POL 138 exam 2
Suppose that we conduct an experiment in which the control group is asked to rate their support for Candidate X on a 0-to-100 scale, and the treatment group is shown a campaign video and then asked to rate their support for Candidate X on a 0-to-100 scale. For the 400 participants in the control group, the mean rating is 25 and the standard deviation of the ratings is 6. For the 400 participants in the treatment group, the mean rating in the treatment group is 25 and the standard deviation is 8. What is a t-statistic for that t-test, to the nearest integer? A. 0 B. -10 C. -20 D. -30
A. 0 No difference in means, so no evidence that the means differ.
Of the p-values below, which p-value is the strongest evidence that an observed difference between the percentage of heads and the percentage of tails from a set of coin flips would have been unlikely to have occurred due to random chance, if the coin were fair and the null hypothesis is that the coin is fair? A. 0.02 B. 0.04 C. 0.50 D. 0.99
A. 0.02 Lower p-values are stronger evidence
Which of the following would be a t-statistic that corresponds to a p-value of 0.05, in sufficiently large samples? A. t=0.00 B. t=0.05 C. t=1.00 D. t=1.96 E. t=2.49
D. t=1.96
Which p-value threshold below would produce fewer false positives, for null hypotheses that there is no association or effect? A. a p-value threshold of 0.01 B. a p-value threshold of 0.10
A. a p-value threshold of 0.01 Lower p-values thresholds indicate that stronger evidence is needed before concluding that there is an association. That lower p-value threshold thus makes it less likely to falsely conclude that there is an association (i.e., less likely to make a false positive inference).
All else equal, for an unfair coin, a p-value for a statistical test of the null hypothesis that coin is fair should ___ as the number of flips of an unfair coin increases. A. decrease B. increase C. not change
A. decrease Lower p-values are stronger evidence
If the drug test for a football player indicates that the football player is on steroids when the football player is not on steroids, then that is a ___ result for the test. A. false positive B. false negative C. true positive D. true negative
A. false positive
It is a ___ to believe that there is an effect when there truly is no effect. A. false positive B. false negative C. true positive D. true negative
A. false positive
Suppose that the mean height in a population is 170 cm and that the standard deviation of height in that population is 10 cm; suppose also that heights in this population follow a normal distribution. Regarding height in this population, a candidate who is 140 cm tall would be in which of the following ranges? A. less than the 2.5th percentile B. between the 2.5th percentile and the 97.5th percentile C. greater than the 97.5th percentile
A. less than the 2.5th percentile More than 2 standard deviations below the mean of a normal distribution
Based on the Law of Large Numbers, if we flip a fair coin over and over again, the more flips we make, the ___ likely it will be that the percentage heads on the total set of flips will approach 50 percent exactly. A. more B. less
A. more
Suppose that we randomly survey 100 people from a population and measure their support for the president. If we then survey 200 additional people from the same population, measure their support for the president, and add their data to the prior 100 participants, what would the addition of the 200 responses do to the standard error for support for the president? A. reduce the standard error B. increase the standard error C. not change the standard error
A. reduce the standard error Lower standard error means more precision, all else equal.
If the p-value is p=0.01 for a single statistical test of a null hypothesis that there is no association, do we have enough evidence to claim that there is statistically significant evidence for the detected association? A. yes B. no
A. yes
standard error
An estimate of the standard deviation of the sampling distribution of a statistic.
t-statistics are used instead of z-scores when ___. A. the population mean is larger than population standard deviation B. the population mean and standard deviation are not 0 C. the population mean and standard deviation are not 1 D. the population mean and standard deviation are not known
D. the population mean and standard deviation are not known
Which is the most common p-value threshold used in political science? A. 0.01 B. 0.05 C. 0.50 D. 1.00
B. 0.05
Suppose that, in an experiment, the mean for the control group was 4, the standard deviation for the control group was 2, the mean for the treatment group was 4, and the standard deviation for the treatment group was 3. What would be the p-value for a test of the null hypothesis that the control group mean equals the treatment group mean? A. 0 B. 1 C. something between 0 and 1
B. 1 No evidence of unfairness: means are the same
If we flipped a coin 12 times and got 6 heads and 6 tails, what would be the p-value for a statistical test of the null hypothesis that the coin is fair? A. 0 B. 1 C. something between 0 and 1
B. 1 no evidence of unfairness
The probability of X happening is 20%, and the probability of Y happening is 20%. X and Y are independent events. What is the probability that X and Y both occur? A. 2% B. 4% C. 20% D. 40% E. 400% F. Cannot be determined from the information provided
B. 4%
Suppose that researchers gave students a political knowledge test on the first day of class. Some students in the class are political science majors, and some students are not political science majors. To assess whether the mean score among students who are political science majors differed from students who are not political science majors, the researchers should use ___. A. a paired t-test B. an unpaired t-test
B. an unpaired t-test
Suppose that the mean height in a population is 170 cm and that the standard deviation of height in that population is 10 cm; suppose also that heights in this population follow a normal distribution. Regarding height in this population, a candidate who is 180 cm tall would be in which of the following ranges? A. less than the 2.5th percentile B. between the 2.5th percentile and the 97.5th percentile C. greater than the 97.5th percentile
B. between the 2.5th percentile and the 97.5th percentile
It is a ___ to believe that there is no effect when there truly is an effect. A. false positive B. false negative C. true positive D. true negative
B. false negative
If the p-value from a single statistical test of a null hypothesis is p=0.0001, do we have enough evidence to claim that the detected association is substantively large? A. Yes B. No
B. no
Below is a list of all 32 possible combinations of five flips of a fair coin: HHHHH HHHHT HHHTH HHTHH HTHHH THHHH HHHTT HHTTH HTTHH TTHHH HHTHT HTHTH THTHH HTHHT THHTH THHHT TTTHH TTHHT THHTT HHTTT TTHTH THTHT HTHTT THTTH HTTHT HTTTH TTTTH TTTHT TTHTT THTTT HTTTT TTTTT Each of these outcomes is equally likely for a fair coin. Suppose that you flip a coin five times and get five tails. If the null hypothesis is that the coin is fair, what would be the p-value for the test of whether the coin is fair? A. p=0.03125 B. p=0.06250 C. p=0.12500 D. p=0.50000 E. None of the above
B. p=0.06250 2/32
The term "statistically significant evidence" refers to ___. A. sufficient evidence that a particular association is large B. sufficient evidence that a particular association exists C. sufficient evidence that a particular association is important
B. sufficient evidence that a particular association exists
Suppose that, for a set of responses from two groups, for Group A, the mean is 10, the standard deviation is 5, and the sample size is 100, and, for Group B, the mean is 10, the standard deviation is 6, and the sample size is 100. Which ONE of the following would be the standard error to two decimal places for a test of the null hypothesis that the means equal each other? A. 0.11 B. 0.61 C. 0.78 D. 0.91 E. 3.00
C. 0.78 SE = sqrt(5*5/100 + 6*6/100) = 0.78
In an experiment, the control group had 100 participants, and the mean response was a 6, with a standard deviation of 2; the treatment group had 100 participants, and the mean response was a 10, with a standard deviation of 4. Which of these could be the t-statistic to one decimal place for a test of the null hypothesis that the control mean equals the treatment mean? A. 0.4 B. 1.3 C. 8.9 D. 10.4 E. 16.3
C. 8.9 t = (6-10)/sqrt(2*2/100+4*4/100) = -8.9 (the t=-8.9 is the same amount of evidence as t=+8.9)
Simpson's paradox item. Below are nine persons, by party (D for Democrat and R for Republican) and gender (F for female and M for male), and an indication of how these persons voted on four issues: A, B, C, and D. For example, Person 2 is a female Democrat who voted on Issue A and did not vote on Issue B, Issue C, or Issue D. Which issue, if any, fits the pattern in which female Democrats were more likely to vote than male Democrats were, female Republicans were more likely to vote than male Republicans were, but males overall were more likely to vote than females overall were? A. Issue A only B. Issue B only C. Issue C only D. Issue D only E. None of the issues F. More than one of the issues
C. Issue C only
Suppose that we conduct an experiment in which the control group is asked to rate their support for Candidate X on a 0-to-100 scale, and the treatment group is shown a campaign video and then asked to rate their support for Candidate X on a 0-to-100 scale. We then conduct a t-test to test the null hypothesis that the mean level of support for Candidate X in the control group equals the mean level of support for Candidate X in the treatment group. If the sample size is sufficiently large and the t-statistic for this test is 1.10, then which of the following should be done? A. The null hypothesis should be rejected and the alternative hypothesis should be accepted B. The null hypothesis should be accepted and the alternative hypothesis should be rejected C. Neither of the above
C. Neither of the above t is not at least 1.96 from zero
Select the ONE statement in the list below that is true. A. The lowest possible p-value is 0. B. The lowest possible p-value is 1. C. The highest possible p-value is 1. D. The highest possible p-value is 10.
C. The highest possible p-value is 1
Suppose that Researcher A has a study, and Researcher B has a study, and that both studies use participants randomly drawn from the same population. Both researchers are measuring support for the president on a 0-to-100 scale. Researcher A's standard error is 5, and Researcher B's standard error is 10. Which researcher's measurement of support for the president has the higher standard deviation? A. Researcher A B. Researcher B C. There is not enough information to correctly answer this question.
C. There is not enough information to correctly answer this question. The standard error formula includes standard deviation and sample size. If all we know is that the standard errors differ, we cannot conclude anything about whether the standard deviations differ, unless we also know the sample sizes.
A researcher conducted a randomized experiment and then compared the mean response of participants in Group A to the mean response of participants in Group B. The p-value for the difference in mean responses was p=0.99. Based on this p-value, the researcher should ___. A. conclude that there is an effect B. conclude that there is no effect C. conclude that there is not enough evidence to conclude that there is an effect
C. conclude that there is not enough evidence to conclude that there is an effect
Suppose that a researcher conducts an online survey to predict the next governor of Illinois. The survey is posted on the website of a popular conservative radio station in a county in downstate Illinois. However, the prediction from the survey does not closely match the results from the election. Of the options below, this weakness in the research design is BEST described as ___. A. a floor effect B. regression toward the mean C. selection bias D. Kelley's paradox E. Simpson's paradox
C. selection bias
If we flipped a coin 12 times and got 0 heads and 12 tails, what would be the p-value for a statistical test of the null hypothesis that the coin is fair? A. 0 B. 1 C. something between 0 and 1
C. something between 0 and 1 Some evidence of unfairness
If we flipped a coin 12 times and got 3 heads and 9 tails, what would be the p-value for a statistical test of the null hypothesis that the coin is fair? A. 0 B. 1 C. something between 0 and 1
C. something between 0 and 1 Some evidence of unfairness
Generally speaking, as p-values get lower, the associated t-statistics ___. A. decrease B. increase C. get closer to zero D. get farther from zero
D. get farther from zero
If the p-value for a test of a null hypothesis is p=0.02, then we should do which of the following? A. accept the null hypothesis and accept the alternative hypothesis B. reject the null hypothesis and reject the alternative hypothesis C. accept the null hypothesis and reject the alternative hypothesis D. reject the null hypothesis and accept the alternative hypothesis E. none of the above
D. reject the null hypothesis and accept the alternative hypothesis
If a pattern within two groups does not hold when the groups are combined, that is an example of ___. A. a floor effect B. regression toward the mean C. selection bias D. Kelley's paradox E. Simpson's paradox
E. Simpson's paradox
If the p-value for a test of a null hypothesis is p=0.98, then we should do which of the following? A. accept the null hypothesis and accept the alternative hypothesis B. reject the null hypothesis and reject the alternative hypothesis C. accept the null hypothesis and reject the alternative hypothesis D. reject the null hypothesis and accept the alternative hypothesis E. none of the above
E. none of the above
Below is a list of all 32 possible combinations of five flips of a fair coin: HHHHH HHHHT HHHTH HHTHH HTHHH THHHH HHHTT HHTTH HTTHH TTHHH HHTHT HTHTH THTHH HTHHT THHTH THHHT TTTHH TTHHT THHTT HHTTT TTHTH THTHT HTHTT THTTH HTTHT HTTTH TTTTH TTTHT TTHTT THTTT HTTTT TTTTT Each of these outcomes is equally likely for a fair coin. Suppose that you flip a coin five times and get 4 heads and 1 tail. If the null hypothesis is that the coin is fair, what would be the p-value for the test of whether the coin is fair? A. p=0.03125 B. p=0.06250 C. p=0.12500 D. p=0.50000 E. None of the above
E. none of the above 12/32
Suppose that the mean height in a population is 170 cm and that the standard deviation of height in that population is 10 cm; suppose also that heights in this population follow a normal distribution. Candidate X is 140 cm tall. What is the z-score that is associated with Candidate X's height in that population? A. -20 B. -2 C. 2 D. 20 E. None of the above
E. none of the above z=-3
The probability of X happening is 20%, and the probability of Y happening is 20%. X and Y are NOT independent events. What is the probability that X and Y both occur? A. 2% B. 4% C. 20% D. 40% E. 400% F. Cannot be determined from the information provided
F. Cannot be determined from the information provided
Suppose that a researcher is assessing the political knowledge among seniors in a political science major at a given university. The researcher emails each senior in the political science major at the university a link to a website that has a 100-item political science knowledge test. Ten percent of students who received the email opened the email, clicked the link, and submitted the test. Suppose that the mean number correct on the submitted tests was 75. Explain whether, considering selection bias, it should be expected that this 75 is an underestimate, an overestimate, or a correct estimate of the political knowledge among seniors in the political science major at the university.
Multiple acceptable responses, such as: Students who submitted the political knowledge test are plausibly more diligent, on average, than students who did not submit the political knowledge test, and diligent students might work harder in school to learn, so the 75 might be an overestimate if the sample is more diligent on average than the population.
Suppose that Antarctica is to be divided into a sufficiently large number of counties, in which half of these counties will have a small population, and the other half of these counties will have a large population. A large number of persons volunteer to move to live in Antarctica, and these volunteers are randomly assigned to live in a small population county or to live in a large population county. These volunteers are not permitted to move away from or travel from their assigned county until they have lived in Antarctica for at least five years. Suppose that, one year after the volunteers started living in Antarctica, the Antarctica government conducted a study of all Antarctica counties. Results indicated that, over the past year, of the 10 Antarctica counties with the highest crime rate, all 10 were small population counties. Crime in this example was measured accurately and with no error. Explain whether this would be sufficient evidence that, at least in Antarctica, having a smaller population in a county causes a higher crime rate, compared to having a larger population in a county.
No. Because of the Law of Large Numbers, it is expected that small samples will be more likely to be outliers than large samples. For the Antarctica example above, it is reasonable to expect that, compared to large population counties, the small population counties will be more likely to have high crime rates *and* to have low crime rates.
law of large numbers
a theorem that describes the result of repeating the same experiment a large number of times
selection bias
bias introduced by the selection of individuals, groups or data for analysis in such a way that proper randomization is not achieved, thereby ensuring that the sample obtained is not representative of the population intended to be analyzed
Simpson's Paradox
patterns within groups do not necessarily hold when the groups are combined
Suppose that, in a sample of 256 persons, the mean support for the president is 60 with a standard deviation of 16. What is a t-statistic for the test of the null hypothesis that this mean support for the president equals 50? Show all of your work.
t = (60-50)/(16/sqrt(256)) = 10
Suppose that we conduct an experiment in which the participants in the control group are only asked to rate their support for Candidate X on a 0-to-100 scale, and participants in the treatment group are only shown a campaign video and then asked to rate their support for Candidate X on a 0-to-100 scale. For the 200 participants in the control group, the mean rating is 60 and the standard deviation of the ratings is 12. For the 220 participants in the treatment group, the mean rating is 70 and the standard deviation of the ratings is 14. What is a t-statistic for a test of the null hypothesis that the control group mean equals the treatment group mean? Show all of your work.
t = (60-70)/sqrt(12*12/200 + 14*14/220) = -7.88
floor effect
the situation in which a large proportion of participants perform very poorly on a task or other evaluative measure
regression toward the mean
the tendency for extreme or unusual scores to fall back (regress) toward their average.
unpaired t-tests
when the same participant does not have data in two groups, such as if students in a night course were compared to different students in a day course.
paired t-test
when the same participant has data in two groups, such as if students in a course took a pretest and a posttest and then this pretest score for each student was then compared to the posttest score for the same student.
