- Is it possible for almost all the numbers in a sequence of measurements to be above their average?
- What's the shortest sequence of measurements that has 90% of the values measured falling below their average?
- For the sequence
1 2 1 1 2 2
compute the mean, mode, and median.
- For the sequence
1 2 6 1 2 2
compute the mean, mode, and median.
- For the sequence
1 2 1 1 2 2 1
compute the mean, mode, and median.
- What's the difference between a sample and a population?
- Is it possible for a group of students in a university to be viewed as a sample for one purpose and as a
population for the other? Give an example.
- A die is thrown 60 times and we obtain the following results:
| Outcome | Number of times obtained |
| 1 | 9 |
| 2 | 10 |
| 3 | 10 |
| 4 | 8 |
| 5 | 12 |
| 6 | 11 |
What is the mode, mean and mode for the set of experiments summarized above?
- A die is thrown 60 times and we obtain the following results:
| Outcome | Number of times obtained |
| 1 | 9 |
| 2 | 11 |
| 3 | 10 |
| 4 | 8 |
| 5 | 12 |
| 6 | 10 |
What is the mode, mean and mode for the set of experiments summarized above?
- Which one is more sensitive to extreme values: the mode, the median, or the mean? Give an example?
- In an experiment a die is thrown 1000 times. How many possible outcomes can the experiment have?
- ABC's 20/20 television broadcast on July 16, 1993 reported on a study in which individuals who had lived
to be 100 years of age or more were queried in the hope of finding common characteristics. The implication was
drawn that if a younger person worked at acquiring the characteristics shared by these centenarians, then the
probability of reaching such an old age increased. Why was this study design inappropriate for the implication
drawn?
- You throw a die a number of times and write down the results as follows:
| Outcome | Relative frequency (%) |
| 1 | 20 % |
| 2 | 20 % |
| 3 | 10 % |
| 4 | 30 % |
| 5 | 10 % |
| 6 | 10 % |
So you know that 4 came up 30% of the times, 1 appeared 20% of the times, and so forth, as listed above.
Can you compute the mode, median, and mean values for this distribution? Why or why not?
- The die in the experiment above seems to be a bit biased. Do you expect the sum of the deviations
to the mean of all the individual experiments for the experiment above to add up to 0 or not? In other
words, do you expect the relationship
to still hold? Is it of any
relevance that the die seems to be biased towards some values? Why or why not?.
- Assume you have the following distributions.
Indicate (with approximation) the relative positions of the mean, mode, and median on the three pictures.
(The relative position matters, the actual precise position does not). Justify your answer.
- During World War II many economists, mathematicians, and statisticians were members of Columbia University's
Statistics Research Group, which did high-level consulting work for the armed services. As part of this group's work,
statistician Abraham Wald was asked where to place armor on planes. It seemed obvious to the aircraft engineers that
armor was needed at the places most frequently hit, as found in a large sample of battle-proven airplanes. After
studying the bullet holes of a sample of returning planes, Wald's conclusion was to place the armor where bullet
holes were least frequently found in these planes. Was his reasoning justified? Was there anything wrong with the
aircraft engineers' sampling design? Did they overlook anything?
- What is a bimodal distribution. Give an example of one.
- An article in Business Week (August 17, 1992) reported on the success of the Saturn automobile. it
gave the following chart of consumer satisfaction:
| Lexus
| Infiniti
| Saturn
| Acura
| Mercedes-Benz
| Toyota
| Industry Average
|
| 179
| 167
| 160
| 148
| 145
| 144
| 129
|
Why is it acceptable to say that Saturn purchasers may be more satisfied than Acura purchasers,
but it is not correct to say that Saturn buyers are 24% more satisfied than the average car buyer?
- Sometimes the business press emphasizes the median and other times the mean as the measure of central
tendency. Why might one be emphasized rather than the other? Give an example of a situation in which the median
might be preferred as a measure of central tendency to the mean. Similarly, describe a situation in which the mean
might be preferred as a measure of central tendency to the median.
- The deVoe Report (June 2, 1980) quoted then U.S. President
Jimmy Carter as saying "half the people in this country are living below
the median income -- and this is intolerable." What is disputable and
what is true in this quote?
- You throw a die 600 times and each of the 6 possible outcomes (1 through 6) comes up exactly 100 times. What
is the mean, mode, median, range, variance, and standard deviation of the experiment?
- If we calculate the variance in a sample of size n we divide by (n - 1).
If we calculate the variance in a population of size n we divide by n. Why? What was the
justification offered in class, and when might you see significant differences between the values computed
with the two formulas?
- Suppose you're measuring temperature in degrees Fahrenheit during a certain year every day. You end up
with 365 measurements once for each day. You then compute the mean, variance, and standard deviations, and you
obtain three numbers. What are the units in which these numbers are expressed?
- Compute the mean, variance and standard deviation for the following sequence:
0 4 0 4
Here are some formulas that you may find useful:
- (Practical Problem) You throw a die six times, and record the obtained values.
You compute the standard deviation (as in a sample) and write it down (2.065). Later
one of the values in your notes is erased:
6 4 ? 5 1 1
How can you find out the third value? What is it?
- (Practical Problem) Solve the previous problem when you look at the measurements as an entire population.
- (Practical Problem) For the following 10 scores:
100 89 81 72 62 49 55 85 75 65
Compute the mean and the standard deviation using only SUM and COUNT.
Then set up a conversion table like this:
Score Letter Points
0 F 0
60 D 1
70 C 2
80 B 3
93 A 4
Then convert the original 10 scores and compute the mean and the standard deviation of the converted scores using only
SUM and COUNT.
- True or False? (If false, state why).
The sum of the deviations around the mean is always zero.
- True or False? (If false, state why).
If the x and y covariance is negative, then the standard deviation
of either x or y must be negative.
- True or False? (If false, state why).
The variance is a measure of the average squared deviation of values around their mean.
- True or False? (If false, state why).
If the correlation coefficient is negative, then the covariance is negative.
- True or False? (If false, state why).
The bigger the coefficient of correlation (r), the stronger the relationship between two variables.
Thus, r = +0.75 shows a stronger relationship than r = -0.85.
- True or False? (If false, state why).
If the correlation coefficient is zero then all the y values must be equal to their mean.
- True or False? (If false, state why).
A negative covariance indicates that for negative deviations from the mean of one variable there tend to be
negative deviations from the mean of the other variable.
- True or False? (If false, state why).
If the covariance is zero, then we know that the coefficient of correlation is zero and that there is no
relationship between the variables.
- True or False? (If false, state why).
The covariance of x and y is a measure of the average variability between x and y.
- For the following four paired measurements draw a scatterplot and discuss whether x and y are
correlated or not. Justify your answer.
Indicate the approximate value of the coefficient of correlation (sign and whether it's closer to 0 or 1).
- For the following four paired measurements draw a scatterplot and discuss whether x and y are
correlated or not. Justify your answer.
Indicate the approximate value of the coefficient of correlation (sign and whether it's closer to 0 or 1).
- For the following four paired measurements draw a scatterplot and discuss whether x and y are
correlated or not. Justify your answer.
Indicate the approximate value of the coefficient of correlation (sign and whether it's closer to 0 or 1).
- (Practical Problem) For the following four paired measurements draw a scatterplot and discuss whether
x and y are correlated or not. Justify your answer.
| x |
2 | -3 | 7 | 8 |
-2 | -5 | -7 | 4 |
-6 | 2 |
| y |
0 | 4 | -5 | -7 |
8 | 9 | 9 | -3 |
5 | -1 |
