Reading Quiz 4

Question 1

logarithmic (exponential) transformation

Answer 1

if (x, y) display approximately exponential shape then graph of (x, logy) will display approximately linear shape.

Question 2

steps in logarithmic transformation

Answer 2

1. graph original data set
2. plot ordered pairs (x, logy). shape should be approximately linear
3. find linear regression equation for logy in terms of x. answer calculator gives is of form logy-hat = ax + b. check correlation coefficient and residual plot to verify that equation is fairly good fit for data
4. take antilogarithm of both sides of equation to solve for y-hat

Question 3

power transformation

Answer 3

if ordered pairs (x, y) display approximate power function graph then graph of ordered pairs is (logx, logy)

Question 4

steps in power transformation

Answer 4

1. graph original data set
2. plot ordered pairs (logx, logy). shape should be approximately linear
3. find linear regression equation for logy in terms of logx. answer calculator gives is of form logy-hat = a + blogx. check correlation coefficient and residual plot
4. take antilogarithm of both sides to solve for y-hat

Question 5

important notes

Answer 5

1. when explanatory variable is years, transform date to years since so values are smaller
2. if function resembles power function then reasonable that point (0,0) should lie on graph
3. can use any type of logarithm in log transformation
4. extrapolation is use of regression line for prediction outside of range of values of explanatory variable x
5. interpolation is use of regression line for prediction inside range of values of x (more trustworthy)

Question 6

important notes cont

Answer 6

6. association does not imply causation
7. a lurking variable is a variable that has an important effect on relationship among variables but is not included in variables
8. a confounding variable is a lurking variable that affects only the response variable but creates a situation where it’s impossible to determine whether the affect on the response variable is caused by the explanatory variable, the confounding variable, or neither

Question 7

two way table

Answer 7

organizes data about two categorical variables

often used to summarize large amounts of data by grouping outcomes into categories

Question 8

row variables

Answer 8

label rows that run across the table

Question 9

column variable

Answer 9

label the columns that run down the table

Question 10

marginal distributions

Answer 10

row totals and column totals in a two way table give the marginal distributions of the two individual variables

Question 11

conditional distribution

Answer 11

look at one row and one column
find each entry in column as percent of column total
conditional distribution of row variable for each column in the table

Question 12

how to describe association between row and column variables when column variable is explanatory

Answer 12

compare conditional distributions

Question 13

how to describe association between row and column variables when row variable is explanatory

Answer 13

compare conditional distributions of column variable for each value of row variable

Question 14

simpson’s paradox

Answer 14

a comparison between two variables that holds for each individual value of a third variable can be reversed when the data for all values of the third variable are combined
example of effect of lurking variables on observed association

Question 15

common response

Answer 15

effect of lurking variables can operate through common response if changes in both explanatory and response variables are caused by changes in lurking variable

Question 16

confounding

Answer 16

cannot distinguish between two variables’ effects on the response variable

Question 17

best evidence that association is due to causation

Answer 17

comes from experiment in which explanatory variable is directly changed and other influences on response are controlled

Question 18

NEED TO DO MORE READING QUESTIONS

Answer 18

OKAY

Question 19

3. True or False: if we have a curvilinear function, and we want to straighten it out to make a linear function, we can’t do that by multiplying or dividing by constants or adding or subtracting constants (i.e. by using linear transformations).

Answer 19

true

Question 20

What are the transformations that are most commonly used, other than linear transformation?

Answer 20

A. Positive and negative powers, and logarithms.

Question 21

5. Linear growth is to adding a fixed amount per unit time as exponential growth is to ______ by a fixed amount per unit time.

Answer 21

multiplying

Question 22

7. Suppose we have a function y=ab^x, where a and b are constants and x is the explanatory or independent variable, and y is the response or dependent variable. Is this an example of an exponential function, or a power function?

Answer 22

exponential

Question 23

8. If y is an exponential function of x, plotting what function of y versus x should result in a linear graph?

Answer 23

the log of y vs x

Question 24

Suppose you do a regression of the log (base 10) of y versus x, and you get a nice linear scatterplot and a high coefficient of determination (r^2) when you do a regression. Now you can use this linear relationship for
prediction. Suppose someone (like a test-maker) asks you what the predicted value is of y (not log y) for a given value of x. How would you find it?

Answer 24

A. You’d use your equation to find the predicted value of log y. Then you take the antilog (or 10 to that number) to get the predicted value of y. In other words, you “untransform” the value back to the original scale. (as shown in Example 4.7 on page 274)

Question 25

0. If a variable grows exponentially, its logarithm grows how?

Answer 25

linearly

Question 26

11. Suppose we have a function y = ax^b, where a and b are constants and x is the explanatory variable and y is the response variable. Is this an example of an exponential function, or a power function?

Answer 26

power function

Question 27

12. To make an exponential function linear, we use the log transformation just with the response variable y. To make a power function linear, we use the log transformation with what?

Answer 27

both the explanatory and the response variable

Question 28

13. If you start with the power function y=ax^p, and take the log of both sides, what result do you end up with?

Answer 28

A. log y=log a +p log x.

Question 29

14. Suppose you have a data set, and its scatterplot is curved. Then you take the log of both explanatory and response variables, and plot them, and you get a line. What do you infer from this?

Answer 29

A. That the original variables were related according to a power function (or power law).

Question 30

15. When you plot the log of y vs. the log of x, do you give any meaningful interpretation to the slope of the line that you get? If so, what is it?

Answer 30

A. According to the equation log y =log a + p log x, the slope of the line is the power to which x is raised in the original power function.

Question 31

16. Suppose you plot the log y vs. the log x and you get a good line, with intercept 2 and slope 3. So log y = 2+3log x. Now you are asked to find the equation for y in terms of x, without logs in it. How do you do this?

Answer 31

A. You take the antilog of both sides (make both sides base 10). You get y = 10^(2+3 log x)  y = 10^2(10^log x)^3  y = 100x^3. That is, y=100 times x cubed.

Question 32

17. What type of variables does a two-way table describe the relationship between?

Answer 32

categorical

Question 33

These give us the distribution for each variable separately, in our sample. These distributions are called what?

Answer 33

marginal distributions

Question 34

21. Suppose you have three age groups, and you have data on how many individuals got educated to each of 4 different levels. Suppose you calculate, just for one of the age groups, the per cent of people in that age group who attained each level. This distribution of per cents for one age group is called what?

Answer 34

a conditional distribution

Question 35

22. Do the percents for a conditional distribution add to 100 for each of the different groups for which you calculate them?

Answer 35

yes

Question 36

23. Do the per cents for conditional distributions equal the per cents for marginal distributions?

Answer 36

not necessarily

Question 37

24. True or False: When describing the relationship between two quantitative variables, the scatterplot and the correlation coefficient are usually the graph and numeric measure of choice; but in describing the relation between two categorical variables, no single graph or numeric measure summarizes the strength of the association. We usually pick and choose among bar charts and pie charts and the reporting of various percents.

Answer 37

true

Question 38

26. There were two AP Statistics teachers. 40% of the 40 students in the first teacher’s classes got 5’s, and 25% of the 40 students in the second teacher’s classes got 5’s. People assumed that the first teacher is better. However, someone then studied the results based on whether or not the students scored above or below a certain cutoff on the SAT, before going into AP Statistics. The first teacher had 80% of students above this cutoff and 20% below. The second teacher had 20% above and 80% below. The first teacher had 50% of the “aboves” get 5’s, and none of the “belows.” The second teacher had 75% of the “aboves” get 5’s, and 12.5% of the “belows.” Now which teacher appears to be better, and why?

Answer 38

A. The second teacher, because a higher fraction of that teacher’s students got 5s from those both above the cutoff and below the cutoff.

Question 39

situation above is whose paradox

Answer 39

simpson’s

Question 40

28. True or False: In Simpson’s paradox, there is a lurking variable (a term you learned in chapter three), which predisposes the results against one of the two groups; controlling for the effects of that lurking variable by looking separately at the subsets formed by the categories of it reveals results in the opposite direction from those obtained when ignoring the lurking variable.

Answer 40

true

Question 41

29. When two variables X and Y are found to correlate with each other, of course two possible explanations for this association are 1) that X causes Y, and 2) (one not diagrammed on page 307) that Y causes X. Please name the other two possible explanations that are good to keep in mind when interpreting findings of associations.

Answer 41

A. Common response (z causes both x and y) and confounding (z, which is associated with x, may cause y).

Question 42

30. Someone finds that the degree of physical fitness in youth (as measured by heart rate recovery from exercise) is correlated with the number of ankle injuries the person has had. But before concluding that we should hurt the ankles of youth in order to make them more fit, a COMMON RESPONSE explanation for the association comes to mind. Can you posit this common response explanation?

Answer 42

A. That both fitness and ankle injuries are associated with more running or more athletic activity – both are responses to this basic causal variable.

Question 43

31. Suppose a researcher studies the effects of a way of teaching children not to be violent. The researcher gives the instruction to all the children in Mrs. Harmony’s classroom, and uses the kids in Mr. Gutsly’s classroom as a comparison group. But then the researcher realizes that Mrs. Harmony has a very different personality and interpersonal style than Mr. Gutsly: she tries to promote kindness and good will, whereas Mr. Gutsly is mainly interested in promoting competitiveness and not being wimps. What would we say about the variables of teacher personality and interpersonal style in this study?

Answer 43

A. That they are CONFOUNDED with the intervention. Thus the effects of these teacher variables can’t be distinguished from the effects of the intervention the study is meant to test.

Question 44

32. What is the strongest type of evidence for causal relations?

Answer 44

A. Well-designed experiments that are meant to control for all lurking variables. (These usually entail randomly assigning individuals to different conditions.)

Question 45

33. What’s the problem with doing a well-designed experiment, for example, to see what the effects of child abuse are?

Answer 45

A. We will never find it ethical to randomly assign children to conditions of child abuse versus nonabuse.

Question 46

34. Is it possible to come to valid causal inferences without doing experiments that randomly assign people to various conditions? Can you give an example of such?

Answer 46

A. Although your text says that “the only fully compelling method” of establishing causality is an experiment, we can and do come to valid causal inferences without randomly assigning people to conditions. The example of smoking and lung cancer is one where the evidence for causation is “overwhelming” despite no study in which people were randomly assigned to smoke or not smoke over many years.

Question 47

35. If a lurking variable can actually reverse the direction of results, do you think it is also possible that a lurking variable could result in lack of an observed association when in fact there is a causal influence?

Answer 47

yes

Question 48

36. Does the fact that lurking variables can obscure influences that are actually present imply that: not only does correlation not imply causation, but lack of correlation does not rule out causation?

Answer 48

yes