Stats and research methods

Question 1

What does the area under the curve of a normal distribution represent?

a. the proportionality of obtaining any given range of values
b. the proportionality of obtaining the highest value in a specified range.
c. the probability of obtaining the lowest value in a specified range
d. the probability that the mean of a population exists in a specified range.

Answer 1

The correct answer is A. The area under the normal curve (the integral) represents the probability of obtaining a range of values. (Technically speaking, it is the area under the density function of the normal distribution.)

Question 2

Imagine a situation in which the null hypothesis is true, and α is (as usual) set to .05. You collect some data, and do your statistical test and get a p-value of .01. If you kept repeating this process of collecting data and doing a statistical test, what percentage of the time would you expect to reject the null hypothesis?

A. 1%

B. 5%

C. 95%

D. 99%

Answer 2

Answer: B. If you refer back to where α was mentioned in the videos you will see that it’s meant to determine the long-term rate of falsely rejecting the a true null hypothesis. The fact that you got a p-value of .01 when you did your first test is irrelevant - the long-term rate of rejecting a true null hypothesis is still meant to be 5%.

By the way, when dealing with psychological data the null hypothesis is almost never true.

Question 3

You have a random sample of 30 data points. Which of these facts would provide the strongest indication of non-normality in the population from which these data points were sampled?

A. The mean and mode and median are all the same

B. The mean is much lower than the median

C. The standard deviation is over 9000

D. There are 0 data points more than 2 standard deviations from the mean

Answer 3

B is a sign of skewness and thus non-normality. In a perfectly normal distribution mean = median = mode and outliers are rare, so A and D can’t be right. C is irrelevant because the value of the standard deviation depends on the scale of the data. The question didn’t tell you whether the range was (for example) 0 to 20,000 or 0 to 60,000, and you can’t really make any sort of judgment about how large “over 9000” is in absolute terms.

Question 4

Which of these variables would be most likely to approximate a normal distribution?

A. Heights of 20-year-old women

B. Populations of human settlements

C. Total sales of single-volume books

D. Values obtained when throwing a fair die (that is, a single dice)

Answer 4

A is the best answer. It won’t be exactly normal, but if you have a large enough sample it will be a decent approximation. The reasons why are not relevant to this course, but simplifying a bit you can expect a (somewhat) normal distribution whenever an outcome is the result of a range of variables acting (somewhat) independently from one another. Regarding those variables, think of all the genetic and environmental factors that contribute to height. It’s true that they don’t necessarily act independently from one another, and that is one of the reasons why the distribution of heights is only approximately normal.

B isn’t right, because there are a whole lot of little townships with small populations, and a small number of colossal cities like Tokyo and Delhi.

C isn’t right for the same reason as B. There are a lot of books with relatively few sales, and then a very small few (e.g. A Tale of Two Cities, Harry Potter and the Philosopher’s Stone) with colossally more sales.

D would not resemble a normal distribution but would be uniform (otherwise known as ‘rectangular’) - each of the six values on the die is as likely as another.

Question 5

Stan Statissian runs a two-sample t-test examining whether science students arrive significantly later than arts students to his class. The p value is 0.4. Which of the following statements is the most reasonable conclusion to draw from this outcome.

A. There is evidence for the proposition that arts and science students arrive at class at the same time.

B. There is evidence against the proposition that arts and science students arrive at class at the same time.

C. There is no evidence for the proposition that arts and science students arrive at class at the same time.

D. There is no evidence against the proposition that arts and science students arrive at class at the same time.

Answer 5

The correct answer is D. The logic of null hypothesis significance testing is somewhat obtuse: we use the obtained data as evidence against the null hypothesis (we never endorse the null hypothesis, we can only fail to reject it). Thus, in this instance, where we have a p value above any reasonable alpha level (conventionally set to 0.05), we fail to reject the null hypothesis meaning that we have no evidence against the proposition that arts and science students arrive at the same time to class.

Question 6

Which of the following statements is implied if the null hypothesis is true?

A. Any observed differences are due to unobserved factors.

B. Any observed differences are due to controlled factors.

C. The experiment was not sensitive enough to pick up a true differences.

D. A false alarm cannot be discounted

Answer 6

The correct answer is A. If the null hypothesis is actually true, then any observed variability is simply due to random perturbations in the data generating process that have not been controlled for or specifically measured.

Question 7

A p-value tells you

Answers:
A. The probability of observing results as (or more) extreme than those actually observed, if the null hypothesis is true

B. The probability that, if the experiment were repeated many times, the results you obtained would be replicated

C. The probability the null hypothesis is false

D. The probability the null hypothesis is true

Answer 7

ANSWER: A. This definition of a p-value is functionally equivalent to the one Simon presented in the previous section. D is always a common answer here, and indeed it would be nice if we could know the probability the null hypothesis is true. However, using the statistics in the present course we do not attempt to get at this.

Question 8

Which of the following is NOT an example of doing statistical inference?

a. Trying to work out a population standard deviation
b. Trying to work out a sample standard deviation
c. Trying to work out whether a single observation could plausibly have been drawn from a certain population
d. Trying to work out whether two samples come from the same or different populations.

Answer 8

ANSWER: B. Working out a sample standard deviation is simply a matter of performing calculations on that sample data. By contrast, the other answers all involve inferring something about a population from a sample. This is most clear in C and D, which were drawn directly from the video. A is a little less obvious but (almost invariably) you will attempt to infer the population standard deviation from the sample standard deviation.

Question 9

What variation does a one-way between-groups ANOVA compare?

A. Between-groups variance and population variance.

B. Between-groups variance and within-groups variance.

C. Sample variance and population variance.

D. Within-groups variance and population variance.

Answer 9

The correct answer is B.

Question 10

Which of the following is the definition of the total sum of squared deviations?

A. It is the sum of the squared difference between each data point and the grand mean.

B. It is the squared sum of the difference between each data point and the grand mean.

C. It is the sum of the difference of the squared value of each data point and the grand mean.

D. Is the squared sum of the difference between the square root of each data point and the square root of the grand mean.

Answer 10

The correct answer is A. The easiest way to work out this question is to write each of the formulae out in full.

The sum of the squared deviations, (X-Xbar)², is also called the sum of squares or more simply SS. SS represents the sum of squared differences from the mean and is an extremely important term in statistics. Variance. The sum of squares gives rise to variance.

Question 11

Which SS measures the variability of each data point from the overall mean?

Answer 11

C. This follows from the definition of the SSTOTAL. In order to calculate the SSTOTAL you ignore what groups people are in and calculate the overall average score on the DV. Then you measure the difference between each data point and the overall average score on the DV, and square those differences. Then you add up those squared differences. By the way, the “Error” SS is another, less common, name for the SSWITHIN.

Question 12

What are the values of the following:

Sum of squares total

Sum of squares within

Sum of squares between

Answer 12

Sum of squares total: Take the difference between each observed scores and the mean score, square them, add them up!

Sum of squares within: Sum of squares of each group, added up

Sum of squares between: sum of squares total minus sum of squares minus the sum of squares within

Question 13

What is the F-statistic?

Answer 13

In general, an F-statistic is a ratio of two quantities that are expected to be roughly equal under the null hypothesis, which produces an F-statistic of approximately 1

Question 14

In ANOVA the different groups are called

A. Groups of the study

B. Levels of the factor

C. Observations of the sample

D. Parameters of the population

Answer 14

ANSWER: B. This is just a matter of terminology. ‘Factor’ is ANOVA-speak for independent variable, while we call the groups ‘levels’.

Question 15

You are replicating a previous study in which the researchers conducted an ANOVA comparing blue-eyed, green-eyed and brown-eyed participants on a measure of anxiety, and obtained a statistically significant result. However, you are using twice as many participants. Which of the following statements is true?

Answers:
A. The effect size would be expected to be bigger in the your study.

B. The null hypothesis will be even more likely to be false in your study than in the previous study.

C. The p-value should be approximately the same as in the previous study.

D. The SSTOTAL would be expected to increase substantially.

Answer 15

ANSWER: D. This follows from the way the SSTOTAL is calculated. As previously mentioned, we don’t get to find out the probability the null hypothesis is true (or false) and hence B can’t be right. Furthermore, the null hypothesis is either true, or it isn’t, and our study doesn’t change the fact of its truth or falsity. The magnitude of effect won’t change in any systematic way due to us taking a larger sample size, so A is wrong. We would expect the p-value to decline with a larger sample size, so C is wrong.

Sum of squares total: Take the difference between each observed scores and the mean score, square them, add them up!

Question 16

You report your ANOVA and conclude that there is a statistically significant difference between the three eye-colour groups, F (2, 500) = 7.72, p = .007. Which of the following statements is true?

A. Because the ANOVA is statistically significant we can proceed to consider pairwise comparisons between groups, so we find out where the statistically significant difference(s) lie. If the ANOVA was non-significant there would be no point doing pairwise comparisons because they would all be non-significant anyway.

B. The reader can conclude that all the group means must be very different from one another.

C. The test statistic is in the tail (rightmost part) of the F-distribution.

D. We have reported a measure of effect size.

Answer 16

ANSWER: C. D is wrong because p-values are not, properly speaking, measures of effect size. They reflect sample size as well as effect size. C is correct, as you can see from the explanation of the F-distribution provided by the video. B is wrong because a significant ANOVA allows us to infer that the group means are not all the same at population level, not that they are all different at population level. A is wrong for many reasons. First, it’s possible for pairwise comparisons to be statistically significant even when the ANOVA is not. Second, saying “so we find out where the statistically significant difference(s) lie” assumes that at least one pairwise difference will be statistically significant. That is likely, but it’s possible that all the pairwise differences will not be statistically significant.

Question 17

Which of the following most accurately describes the Student’s t distribution?

A. It is the distribution of sample means obtained when sampling a set number of times from a population with a fixed mean and an estimated standard deviation.

B. It is the distribution of sample means obtained when sampling a set number of times from a population with no fixed mean but with an estimated standard deviation.

C. It is the distribution of sample scores obtained when sampling a set number of times from a population with a fixed mean and an estimated standard deviation.

D. It is the distribution of sample scores obtained when sampling a set number of times from a population with no fixed mean but with an estimated standard deviation.

Answer 17

The correct answer is A. The null hypothesis provides a hypothetical distribution of sample means drawn from a distribution with a set mean and a standard deviation estimated from the sample standard deviation.

Question 18

For a single-sample t-test, Dan gave the formula t = M - μ / SM. Which of the following statements is true?

Answers:
A. If you assume the null hypothesis is true, then 1 is the most likely value of t

B. If you assume the null hypothesis is true, then 1 is the most likely value of (M - μ)

C. M - μ is an estimate of the standard deviation of the distribution of SM

D. SM is an estimate of the standard deviation of the distribution of (M - μ)

Answer 18

ANSWER: D. A and B are both wrong for the same reason. If the null hypothesis is true, then the most likely value of the sample mean minus the population mean is zero, and hence the most likely value of t is also zero. D is correct, though it’s a tricky question because you needed to remember that during the video Dan mentioned that SM was the standard error of (M - μ), and thus it is the standard deviation of the distribution of (M - μ). C is just factually incorrect.

Question 19

What does μ stand for?

Answers:
A. Population mean

B. Population standard deviation

C. Sample mean

D. Sample standard deviation

Answer 19

ANSWER: A. This is just a brute fact, though you could have got a hint if you remembered that sample statistics tend to be in Latin letters while population parameters tend to be in Greek letters.

Question 20

We know that sample means fluctuate from study to study. What do we use as a measure of how much variability we should expect between sample means?

Answers:
a. p-value

b. Standard error
c. Standard deviation
d. t-test

Answer 20

ANSWER: B. This is basically a definitional question. If you repeatedly sample from a distribution and record the means, and then calculate the standard deviation of those means, then you get the standard error of the mean. Since that makes the standard error a kind of standard deviation, C is a valiant guess, but still not as precise an answer as B.

Question 21

What is standard error?

Answer 21

a measure of the statistical accuracy of an estimate, equal to the standard deviation of the theoretical distribution of a large population of such estimates.

Question 22

What is standard deviation?

Answer 22

a quantity expressing by how much the members of a group differ from the mean value for the group

Question 23

In the context of a single-sample t-test, what will tend to decrease the standard error?

Answers:
A. Increased sample sizes

B. Increased variance in population scores on the DV

C. Both A & B

D. Neither A nor B

Answer 23

ANSWER: A. Increased sample sizes will tend to push the standard error down because it means we’re less likely to have randomly sampled a bunch of scores from one extreme end of the population distribution. B is wrong because increased variability within population scores would make sample means fluctuate more, not less. It’s likely that that higher variability in population scores would be reflected in higher variance in the sample scores on the DV.