you learn?

f. Why is it not possible to conduct a t test in this situation?

g. Why is it not possible to use a within-groups design for this study? 

Iranian researchers studied factors affecting patients’ likelihood of wearing orthodontic appliances, noting that orthodontics is perhaps the area of health care with the highest need for patient cooperation (Behenam & Pooya, 2007). Among their analyses, they compared students in primary school, junior high school, and high school. The data that follow have almost exactly the same means as they found in their study, but with far smaller samples. The score for each student is his or her daily hours of wearing the orthodontic appliance.

Primary school: 16, 13, 18

Junior high school: 8, 13, 14, 12

High school: 20, 15, 16, 18

a. What is the independent variable? What are its levels?

b. What is the dependent variable?

c. Conduct all six steps of hypothesis testing for a one-way between-groups ANOVA.

d. How would you report the statistics in a journal article?

e. Conduct a Tukey HSD test. What did you learn?

f. Calculate the appropriate measure of effect size for this sample.

g. Based on Cohen’s conventions, is this a small, medium, or large effect size?

h. Why is it useful to know the effect size in addition to the results of a hypothesis test?

i. How could this study be conducted using a withingroups design?