Sunday, January 27, 2008

Sample midterm questions - Part 1

The midterm questions from last quarter's midterm have been posted with the answers. However, there's no explanation of the answers. I'll endeavor to explain how I think these answers are derived. If you think I got one wrong or if you have a better/easier way to get the answer, please let me know in a comment or email.

Since chapter 6 (normal distribution) is not included on our midterm, I'm going to skip those questions for now and focus on the questions that deal with chapters 1-5.

Question 1 & 2 - Skip

Question 3:
Which of the following is not a property of a binomial experiment?
a. the experiment consists of a sequence of n identical trials
b. each outcome can be referred to as a success or a failure
c. the probabilities of the two outcomes can change from one trial to the next
d. the trials are independent
Answer:Think of the classic example of a binomial experiment – flipping a coin. All the flips are identical. Each outcome has two possibilities, heads/tails. Each flip is independent of the previous flips. The probabilities do not change from one flip to the next. So C is not true.

Exhibit 5-8: The random variable x is the number of occurrences of an event over an interval of ten minutes. It can be assumed that the probability of an occurrence is the same in any two time periods of an equal length. It is known that the mean number of occurrences in ten minutes is 5.3.

Question 4. Refer to Exhibit 5-8. The probability that there are 8 occurrences in 10 minutes is
a. .0241
b. .0771
c. .1126
d. .9107
As soon as I see that we’re talking about events occurring over a period of time, I think “Poisson Distribution”. It’s the whole “area of opportunity concept. Therefore, apply the Poisson Formula (or use a lookup table) for X=8 and mean=5.3:
F(x=8) = (e^-5.3)(5.3^8)/8! = (0.00499)(622,596.9)/(40320) = 0.0771
So the answer is B.

Question 5: Refer to Exhibit 5-8. The probability that there are less than 3 occurrences is
a. .0659
b. .0948
c. .1016
d. .1239
We apply the Poisson Distribution formula again, but this time to determine
F(x<3)= F(x=0) + F(x=1) + F(x=2)
You can apply the Poisson formula individually for these 3 values, but I prefer to look them up in the table and get:
F(x<3)= 0.0050 + 0.0265 + 0.0701 = 0.1016
So answer C is correct.

Exhibit 5-5
Probability Distribution
x f(x)
10 .2
20 .3
30 .4
40 .1

Question 6: Refer to Exhibit 5-5. The expected value of x equals
a. 24
b. 25
c. 30
d. 100
Answer: Apply the formula for the expected value: multiply each value by its probability and add them up.
E(x) = (10)(0.2) + (20)(0.3) + (30)(0.4) + (40)(0.1) = 2 + 6 + 12 + 4 = 24< style="font-weight: bold;">
Question 7:
Refer to Exhibit 5-5. The variance of x equals
a. 9.165
b. 84
c. 85
d. 93.33
Just apply the standard formula for variance: sum the squares of the difference to the mean times the probability of each value. (Note: You don’t divide by the number of values as you would in a data sample of population. When you have a distribution, you multiply each discrete value by it’s probability.)
Var = (10-24)^2(.2) + (20-24)^2(.3) + (30-24)^2(.4) + (40-24)^2(.1)
= (196)(.2) + (16)(.3) + (36)(.4) + (256)(.1)
= 39.2 + 4.8 + 14.4 + 25.6
= 84

Correct answer is B.

Question 8: 20% of the students in a class of 100 are planning to go to graduate school. The standard deviation of this binomial distribution is
a. 20
b. 16
c. 4
d. 2
Answer: This is pretty straightforward, except that the formula sheet has the formula for variance. You need to remember to take the square root to get the standard deviation.
Std. Dev. = SQRT(np(1-p))
= SQRT ((100)(.20)(.80))
= SQRT (16)
= 4
Correct answer is C.

Questions 9, 10 and 11 - Skip

Question 12: Of five letters (A, B, C, D, and E), two letters are to be selected at random. How many possible selections are there?
a. 20
b. 7
c. 5!
d. 10
Answer: This question is actually rather vague. It doesn't tell us if the order of the selection matters. I.e. is (A,B) considered the same selection as (B,A)? From the answer key, it's pretty clear that they are considered the same, so we'll go with that.

Apply the binomial coefficient formula:
nCx = n! / (n-x)!x!
= 5! / 3!2!
= 120 / 12

Correct answer is D.

BTW, this problem could easily be solved by inspection (i.e. by just looking at it): You're selecting 2 letters out of a pool of 5. For the first letter there are 5 possibilities, for the second letter there are 4 possibilities (since the first letter, whatever it was, is no longer in the pool). 5x4 = 20. Then divide by 2 because each two-letter pair has a duplicate in the reverse order. 20/2 = 10.

Question 13: If A and B are independent events with P(A) = 0.2 and P(B) = 0.6, then P(A U B) =
a. 0.62
b. 0.12
c. 0.60
d. 0.68
Answer: This question requires us to use the Addition Law and the multiplication rule of independent events (both are on the formula sheet):
Addition Law gives us: P(A U B) = P(A) + P(B) - P(A intersect B)
Multiplication rule of independent events says: P(A intersect B) = P(A)P(B)
P(A U B) = P(A) + P(B) - P(A)P(B)
= (0.2) + (0.6) - (0.2)(0.6)
= (0.8) - (0.12)
= 0.68

Correct Answer is D.

Question 14: The variance of the sample
a. can never be negative
b. can be negative
c. cannot be zero
d. cannot be less than one
Answer: This one is right out of the book. See page 82 where it says (in italics): neither the variance nor the standard deviation can ever be negative. But even if you somehow didn't remember that tidbit of information, you could just eyeball the formula for variance and see that the numerator is a squared quantity which is always positive and the denominator is the number of events in the sample, which is also always positive. So, Answer A is correct.

Incidentally, the variance can be zero if the data points are all equal to the mean, in which case the numerator is zero and therefore the variance is 0. It can also be less then one if the data points are all relatively close to the mean.

Question 15: The value of the sum of the deviations from the mean, must always be
a. negative
b. positive
c. positive or negative depending on whether the mean is negative or positive
d. zero
Answer: This one is similar to the question above and it's also right out of the book, page 84, where it says this sum will always be zero. But again, if you missed that factoid, you could easily see that it's true by definition of the mean. Correct answer is D.

Question 16: If A and B are independent events with P(A) = 0.38 and P(B) = 0.55, then P(A|B) =
a. 0.209
b. 0.000
c. 0.550
d. 0.38
Answer: This is another tricky question. No formulas needed here. Just think about it: If A and B are independent, then by definition, the probability of A is not influenced by B. So the fact that B is "given", makes no difference - the probability of A is still 0.38. So P(A|B) = P(A) = 0.38. Answer D is correct.

Question 17: A six-sided die is tossed 3 times. The probability of observing three ones in a row is
a. 1/3
b. 1/6
c. 1/27
d. 1/216
Answer: Come on! Is this really graduate school?

3 independent events. Get the total probability by taking the product of the 3 events. Each one has a 1/6th probability of rolling a one. Therefore, total probability is (1/6)^3 = 1/216. Answer D is correct.

Question 18: If the coefficient of variation is 40% and the mean is 70, then the variance is
a. 28
b. 2800
c. 1.75
d. 784
Answer: This one is a bit tricky in that you need to remember that the standard deviation is the square root of the variance. First, use the formula for Coefficient of Variation:
CV = (StdDev / mean) x 100 %
rearrange to get:
Std Dev = CV x mean/100
= (40)(70)/100
= 28

Now since Var = StdDev^2, Var = 28^2 = 784
Answer D is correct.

Question 19: If two groups of numbers have the same mean, then
a. their standard deviations must also be equal
b. their medians must also be equal
c. their modes must also be equal
d. None of these alternatives is correct
Answer: Hmm. Does this one need explanation? If you understand mean, median, mode and standard deviation, you should understand that they're completely independent ways of describing data. I suppose someone could take the time to show examples of data sets with the same mean, but different median, mode and stddev, but I think it's pretty much intuitively obvious that Answer D is correct.

Has anyone noticed that D is the correct answer for questions 11-19, except for 14? What's the probability of that happening?

Exhibit 3-3:
A researcher has collected this sample data. The mean of the sample is 5.
3 5 12 3 2

Question 20: Refer to Exhibit 3-3. The standard deviation is
a. 8.944
b. 4.062
c. 13.2
d. 16.5
Answer: Plug the numbers into the formula for standard deviation
StdDev = SQRT((3-5)^2+(5-5)^2+(12-5)^2+(3-5)^2+(2-5)^2)/(5-1))
= SQRT((4+0+49+4+9)/4) = SQRT(66/4) = SQRT(16.5)
= 4.062

Don't forget that last step of taking the square root! Otherwise you're giving the variance which, of course, is one of the choices (D). B is the correct answer. Answer C (13.2) is the variance if this data were for an entire population, in which case you would divide by N=5. Remember to divide by n-1 for a sample.

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