The Joy of Stats

Chapter One

Claudio is studying palm tree heights in the state of Pernambuco, Brazil. In this study, which of the following is a variable?

a. Location = Pernambuco.b. Tree height.c. The height of any specific tree—for example, 3 metres.

(Answer: b; height)

In this study, which of the following is the value of a variable?

a. Location = Pernambuco.b. Tree height.c. The height of any specific tree—for example, 3 metres.

(Answer: c; a value for a specific observation)

In this part of the study, what are the cases or units of analysis?

a. Location.b. Tree heights.c. Metres.d. Palm trees.

(Answer: d; the variables are measured on trees.)

Claudio discovers that the average height of palm trees is different in 10 distinct areas depending on the average yearly rainfall of the area. In this part of the study, the units of analysis are:

a. Rainfall in inches.b. The 10 areas.c. Pernambuco.d. Palm trees.e. Tree height in metres.

(Answer: b; the 10 areas. The concept is place vs. individual level units of analysis.)

In the part of the study in which Claudio looks at the average height of palm trees and the average yearly rainfall in 10 different areas, what is the best statement of his research question?

a. Is there variation in the average yearly rainfall of Pernambuco?b. Which forest area has the highest average annual rainfall?c. Is average tree height in an area related to average annual rainfall in the area?d. Does a heavy rain cause trees to grow higher?e. Do taller trees cause heavier rains?

(Answer: c; “related”—remaining initially cautious about causality and the direction of effects)

Which of the following is a constant, not a variable?

a. The provinces of Canada, such as Alberta, Manitoba, Quebec, etc.b. Children’s heights in the nation of Laos.c. Vietnam.d. Suicide rates of European countries.e. 5 different species of mould found in Texas.

(Answer: c)

Short EssayPetra wants to know if countries characterized by a high level of social inequality have higher rates of homicide than countries with low levels of inequality.

Question 1: Which is the better choice of independent variable for this question and why? Question 2: Both of the variables in the study present problems of operationalization.

Why? And what would you do to operationalize the variables?

For each pair, identify the independent variable for the research question OR indicate that either variable could be the independent variable:

a. An individual’s sex category at birth and height. b. Ambient humidity and extent of mould damage in buildings. c. U.S. cities’ unemployment rates and their poverty rates. d. Level of truck traffic in cities and level of air pollution. e. Level of air pollution in cities and level of water pollution in cities.

(Answers: a) sex category, b) humidity, c) either one, d) truck traffic, e) either one)

Short Essay or Discussion QuestionFor which of the preceding pairs of variables could there be a causal relationship? (Answer: For a, b, and d, we could make a good case for a causal relationship, but this relationship is less evident for c and e.)

Identify the level of measurement for the following variable: weight of individual human beings measured in kilograms.

a. Nominal.b. Kilograms.c. Ordinal.d. Interval-Ratio.e. None of the above.

(Answer: d; interval-ratio)

As a Fill-in-the-Blanks ItemWeight measured in kilograms is considered to be at the ___________level of measurement. (Answer: interval-ratio)

Identify the level of measurement for the following variable: Rank order of cities as tourist destinations in Mexico (first, second, third, etc.).

a. Nominal.b. Ordinal.c. Interval-Ratio.d. Continuous.e. None of the above.

(Answer: b; ordinal)

Identify the level of measurement for a variable defined on students at the University of Toronto so that all students with a Y chromosome are defined as male and all others as not-male.

a. Nominal.b. Ordinal.c. Interval-Ratio.d. Dichotomous.e. All of the above.

(Answer: e; all of the above)

Discussion QuestionUp until the 1960s Olympic athletes were identified as men or women by an inspection of external genitalia. Over the years, testing procedures have become increasingly complex, multifaceted, and controversial. What are the issues? (For example, is it fair for women to compete against someone who has many male physical characteristics?) What do you think is the best operational definition of sex categories in athletic competition?

Note to the instructor: Students may find it interesting to read about Caster Semenya, a South African runner, featured in an article by Ariel Levy, “Either/Or Sports, Sex, and the Case of Caster Semenya,” The New Yorker, November 30, 2009.

Fill in the BlanksProvide examples of variables at each level of measurement.

Franco is studying beauty ideals, and part of his study involves comparing the average age in years of women featured in celebrity magazines in three global regions: Europe, Asia, and South America. In this part of the study, the units of measurement are:

a. Countries.b. Celebrities.c. Magazines.d. Years.e. Nominal-level data.

(Answer: d; years)

For the study of beauty images in the preceding question, Franco defines a variable called “estimated body mass index” by looking at photos of the celebrities and estimating their body

mass index from the information in the photographs. Creating this variable is an example of ______ a variable:

a. Deducing.b. Operationalizing.c. Dichotomizing.d. Theoretically conceptualizing.e. Maximizing.

(Answer: b; operationalizing)

In the study of beauty ideals, which variable pertaining to the celebrities would be the most difficult to operationalize?

a. Height.b. Weight.c. Beauty.d. Age.e. Skin tone.

(Answer: c; beauty)

Saba is studying the AIDS death rates of 60 countries. The units of analysis or cases in her study are:

a. Individuals.b. Places (countries).c. Number of deaths.d. Operational variables.e. All of the above.

(Answer: b; places/countries)

The data file for the study of countries’ AIDS death rates would have how many cases?

a. One for each individual who died.b. A case for each person in the total population of the 60 countries regardless of whether

they died or not.c. One for each person who contracted AIDS in the 60 countries.d. 60 cases, one for each country in the study.e. Can’t tell; not enough information is available to answer the question.

(Answer: d; 60 cases. Each country is a case.)

If Teaching SPSS/PASWAll information for a case would be located in a ________ of the data file:

a. Column.b. Row.c. Cell.d. Variable definition.e. None of the above.

(Answer: b; row)

Chapter Two

In his study of palm trees, Claudio found that, in one cluster of trees, 343 are healthy, 25 are in early stages of a disease, and 20 are in an advanced stage. In the frequency distribution of tree health, what percentage are in the early stage of disease? ____________

a. 10.1b. 5.7c. 6.4d. 11.2e. Can’t answer the question with the available information.

(Answer: c; 6.4%)

For the variable tree health, defined in the preceding problem, which summary measures can be obtained?

a. Mean.b. Median.c. Mean, median, and mode.d. Mean, median, variance, and standard deviation.e. None of the above.

(Answer: b; median. It is ordinal data.)

To display tree health, as defined in the preceding problems, which graphs would be appropriate?

a. Bar chart.b. Bar chart and histogram.c. Bar chart and pie chart.d. Histogram.e. Histogram and boxplot.

(Answer: c; bar chart and pie chart for nominal and ordinal data)

Which statement best characterizes the variance of a distribution?

a. It is the square of the standard deviation.b. It is a measure of variability in a distribution.c. It is the average of the squared deviations of observations from the mean of the

distribution.d. It is never a negative number.e. All of the above.

(Answer: e; all of the above)

When is the median a better summary measure than the mean to represent a distribution?

a. When it is larger than the mean.b. When the distribution is skewed.c. When a distribution has two long symmetrical tails.d. When the level of measurement is nominal.

e. When it is smaller than the mean.(Answer: b; skewed)

A measure of _________________ should always be reported along with the measure of central tendency.

a. Similarity.b. The average of the distribution.c. Magnitude.d. Variability.e. None of the above.

(Answer: d; variability)

The administration at Big U has computed the mean and median of the family incomes of incoming students. In order to represent the income distribution accurately, they should also compute the ____________ of family income.

a. Similarities.b. Standard deviation.c. 50th percentile.d. Absolute value.e. Central tendency.

(Answer: b; standard deviation)

Which of the following are measures of central tendency?

a. Median.b. Mode.c. Mean.d. All of the above.e. None of the above.

(Answer: d; all of the above)

Where is the line indicating the median placed in a boxplot?

a. In the middle of the box.b. In the box but not necessarily in the middle.c. Always at the upper edge of the box.d. Always at the lower edge of the box.e. It cannot be specified—depends on the shape of the distribution.

(Answer: b; in the box but not necessarily in the middle)

Which statement best describes a stem-and-leaf chart?

a. The raw scores are preserved in the chart.b. It is organized similarly to a histogram.c. It is used with interval-ratio data.d. The “leaves” represent digits that are further to the right in the values of the observations.e. All of the above.

(Answer: e; all of the above)

In a histogram:

a. The horizontal dimension refers to the frequency of each value, and the vertical dimension shows the range of values of the variable.

b. The vertical dimension refers to the frequency of each value, and the horizontal dimension shows the range of values of the variable.

c. The vertical dimension refers to the percentiles of the distribution.d. Each bar represents a category of a categoric variable.e. All of the above.

(Answer: b; vertical displays frequency, horizontal displays values)

Which statement is true of the mean of a distribution?

a. It can never be 0.b. All values or observations must be included in the calculation.c. It is a measure of dispersion.d. It cannot be computed for a binary variable.e. All of the above.

(Answer: b; all values must be included in the calculation)

Claudio has created a data file for his palm tree data. Each of 1500 trees is characterized by 4 variables: height in metres, state of health (healthy, early stage disease, advanced stage disease), location in one of 10 forest areas, and tree diameter in metres. For which variable(s) would a mean be meaningful?

a. Height and diameter.b. Height.c. Height and location.d. Health and height.e. All of the above.

(Answer: a; height and diameter)

Claudio creates boxplots of tree height for trees in each of the three states of health. This graph displays

a. One variable: health.b. Two variables: health and height.c. Three variables: one for each of the three states of health.d. One variable: height.e. Nothing: it cannot be made because health is not an interval-ratio variable.

(Answer: b; two variables, health and height. However, when this chart is made, there will be three boxplots of tree heights—one for each state of health—although there are only two variables. Before answering this question, students might need help to create these types of boxplots that use a categoric variable and show boxplots of the interval-ratio variable displayed for each category of the categoric variable.)

For which of the following variables would an index of diversity and an index of qualitative variation NOT be appropriate summary measures?

a. Height in metres of students in Canadian high schools.b. Types of sports in which students in a school choose to participate.c. Species of predators found in a province of Canada.d. Types of mental illness diagnosed in a community mental health centre during the course

of a month.e. The two indexes can be computed for all of the above variables.

(Answer: a; height in metres. This is an interval-ratio variable not a categoric one.)

According to Chebycheff’s Inequality, in a distribution with a finite mean and variance, we would expect a score or value that is five standard deviations from the mean to appear with a probability of no more than:

a. 5b. 1/5c. 0 (never)d. 68%e. .997

(Answer: b; 1/5)

Claudio discovers the mean height of palm trees in the region is 5 metres with a standard deviation of .5 metres. He finds one tree that is 5.5 metres tall. Its Z-score in the height distribution is:

a. +5.5 b. +.5c. 1d. 0e. Can’t tell from the information provided.

(Answer: c; 1. Its height is one standard deviation above the mean.)

Which statement is NOT true about Z-scores for any distribution with a finite mean and variance?

a. Their mean is always 0.b. A Z-score can be a negative number.c. Their standard deviation is always 1.d. Values larger than the mean of the distribution have positive Z-scores.e. Each case’s Z-score is invariant for all distributions in which the case appears.

(Answer: e; Z-scores for a case vary depending on the distribution.)

Easier Version of Preceding QuestionCassie tells us that she has a Z-score of –1 for weight in the weight distribution of her graduating class. Identify the correct statement based on this information:

a. This is not possible because Z-scores can never be negative.b. Her weight is below the mean of the class’s weight distribution.c. This Z-score represents her weight in any weight distribution in which she is a case.d. This indicates that she has an extremely low weight—it is an outlier in the distribution.e. Her weight is exactly 5 standard deviations below the mean of the class weights.

(Answer: b; her weight is below the class mean weight)

The algorithm for a Z-score is:

a. Z = unstandardized score minus the mean of the distribution.b. Z = divide the number of cases into the difference between the unstandardized score and

the mean of the distribution.c. Z = unstandardized score divided by the standard deviation.d. Z = divide the standard deviation into the difference between the unstandardized score

and the mean of the distribution. e. Z = unstandardized score divided by the variance of the distribution.

(Answer: d; divide the standard deviation into the difference between the unstandardized score and the mean of the distribution)

Another term for Z-score is:

a. Chebycheff number.b. Standardized score.

c. Mean score.d. Hinge score.e. Parameter.

(Answer: b; standardized score)

Shirley weighs 120 pounds and Dalia weighs 140 pounds. We consider their weights in the distribution of weights for 220 women in their graduating class. What can we conclude based on the information provided?

a. Shirley’s weight Z-score is 12 and Dalia’s is 14.b. Shirley’s weight Z-score is lower than Dalia’s.c. We cannot say anything about the Z-scores because the information about the mean and

standard deviation of the distribution is not provided.d. Dalia’s score is closer to 0 than Shirley’s.e. Dalia’s Z-score is closer to +1 than Shirley’s.

(Answer: b; Shirley’s Z-score is lower than Dalia’s)

Difficult Short Essay Question or Question for Class DiscussionHow is an SPSS/PASW data file different from a frequency distribution of a variable? (Answer: Students are sometimes confused about this, so stress that the frequency distribution displays only one variable. When we see it, we cannot identify the cases to which the values pertain; it only provides a summary of the distribution, not the full information or “raw data.” We do not know which value pertains to which case. In the data file we can see this information. The frequency distribution is a summary of information in the data file. It tells us how many cases are associated with a specific value of the variable.)

We have a data set of 1500 cases from a simple random sample of Cape Town residents. The variable is height measured to the nearest millimetre. Which of the following would be a good way of summarizing the variable distribution using SPSS/PASW?

a. A frequency distribution table for heights measured to the nearest millimetre.b. A pie chart displaying all the heights of the 1500 individuals.c. A bar chart.d. Descriptives and a histogram.e. All of the above.

(Answer: d; descriptives and a histogram. This is interval-ratio data.)

Note to the instructor: Unless the problem is called to their attention, many students would go ahead and print out a frequency table for this type of data, remaining “unfazed” by the fact that the table goes on for page after page! Emphasize that they can obtain a frequency distribution only if the data are categoric or if they are grouped into intervals—SPSS/PASW does this automatically for the histogram but not for the frequency table.

Chapter Three

The formula for the estimated standard error of the mean is:

a. Estimated SE = σb. Estimated SE = S/Nc. Estimated SE = S/ d. Estimated SE = S2/Ne. Estimated SE = S

(Answer: c; sample standard deviation divided by the square root of sample size)

When sample size is 2 or more, the standard error of the mean is _______ the standard deviation of the underlying variable.

a. Greater than.b. Less than.c. Equal to.d. Equal to 1 – s.e. Could be any of the above; it depends on the situation.

(Answer: b; less than)

Which of the following is NOT a characteristic of the sampling distribution of the mean?

a. It is normally distributed.b. Its mean is equal to the mean of the underlying distribution.c. It presents an exception to Chebycheff’s Inequality.d. Its standard deviation is sigma divided by the square root of N.e. Its standard deviation is less than sigma, provided N > 1.

(Answer: c; exception to Chebycheff’s Inequality)

The standard deviation of the sampling distribution is called:

a. Measurement error.b. Estimated error term.c. The variance of the sampling distribution.d. Standard error.e. Bias.

(Answer: d; standard error)

Standard error represents:

a. Variability in sample outcomes as the result of randomness in sampling.b. Errors a researcher made in measuring.c. Errors that result from researcher bias.d. Errors that result from inaccurate operationalization of variables.e. All of the above.

(Answer: a; variability in sample outcomes as the result of randomness)

If we roll a pair of dice and sum the numbers that turn up on the upper face of each die, what can we say about the sum?

a. All numbers from 1 to 12 are equally likely as the value of the sum.b. All numbers from 2 to 12 are equally likely as the value of the sum.c. 7 is the expected value.d. 12 is the expected value of the distribution and the maximum likelihood estimate.e. We cannot draw any conclusion about the sum because it is a random process.

(Answer: c; 7 is the expected value)

The standard error of the mean represents:

a. Qualitative variation in the underlying empirical variable.b. The variability of sample outcomes.c. Researcher bias in sample selection.d. Mistakes in sampling procedures.e. Problems in operationalizing the variable.

(Answer: b; the variability of sample outcomes)

Which of the following is an example of Type I error?

a. A vaccine fails to lower the rate of infection, but, on the basis of sample data, we conclude it is effective.

b. A vaccine actually lowers the rate of infection, but, on the basis of sample data, we conclude it is not effective.

c. We reject a true null hypothesis of vaccine effectiveness. d. We have sample results that are too inconclusive to make a decision about the vaccine.e. All of the above are examples of Type I error.

(Answer: a; we conclude it is effective on the basis of sample data, but it is not)

Type I error is sometimes referred to as:a. Alpha error.b. Beta error.c. Gamma error.d. Delta error.e. None of the above.

(Answer: a; alpha error)

By the conventional standards of research, which of the following p-values would lead to the conclusion that the results are significant?

a. .50b. .04c. .899d. +1e. –1

(Answer: b; .04)

For a calculated t to allow rejection of the null hypothesis, it must:

a. Be exactly equal to the critical value.b. Have an absolute value that exceeds the critical value.c. Have an absolute value less than the critical value.d. Fall outside of the critical region.e. Exceed 120 as the sample size.

(Answer: b; the absolute value exceeds the critical value)

A t-test is performed when:

a. Sample size is over 120.b. Sigma, the population standard deviation, has to be estimated from the sample standard

deviation.c. We don’t know the population mean and have to estimate it.d. The sample mean and variance are not finite.e. The population mean exceeds the critical value.

(Answer: b; when sigma has to be estimated from the sample data)

When we see a normal curve in a stats book, the author is most likely using the graph to refer to:

a. The normal distribution of sampling error.b. The normal distribution of every empirical variable.c. The normal distribution of variables in the population.d. The normal distribution of observer bias.e. The normal distribution specified by Chebycheff’s Inequality.

(Answer: a; the normal distribution of sampling error)Constructing confidence intervals for means requires us to have which of the following items of information before we can solve the problem?

a. Sample size.b. Sample mean, sample size, and the estimated standard deviation.c. Sample size and sample mean.d. Only the estimated standard deviation is needed.e. The UCL and the LCL.

(Answer: b; sample mean, sample size, and estimated standard deviation)

Which of the following does NOT contribute to making the confidence interval more precise (narrower)?

a. Low variability in the population.b. Higher confidence level (e.g., 99% rather than 95%).c. Larger sample size.d. Lower confidence level (e.g., 95% rather than 99%).e. Right answer is both a and b.

(Answer: b; higher confidence level)

Note to the instructor: This question requires discussion in class—students are unlikely to get it from reading alone.

In the computation of the standard error of a proportion for constructing a confidence interval, some texts suggest always using as the value of the standard deviation:

a. 1 because it will maximize the standard deviation.b. .5 because it will maximize the standard deviation.c. –1 because it will minimize the standard deviation.d. 0 because it will minimize the standard deviation.e. None of the above.

(Answer: b; .5 will maximize the standard deviation and is therefore “safest”)

Which of the following sample sizes, all other things being equal, will produce the most precise (narrowest) confidence interval?

a. 900b. 1200c. 25d. 100e. All will produce the same interval, because the calculation depends only on the mean and

the standard deviation.(Answer: b; 1200)Which of the following standard deviations will produce the most precise (narrowest) confidence interval, all other things being equal?

a. 25b. 15c. 30d. 500e. All will produce the same interval because the calculation depends only on sample size.

(Answer: b; 15)

How would we change the formula for the UCL in constructing a confidence interval if we wanted a 99% confidence level instead of a 95% confidence level?

a. Substitute 99% for 95% in the formula.b. Substitute 2.58 for 1.96.c. Substitute .99 for .95.

d. Substitute 2 for 1.96.e. None of the above.

(Answer: b; substitute 2.58 for 1.96 in the formula)

Which of the following is the most plausible answer to this question: What are the UCL and LCL (expressed as percentages) for an interval with a 95% confidence level for candidate Ahmadinajad’s voter support based on a poll of 1200 voters?

a. 10% to 90%.b. .0001 to .0002.c. 58% to 62%.d. 0 to 100%.e. Could be anything; the question is too indefinite to answer without further information.

(Answer: c; 58% to 62%. This question reinforces the instructor’s encouraging students to always estimate plausible answers before a calculation. Only this answer is plausible.)

Shortweighting ProblemsA supplier claims that the mean weight of the vegetable packs is 200 grams (about half a pound). We test the claim with a sample of N packages and find a mean weight of for the packages with a standard deviation of s.

For each set of information, determine whether or not the likelihood that shortweighting has occurred is statistically significantly.

N = 400, = 199 grams, s = 2.N = 100, = 199.5 grams, s = 5.

Note to the instructor: You can vary the sample size, the observed mean weight, and the standard deviation. Problems such as these require the students to be able to look at the t-distribution. (The table can be copied from the text and passed out.) It is also a good idea to go through this exercise in class. First give them the one-sample t-test formula; after a number of runs, encourage them to memorize the formula and to explain the steps in the procedure (i.e., the formula, the substitutions, the meaning of the result, looking it up in the t-table, correct interpretation in terms of the null hypothesis). Don’t give them this problem on a test until similar examples have been worked through in class.

When we compute the test statistic t for a one-sample t-test with a test value, what are we doing?

a. Comparing two sample means.b. Asking if the test value is equal to the null hypothesis.c. Asking if the sample mean is highly discrepant from the test value.d. Asking if the sample standard deviation is different from the observed standard deviation.e. None of the above.

(Answer: c; asking if the sample mean is discrepant from the test value specified by the null hypothesis)

The test value in a one-sample t-test represents:

a. The mean found in the sample.b. The value posited in the null hypothesis.c. The standard deviation found in the sample.d. The critical value of t.e. All of the above.

(Answer: b; the value posited in the null hypothesis)

Note to the instructor: One-sample t-tests are difficult to understand because we have to supply the test value, whereas, in most hypothesis tests, the null hypothesis test value is given by the logic of the problem.

What is meant by “the critical value” of a test statistic such as Z or t?

a. The value that is associated with a specified level of risk of Type I error.b. A value that, if exceeded by the calculated value of the test statistic, allows us to reject

the null hypothesis.c. A value that represents a specified number of standard deviation units of the sampling

distribution.d. A value that falls in the values of the sampling distribution and does not pertain to the

range of the underlying empirical variable.e. All of the above

(Answer: e; all of the above)

The units of a calculated t statistic are:

a. Empirical units like metres or dollars.b. Estimated standard errors.c. Values of the sample mean.d. The square root of sample size.e. It is not expressed in units of anything.

(Answer: b; estimated standard errors)

Characteristics of the t-distribution include:

a. It is virtually identical to the Z-distribution when N > 120.b. When sample size is small, it has “fatter tails” and a lower peak than the Z-distribution.c. It assumes a near normal distribution of the underlying empirical variable, but many

researchers are “careless” about this condition.d. It is used when sigma has to be estimated from s.e. All of the above.

(Answer: e; all of the above)

All the “rigmarole” about hypothesis testing is necessary because:

a. It makes our observed data more precisely measurable.b. It makes our findings more mathematically precise.c. It helps prevent mistaking sampling variability for real differences in outcomes.d. a and b.

e. a and c.(Answer: c; it helps prevent mistaking sampling variability for differences in outcomes)

Candidate Candy Date’s polls show her with support from 60% of a random sample of 900 voters. Which of the following is a plausible estimate for a 95% confidence interval for her support? (Hints: Use 50% (.5 after the percentages are converted into decimal fractions) as the estimated standard deviation of the sample because it provides the maximum—safest—estimate of variability. Convert the percentages into decimal fractions. Use 2 as a “safe” round off of 1.96.)

a. UCL = 65%, LCL = 55%b. UCL = 67%, LCL = 77%c. UCL = 90%, LCL = 30%d. UCL = 60.5%, LCL = 59.5%e. UCL = 450, LCL = 440

(Answer: a; 55% to 65%)

Note to the instructor: You may want to explain in class that polling organizations often use sample sizes of about 1200 respondents, which yield 95% confidence intervals with a plus or minus 3% margin of error, typically considered acceptable. In this problem, the margin of error is a little larger. The problem also reiterates the importance of estimating what an answer is likely to look like—thus eliminating immediately the wrong answers.

A light bulb manufacturer claims that the average life of its light bulbs is 800 hours. We test a sample of 800 randomly selected light bulbs and find that their mean life is 798 hours with a standard deviation of 20 hours.

The test value for the null hypothesis is:

a. 20 hours.b. 800 hours.c. 800 bulbs.d. The square root of 800.e. 798 hours.

(Answer: b; 800 hours, the manufacturer’s claim)

In this problem, N =

a. 20 hours.b. 800 hours.c. Square root of 800 bulbsd. 798 hours.e. 800 bulbs.

(Answer: e; 800 bulbs)

In this problem, the standard error of the mean is:

a. The square root of 800.

b. 20 divided by the square root of 800.c. 798 – 800.d. 800 divided by the square root of 800.e. Can’t be computed; we do not have enough information.

(Answer: b; 20 divided by the square root of 800)

In a one-sample t-test, which of the following conditions make it more likely that the null hypothesis will be rejected?

a. A large sample.b. Low variability in the distribution.c. A large discrepancy between the sample mean and the null hypothesis test value.d. Conditions a and c.e. Conditions a, b, and c.

(Answer: e; conditions a, b, and c)

Kay’An tests 36 light bulbs and fails to reject the manufacturer’s claim of a mean life of 800 hours. But she remains convinced that the bulbs have a shorter mean life than advertised. She could:

a. Retest with a larger sample.b. Change the test value.c. Change the sample standard deviation.d. Both a and c.e. None of the above; there is nothing she can do.

(Answer: a; retest with a larger sample)

Which of the following procedures is best for producing a simple random sample from the student population at a university?

a. Obtain an alphabetical list of all students and select every 10th name.b. Obtain an alphabetical list of students, assign numbers, and use a random-number

generator on a computer to identify which students to select.c. Carefully make a list of the key variables in the study, identify the categories for each

variable, and then make sure that there are enough individuals in each category of each variable that the sample accurately and precisely reflects their proportions in the student population.

d. Invite people whom you know to be in the study, and ask each of them for the names of two additional individuals to include because personal relationships guarantee a robust response rate.

e. Ask for volunteers by advertising in the campus newspaper, through the campus radio station, on residence hall bulletin boards, and on university web pages.

(Answer: b; use an alphabetical list of students, then assign numbers to cases, and finally select numbers randomly)

Ilana wants to challenge a food company claim that each package of Mint-Os contains an average of 100 pieces. She tested 400 packages and found an average of 99 pieces per package. What can she conclude at this point?

a. She should go ahead with her challenge—the mean weight is well below the claim.b. Nothing—she is still missing a vital piece of information.c. She needs to test more packages.d. Her suspicions are not supported by the data; 99 pieces is well within the acceptable

range.e. She needs to identify how many packages actually contained more than 100 pieces before

reaching any conclusion.(Answer: b; she is missing a vital piece of information)

Comment for discussion: In the previous problem, Ilana cannot go ahead with her challenge because she is missing a key piece of information—the standard deviation of the distribution of the candy pieces per package. Explain why this information is necessary for the t-test and the conclusion.

Answering “by the formula” We need to know s in order to compute t.

Answering by the concept High variability produces a larger standard error, which, in turn, shrinks the value of t and makes it less likely to exceed the critical value.

Answering by intuitive thinking More variability in the distribution means that the mean can be more “all over the map” and that it is harder to hone in on the expected value.

Chapter Four

A bivariate relationship means a relationship between _______ variables:

a. Two binary.b. Two.c. Binary.d. Dichotomous.e. All of the above.

(Answer: b; two)

Identifying a relationship among variables means comparing patterns of _______ in the variables.

a. Operationalization.b. Inference.c. Variability.d. Hypotheses.e. Induction.

(Answer: c; variability)

When the independent and dependent variables are categoric, the technique of data analysis is usually_________:

a. Crosstabs.b. F-test.c. Interval-ratio.d. Regression analysis.e. ANOVA.

(Answer: a; crosstabs)

With _________ variables, we often carry out a linear regression analysis:

a. Nominal.b. Interval-ratio.c. Categoric.d. Multi-category.e. None of the above.

(Answer: b; interval-ratio)

For ANOVA, the variables need to be:

a. Categoric IV, interval-ratio DV.b. Binary outcome, nominal predictor.c. Categoric IV, categoric DV.d. Interval-ratio IV, multi-category DV.e. None of the above.

(Answer: a; categoric IV, interval-ratio DV)

R, Pearson’s correlation coefficient, has which of the following properties?

a. It ranges from 0 to infinity.b. It can never be equal to 0.c. It ranges from –1 to +1.d. It can be read as a percentage.e. It can never be a negative number.

(Answer: c; it ranges from –1 to +1)

An R of –1 is interpreted as indicating:

a. A weak negative relationship.b. No relationship at all.c. A strong negative relationship.d. That there are too many outliers in the distribution.e. None of the above because R can never have a value less than 0.

(Answer: c; a strong negative relationship)

Pearson’s correlation coefficient, R, is computed as:

a. ∑ ZxZy

b. ∑ ZxZy/Nc. ∑ ZxZy/Sxy

d. ∑ ZxZy/Sx

e. None of the above.(Answer: b; the average of the Z-score products)

When an increase in the independent variable is associated with a decrease in the dependent variable—

a. The relationship is said to be inverse.b. The sign of R is negative.c. The OLS regression line for the scatterplot slopes from lower left to upper right.d. a and b.e. a, b, and c.

(Answer: d; a and b. The answers c and e are incorrect because c is wrong—the line slopes the other way.)

R expresses the _____ and ______ of a relationship.

a. Strength and direction.b. Correlation and regression.c. Dispersion and centrality.d. Variability and magnitude.e. None of the above.

(Answer: a; strength and direction)

Another formula for R is:

a. The covariance of X and Y.b. The covariance of X and Y, divided by the product of their standard deviations.c. The covariance of X and Y, divided by the standard deviation of X.d. The covariance of X and Y, divided by the standard deviation of Y.e. The covariance of X and Y, divided by the product of their variances.

(Answer: b; the covariance of X and Y, divided by the product of their standard deviations)

OLS stands for:

a. Ordinary linear summation.b. Ordinary lesser sum.c. Ordinal lesser sum.d. Ordinary least squares.e. Orthogonal lambda score.

(Answer: d; ordinary least squares)

The slope coefficients are tested for significance, with the null hypothesis usually being:

a. b = 1.b. b = 0.c. b = the standard deviation of the X-distribution.d. b = the standard deviation of the Y-distribution.e. None of the above.

(Answer: b; b = 0)

An OLS regression should be carried out only if the scatterplot looks:

a. Exponential.b. Linear.c. Logarithmic.d. Like a funnel.e. Like a series of chimneys above selected points on the x-axis.

(Answer: b; linear)

In a regression, “beta” or “beta-weight” refers to a:

a. Slope coefficient computed using Z-scores.b. The significance test for the constant term.c. The value of R2.d. The unstandardized slope coefficient.e. All of the above.

(Answer: a; slope coefficient computed using Z-scores. However, be aware that “beta” is also the symbol for the population parameter for the slope coefficient—this can be confusing.)

R2 is called the __________________________

a. Coefficient of predictability.b. Coefficient of determination.c. Slope coefficient.d. Correlation coefficient.e. Root-squared deviation.

(Answer: b; coefficient of determination)

R2 can be read as a _______________

a. Proportion.b. Slope coefficient.c. Number between –1 and +1.d. Standardized coefficient.e. None of the above.

(Answer: a; proportion)

R2 expresses:

a. The direction—positive or negative—of a relationship.b. What proportion of the variance in the IV is linearly predictable from the DV.c. What proportion of variation in the DV is linearly predictable from the IV.d. The observed value of X for a given Y.e. The estimated value of X for a given Y.

(Answer: c; proportion of variation in the DV predictable from the IV)

Examining data from a set of 120 countries, Carlos discovers that R2 for the infant mortality rate (as the DV) estimated from the women’s literacy rate is .6. The best way to word the interpretation of this result is:

a. 60% of the IMR in a country is caused by the level of women’s literacy.b. 60% of variation among countries in their IMR can be predicted from their women’s

literacy rates.c. Having a lot of literate women in a country reduces the IMR by 60%.d. If 60% or more of the women in a country are literate, the country’s IMR will be lower.e. None of the above.

(Answer: b; 60% of variation in the IMR among countries can be predicted from women’s literacy rates.)

Camilla is using a data set of 180 countries to analyse whether countries’ per capita incomes can be used to estimate their infant mortality rates. Which of the following statements does NOT apply to her data analysis?

a. In a scatterplot, she would place per cap income on the x-axis.b. In a scatterplot, she would place infant mortality rate on the x-axis.c. She will use the data to see if she can calculate slope coefficients for an OLS regression

line.d. She will look at ANOVA output and an F-test for the significance of R2.e. Statement b and statement d do NOT apply, but a and c do apply.

(Answer: b; infant mortality is placed on the y-axis, not the x-axis.)

Note to the instructor: This question is a bit of a trick because the fact that an ANOVA for R2 is indeed included in regression analysis can be confusing. Be sure to draw students’ attention to the use of ANOVA and an F-test for the significance of R2 as part of regression analysis.

Which set of variables is best suited for an ANOVA as a data analysis technique?

a. Race/ethnicity as the IV and type of pet (large, medium, small) as the DV.b. College majors as the IV and GPA as the DV.c. High school GPA, family income, and SAT score as IVs, and whether the individual

highest degree is elementary school, high school, college, or post-BA. d. High school GPA as the IV and college GPA as the DV.e. GPA as the IV and pet type (large, medium, small) as the DV.

(Answer: b; college majors and GPA)

Note to the instructor: An alternative wording of the question is to provide the list of answers and ask students to identify the best data analysis technique for each pair of variables. As a fill-in-the-blanks question, ask students to identify two variables, an IV and a DV, suitable for ANOVA.

Janice has collected the following data from a sample of 300 college students: family income in dollars, whether or not the student owns a pet, college grade point average (GPA), high school GPA, Scholastic Aptitude Test (SAT) score, and family place of residence (urban, suburban, or rural).

Short Essay QuestionFormulate a plausible research question for Janice and suggest an appropriate data analysis technique to answer it. (Answers can vary.)

Janice recodes the family income variable into position in quartiles (quartile 1 having the lowest incomes and quartile 4 the highest). Using this new variable as the IV, which of the following variables can Janice NOT use as the DV in an ANOVA?

a. Pet ownership, coded dichotomously.b. High school GPA.

c. SAT score.d. College GPA.e. Family place of residence.

(Answer: e; family place of residence. The dichotomous variable—pet ownership—can be used because the mean of a dichotomous variable distribution can be computed.)

Janice is preparing a crosstab for two variables in her study of college students: race/ethnicity and attendance at university events such as lectures and films. She codes attendance as “never, sometimes, and often.” The best table layout for the data analysis is:

a. Race/ethnicity in the rows and attendance in the columns.b. Race/ethnicity in the columns and attendance in the rows.c. Either one of the above because either variable could be the IV.d. Race/ethnicity in the marginals and attendance in the cells.e. All of the above are acceptable, depending on the purpose of the study.

(Answer: b; race/ethnicity in the columns and attendance in the rows)

After Janice has examined the crosstab for race/ethnicity and event attendance, she wants to add a third variable as a layer in the table to elaborate the analysis. Which of the following would NOT be a good choice?

a. Students’ majors—commerce, liberal arts and sciences, theatre.b. Family income in dollars.c. Location in a quartile of the family income distribution.d. Whether or not the student is employed.e. Whether or not the student lives on campus.

(Answer: b; family income in dollars. It is an interval-ratio variable.)

Note to the instructor: As a short answer question, you could have students explain which variable would not be a good choice and why not.

In Janice’s study, she finds that Latino students are less likely to attend campus events than students of other ethnic backgrounds because they are more likely to be commuters and less likely to live on campus in the residence halls than other students. In this elaboration of the bivariate analysis—

a. Residence is best thought of as explaining away a spurious relationship between ethnicity and attendance.

b. An interaction effect between ethnicity and attendance is revealed.c. Residence is best thought of as an intervening variable that helps to interpret the

association of ethnicity and attendance.d. The three-variable analysis replicates the bivariate analysis.e. None of the above.

(Answer: c; intervening variable)

Note to the instructor: The challenge in many of these third variable elaborations is to distinguish the intervening variable situation from the spurious relationship. Get students to think of a diagram indicating time order.

Difficult QuestionFor some features of a crosstab, the choice of which variables to put in the rows and which in the columns makes a difference, and, for others, it does not. For which of the following does the row/column choice make NO difference?

a. For the computation of lambda.b. For the title of the table in SPSS/PASW output.c. For the computation of chi-square, its level of significance, and its degrees of freedom.d. No difference for a and c.e. No difference for a and b.

(Answer: c; chi-square is symmetric. The other elements of the crosstab are not.)

In a large data set in which the cases are 120 countries, we find a .896 correlation between male and female literacy rates. The best way to interpret this result is:

a. Women’s literacy is on average 89.6% of men’s literacy.b. The two variables are strongly and positively related.c. The higher men’s literacy rate, the lower women’s literacy rate.d. Almost 90% of variation in men’s literacy in each country can be predicted from

women’s literacy in that country.e. In only 10% of the countries are men’s and women’s literacy rates very different.

(Answer: b; the variables are strongly and positively related)

For the finding described in the previous question, the coefficient of determination is:

a. 89.6.b. 100 – 89.6 = 10.4.c. About 81%.d. 89.6/10 = 8.96.e. Cannot be calculated from the information given.

(Answer: c; about 81%)

What proportion of the variation in men’s literacy rates has to be predicted by “residual” variables that are not specified by the regression?

a. 10.4%.b. 89.6%.c. 1.04%.d. 81.4%.e. None of the above.

(Answer: a; 10.4%)

Using a sample of 100 countries, Paola performs an ANOVA for homicide rate by the quartile of the Gini coefficient of income inequality into which the country falls. (The Gini coefficient is a measure of inequality that ranges from 0 to 100, with 0 meaning complete equality and 100 meaning that all income goes to one individual. Values of the Gini coefficient typically fall between 20 and 60.) For the ANOVA, Paola finds that the p-value of F is .04. What is the most precise statement of her conclusion?

a. There are no significant differences among the quartiles in their mean homicide rate.b. High homicide rates cause inequalities.c. There are significant differences among quartiles of the Gini coefficient variable in their

mean homicide rate.d. There are no significant differences among countries in their homicide rates.e. Countries differ significantly in their homicide rates.

(Answer: c; there are significant differences among the Gini quartiles in the homicide rate means)

In the study described in the preceding question, what is the next step in the analysis?

a. Repeat the study with a larger sample in order to obtain a significant result.b. Perform a post-hoc (multiple comparison) test to see which of the four quartiles have

significantly different mean homicide rates from each other.c. Perform a post-hoc (multiple comparison) test to see which countries have significantly

different levels of inequality.d. Recalculate F to see if a higher p-value can be obtained.e. None of the above.

(Answer: b; post-hoc test to see which quartiles have significantly different mean homicide rates from each other)

Paola could recode her data from the study described in the preceding question into two new variables defined on countries: homicide rate (high, medium, low) and level of inequality (high, medium, low). When she does that, she would probably carry out ______ to examine the relationship between the variables.

a. An ANOVA of homicide rate by inequality.b. A crosstab of homicide rate by inequality.c. A crosstab of inequality by homicide rate.d. A regression of homicide rate by inequality.e. Any of the above would be good techniques.

(Answer: b; crosstab of homicide rate by inequality)

In the recode described in the preceding question, Paola created:

a. Two ordinal variables.b. Two nominal variables.c. Two interval-ratio variables.d. Two dichotomous variables.e. Two continuous variables.

(Answer: a; two ordinal variables)

Paola would like to know if the Gini coefficient (the measure of inequality described in the preceding questions), national per capita income, and the literacy rate can serve as predictors of the crime rates of countries. The technique she would probably use is:

a. Crosstabs.b. Crosstab with all four variables.c. Logistic regression with the crime rate as the outcome variable.d. Linear regression with three predictor variables and crime rate as the outcome variable.e. ANOVA with crime rate and literacy rates as the dependent variables.

(Answer: d; linear regression, crime rate as the outcome variable)

If Paola had recoded crime rate as a binary variable (high or low) she could redo the analysis in the preceding question as a:

a. Crosstab.b. ANOVA with crime rate as an independent variable.c. Crosstab with four variables.

d. Logistic regression with crime rate as the outcome variable.e. None of the above.

(Answer: d; logistic regression with crime rate as the outcome variable)

Which statement best describes what is accomplished in a bivariate linear regression?

a. We are trying to estimate the values for the dependent variable because our observed data only includes the values of the independent variable.

b. We are trying to create a mathematical model that will be valid regardless of the observed data.

c. We are testing how well our model predicts the observed values of the dependent variable from information about the observed values of the independent variable.

d. We are testing whether there is a causal relationship between the variables.e. All of the above.

(Answer: c; we are testing how well the model predicts the observed values of the DV given the information about the observed values of the IV)

Note to the instructor: This question requires prior class discussion of what we are doing in a linear regression. We are

Creating a model that is linear; Looking at observed data for both the IV and DV, and using observed data as the

basis of the model—the model we create relies on these data; Testing the model against observed data; and Not making any causal claims, only testing the strength of relationships.

See Kennedy (2003), cited in the bibliography of The Joy of Statistics, for more on models and what we are trying to accomplish in regression analysis.

Leila creates a crosstab for two categoric variables: gender (men and women) and interest in electronics shopping (high, medium, and low). The df for this table would be:

a. 6b. 2c. 3d. 4e. Can’t tell unless we know sample size.

(Answer: b; 2. (3 – 1)(2 – 1) = 2 is the equation.)

Which of the following characteristics in NOT a correct statement about the chi-square statistic?

a. Its degrees of freedom are (r – 1)(c – 1).b. It is never less than 0.c. It is sensitive to sample size—a large sample makes it more likely to fall in the critical

region.d. It is symmetric: It does not matter which variable is put in the rows and which in the

columns.e. All of the above are correct statements.

(Answer: e; all the statements are correct)

Jeff claims that the percentages in a crosstab should always be run in the direction of the independent variable. What is the best response to his claim?

a. Yes, but remember that this rule is just a convenient convention.b. Yes, because otherwise the relative sizes of the IV categories will confound the answer.c. No, the percentages must be run in the direction of the DV.d. No, the percentages should always be computed on the basis of the total number of cases.e. No, percentages should always be computed in three ways: in the direction of the IV, the

direction of the DV, and on the basis of the total number of cases.(Answer: b; yes, otherwise the relative sizes of the IV categories will confound the answer)

We want to elaborate a bivariate crosstab with a third variable. Our best choice would be:

a. A variable measured at the interval-ratio level.b. A continuous variable.c. A categoric variable with few categories—2 or 3.d. Either a or b.e. All of the above are good choices.

(Answer: c; categoric variable with few categories)

In the F-test, a large value in the numerator of F relative to the denominator means that:

a. A lot of variation in the IV means is associated with the DV categories.b. A lot of variation in the DV means is associated with the IV categories.c. The within-groups variance is large compared to the between-groups variance.d. The result of the F-test is not likely to be significant.e. The p-value of the F-test is likely to be high.

(Answer: b; a lot of variation in the DV means is associated with the IV categories)

An F-test and an independent-samples t-test will have identical results, and either one could be used when:

a. N = 2.b. The t-test value is at least 120.c. N > 2.d. There are two IV groups.e. The post-hoc results indicate this condition is acceptable.

(Answer: d; there are two IV groups)

In SPSS/PASW output for a regression, in the table of coefficients, we see the results of t-tests. What null hypothesis is being tested?

a. That the slope coefficient is 0.b. That the absolute value of Pearson’s correlation coefficient is greater than .8.c. That the absolute value of Pearson’s correlation coefficient is greater than 1.d. That R2 is 1.e. None of the above.

(Answer: a; that the slope coefficient is 0)

A Question for Measures of AssociationWe obtain lambda, and it equals 0 for a specific crosstab. What can we conclude?

a. That the result is highly significant.b. That all the IV categories have the same DV mode.c. That the observed and expected counts in the cells are virtually identical.d. That the sample size was too small.e. All of the above.

(Answer: b; that all the IV categories have the same DV mode)

Which of the following variable relationships is almost certainly negative?

a. Individuals’ weights and the distances that fire fighters can carry them.b. Heights and weights.c. IQ and shoe size among adults.d. The distances students live from campus and how long it takes them to commute to their

classes.e. Heights of buildings and the square feet (or metres) of office space available in them.

(Answer: a; individual weights and distances fire fighters can carry them)

Which of the following statements is most likely to be the null hypothesis in a study of water pollution and frog survival in lakes and ponds?

a. The higher the pollution level, the larger the frog population.b. The ponds’ pollution levels are not related to differences in frog survival rates.c. The higher the pollution level, the lower the frog population.d. The higher the pollution level, the lower the frog survival rate.e. The lower the frog population, the higher the pollution rate.

(Answer: b; no differences in survival rates by pollution levels)

Discussion QuestionWhen we say a relationship (association or correlation) is causal—what do we mean? (Answer: This is a very complex question—some students will be impatient with the conversation. The answer usually involves three conditions: that the variables are indeed associated or correlated, that the candidate for the causal variable does not follow the other variable but either is prior or concurrent (i.e., time order), and that we cannot find a third variable that explains away the relationship as spurious. Some people might argue that we must also be able to specify the “causal mechanism”—how the cause actually produces the effect.)

Discussion QuestionGive examples that show the difference between the situation in which the third variable explains away the spurious relationship between the other two variables and the situation in which the third variable is an intervening variable? Can we reach any conclusions about causality in the case of the intervening variable?