# Answers to Sample Statistics Questions

Correct answers are marked in Red Bold, with reasons in blue. Incorrect choices are black, with reasons they are wrong in green.
1. In a truly normal frequency distribution
a. the mean always is the same as the standard deviation
Not at all; could have a negative mean, but standard deviation is always positive
b. the mean is never the same as the mode
Actually, it is always the same
c. the mode is never the same as the median
Actually, it is always the same
d. the mean always is the same as the median
Mean, median, and mode are all the same; skew = 0

2. How might the standard deviation (S) of a normal distribution be greater  than the mean?
a. S is given by a square root, and the square root is larger than the fraction.
What does that have to do with anything?
b. In a normal distribution, the variance must equal the mean.
Not true, although it is true of a Poisson distribution (which we didn't get to)
c. If some scores are negative, the mean could be very small despite a large S.
The mean could be anything, even negative. Standard deviation is always  positive...
d. The median would have to be less than the skew.
Since the skew of a normal distribution is zero, this is only true if the mean is negative

3. In a class of 100, the mean on a certain exam was 50, the standard  deviation, 0. This means
a. half the class had scores less than 50
For the mean to be 50, the others would need scores above 50. If the scores are not all the same, there will be differences from the mean; those differences squared will add to a non-zero sum so standard deviation will be greater than zero.
b. there was a high correlation between ability and grade
You can't know about correlation from the distribution of a single variable -- what would it correlate with?
c. everyone had a score of exactly 50
A zero standard deviation means all scores are the same, and equal to the mean (what else could the mean be?)
That would make the mean = 25, and the standard deviation > 625 (25 squared for everyone)

4. The null hypothesis in an experiment would be
a. there is a high correlation between the independent and dependent variables
That could be an expectation, but the null hypothesis says there is no effect
b. changing the independent variable has no significant effect on the dependent variable
That is, every group (experimental or control or whatever) is a sample of the same population, even though the groups are differentiated by the independent variable.
c. changing the dependent variable causes a significant change in the independent variable
That's the opposite of the null hypothesis
d. the standard error of the dependent variable is greater than the mean of the independent variable
Isn't it amazing how random words in random order can sound like they mean something?

5. Suppose the mean on the final exam is 24 (of 40), with a standard deviation of 1.5.  If you get a 21, how well do you do (relative to the rest of the class)?
a. very poorly--perhaps the lowest score
That's 2 standard deviations below the mean (z = -2.0). The fraction in the lower tail is 0.0228 only 2 1/4% did worse! (Assume it's roughly a normal distribution....)
b. not well, but somewhere in the C's
If only 2 1/4% did worse, there won't be many D's or E's, will there?
Average is a z near 0
d. nicely--better than the median
Assuming an approximately normal distribution, median is close to the mean...

6. You can claim that there is a significant difference between scores from two  groups if
a. the difference between the means is large compared to the standard error
This is basically the definition of t (difference in means divided by standard error). A large t means you can reject the null hypothesis.
b. the means are large compared to the standard error
Size of means is irrelevant -- it's the difference that matters. You can't claim two groups differ if their means are the same, even if the means are 1,000,000 and standard errors are around 0.5.
c. the means are small compared to the standard error
As in (b) -- it's the difference between the means that matters
d. the difference between the standard deviations is large compared to the means
That's backwards -- it's difference between means, not standard deviations....

7. The correlation between a person's hair length and score on the midterm is very  nearly zero. If your friend has a crewcut, your best guess as to what he got on  the midterm is
a. the standard deviation of scores on the midterm
Why would standard deviation predict a score? The distribution of GRE scores has a mean of 500 and standard deviation of 100 -- 100 is not even a possible score!
b. the mean minus the standard deviation
Even less sensible that (a)
c. the mean plus the standard deviation
Same as (b)
d. the mean score
If correlation is zero, there is no added information from the other variable (hair length. Your best guess is the Expected Value, or the mean.

8. There is a low (but real) negative correlation between the amount of rain in  a given summer and the amount the summer before. In the absence of any  information except that this summer is wetter than usual, you are asked to  guess next summer's rain. Your best guess:
a. somewhat more than the average summer rainfall
Negative correlation means you expect less
b. the average summer rainfall
Correlation was real (different from 0) so regression will do better than the mean
c. somewhat less than the average summer rainfall
Since the correlation is negative, more rain this year means less next. The correlation is weak (small), so the regression line has a low slope; that is, it won't be very different from the mean.
d. the standard deviation of the rainfall