MCO302 Final Exam Answers



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 |test negative |total |

|actually have it | 990 |(false negatives) | 1,000 |

| | |10 | |

|Don’t have it |(false positives) | 989,010 | 999,000 |

| |9,990 | | |

|total | 10,980 | 989,020 | 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.

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