JMC465/MCO510 Math/Stats practice questions
Imagine that in 1998 (when the Consumer Price Index was 163.0) your older
brother regularly paid you $5 to get lost when his girlfriend came over to the house.
What would be the going rate today (with the CPI at 237.4) if you had to buy a
similar favor from a younger sibling?
(237.4 / 163.0) * $5 = $7.28
If you throw three dice, what is the probability that you will roll 1-1-1??
You use Probability Rule 3 because the dice are independent; the face each shows doesn’t
affect any of the others. So you multiply the probabilities of getting a 1 on each coin:
.167 * .167 * .167 = .0046.
You buy a ticket for a fund-raising raffle. The tickets are $5 each. The chance of
winning the first prize of $1,000 is 1/5000. The chance of winning the second prize of
$500 is 1/1000. The chance of winning the third prize of $100 is 1/500. What is the
expected value of this game?
First, convert the fractions to decimals. 1/5000 = .0002. 1/1000 = .001. 1/500 = .002.
Those are the probabilities of winning something. Add them up (.0032) and subtract from
1 to get the probability of losing your $5. 1-.0032 = .9968.
EV= ($995 * .0002) + ($495 * .001) + ($95 * .002) + (-$5 * .9968) = -$4.10
Note well in this formula that you have to subtract the cost of the ticket from each prize
amount, and you have to include the cost of a losing ticket as a negative amount.
This means ticket buyers overall will lose $4.10 for every $5 ticket sold. Another way to
think of it is “You give me $5 and I’ll give you 90 cents.” Fun game!
In one recent class, there was a .36 probability that any randomly-picked student
had a GPA greater than 3.5, and a .40 probability of having a GPA of 3.0-3.5. What
was the probability of having a GPA below 3.0?
Under Probability Rule 2, the formula for students who GPAs of 3.0 or greater is .36 +
.40 = .76. Under Probability Rule 1, the formula for students with GPAs less than 3.0 (the
only other possibility) is 1 - .76 = .24.
Local garbage collectors are threatening to strike on the grounds that since 2000 the
city council hasn’t given a real raise. The mayor retorts that the median salary for
garbage collectors in 2000 was $32,500, and that today the median salary is $42,300.
After accounting for the effects of inflation (CPI in 2000 was 172.2 and today it's
237.4), how much additional buying power – if any – do garbage collectors at the
median salary have today?
(237.4 / 172.2) * $32500 = $44,805 in 2014 dollars. And 42300 - 44805 = - $2,205,
which means the garbage collectors actually have lost $2,205 in spending power over the
years.
A survey of a sample of 100 respondents will have what error margin?
Error margin is 1/SQRT(N). The square root of 100 is 10, so 1/10 = .10, or 10 percentage
points.
Last year there were 22 murders in your city, and during the same period this year
there have been 17 murders. What has been the percentage change in murders?
(NEW - OLD) / OLD = (17 - 22) / 22 = -5 /22 = -.227, or a 23% DECREASE in murders.
You’re working on a story about complaints that minority drivers are being
targeted for traffic tickets by the police in your town. License records tell you that
20% of the drivers in your town are minorities. You examine a properly-drawn
random sample of 280 tickets, out of the 8,322 tickets issued last year, and find that
84 of the tickets in your sample were given to minority drivers.
Driver question 1: What proportion of the tickets in your sample were given to
minority drivers?
84 / 280 = .30, or 30%
Driver question 2: Knowing that minority drivers should get about 20% of the
tickets if there was no bias, calculate the relative risk of minority drivers getting
tickets compared to what they should get.
From question 1, you know that minority drivers are getting 30% of the tickets, compared
to their 20% share of the driver population. So .30 / .20 = 1.5. The relative risk of 1.5
means that minority drivers face a 50% increased risk of getting a ticket.
Driver question 3: What is the error margin (the 95% confidence interval) of the
proportion of minority drivers in your sample?
1/sqrt(280) = .06, or +/- 6 percentage points. Therefore, you can be 95% confident that if
you examined all, 8,322 tickets, you would find the proportion given to minority drivers
was somewhere between 24% and 36%. In other words, you can be pretty sure that the
added risk for minority drivers isn’t just a fluke of sampling.
You have an odd rash, and your doctor tests you for a particularly horrible STD
that afflicts about one in a thousand people. The results come back positive! The test
is 99% accurate (both in sensitivity and specificity), and you are one of a million
people who have been tested for this dread disease.
Fill out a "confusion of the inverse" table and answer the next four questions.
STD Question 1: How many actual sufferers of this disease should be found among
the 1,000,000 people tested?
First, here is the table you need to create:
test positive
actually have
it
Don’t have it
total
test negative
(false negatives)
990
10
(false positives)
9,990
989,010
10,980
989,020
total
1,000
999,000
1,000,000
The key calculations:
 One in 1,000 people actually have the disease, so 1,000,000 * 1/1000 = 1,000
actual users. Therefore 1,000,000 - 1,000 = 999,000 actual disease-free.
 Of the 1,000 real sufferers, 99% will correctly test positive. That’s 990. The other
10 will falsely test negative.
 Of the 999,000 non-sufferers, 99% will correctly test negative. That 989,010. The
other 9,990 will falsely test positive.
So the answer to Question 1 is 1,000
STD question 2: About how many people who are NOT sufferers will test negative
(i.e., be true negatives)?
The answer is 989,010 true negatives.
STD question 3: How many people who ARE sufferers will test positive (i.e., be true
positives)?
There are 990 true positives.
STD question 4: What percent of all the people who test positive actually will NOT
be STD sufferers (i.e., they are false positives)? In other words, what is the chance
that you actually don’t have the disease?
Of the 10,980 who test positive, like you, 9990/10980 = .91, are false positives. This
means that 91% of the positive results are false. Therefore there is an extremely high
chance that you don’t actually have the dread disease.
Regression Question 1: You use SPSS to do a regression analysis of school test score
data, comparing last year’s scores with this years scores. SPSS tells you the
regression formula is
PREDICTED_2014_SCORE = .907 * ACTUAL_2013_SCORE + 5.6.
Cronkite Elementary had a 2013 score of 44. What is that school’s predicted score?
=.907 * 44 + 5.6 = 45.5, so the predicted score is 45.5
Regression Question 2: Cronkite Elementary actually scored 49 in 2014. What is the
residual?
Residual = Actual – Predicted.
Therefore 49 - 45.5 = a residual of +3.5. If the Actual score had been 42, the residual
would have been 42 - 45.5 = -3.5
You are told that the chance of getting a parking ticket for not putting money in the
meter is only 1/50. Your two-hour class meets 26 times during the semester. What is
the chance that you will get a ticket at least once during the semester if you don’t
feed the meter.
Chance of something occurring in n trials: 1 – (1 – p) n
A 1/50 chance is p = 0.02
1 – (1 – 0.02)26 = 1 – (0.98)26 = 1 – 0.59 = 0.41. This means there is a 41% chance that
you will get at least one ticket.