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Math 10 - Compilation of Sample Exam Questions + Answers
Sample Exam Question 1
We have a population of size N. Let p be the independent probability of a person in the population devel- oping a disease. Answer the following questions in terms of N and p.
a) What is the probability that a person in the population NOT developing the disease? (1 pt)
b) If N = 2, what is the probability that no one develops the disease? If your answer involves N , please replace it by the number 2. (1 pt)
c) For a general size N , what is the probability that no one develops the disease? (1 pt)
d) What is the probability that, for a general size N , at least one person develops the disease? (1 pt)
Answer: a) 1 − p, b) (1 − p)^2 , c) (1 − p)N^ , d) 1 − (1 − p)N^.
Sample Exam Question 2
Suppose 3 fair dice are rolled independently. Let their outcomes be D 1 , D 2 , D 3. Please simplify your answer as much as possible.
a) Suppose that D 1 = 5 and D 2 = 6, what is the probability that D 3 is a different outcome? (1 pt)
b) In general, what is the probability that D 3 is different from D 1 and D 2 GIVEN that D 1 6 = D 2? (1 pt)
c) In general, what is the probability that none of the rolls had the same outcome? (2 pts)
d) What is the probability that at least two of the rolls had the same outcome? (1 pt)
Answer: a) 23 , b) 23 , c) 5646 = 2036 = 59 , d) 1 − 59 = 49.
Sample Exam Question 3
Police officers in a town are stopping drivers at random and subjecting them to a breathalyzer test that detects alcohol intoxication. During the operation, there were 5000 drivers on the road.
One of those drivers were chosen at random, say Tommy, and he tested positive. The police were using breathalyzers that detects drunkenness with 100 % probability of IF the driver is drunk. (positive result) If the driver is sober, it has a 99 % probability of reporting that the driver is NOT drunk. (negative result)
a) The police wants to deport Tommy because they think the probability that he is driving drunk is 99 %. Is their claim true or false based on the information so far? Explain. (2 pts)
Now you are told that out of the 5000 drivers on the road, 5 of them are drunk, while the rest are sober.
b) If all 4995 sober drivers are given the breathalyzer test, approximately how many would get a positive result on average? (1 pt)
c) The police was picked a driver at random from the 5000. What is the probability that he or she is sober? You do not have to simplify your answer. (1 pt)
d) Using Bayes Theorem, write down a numerical fraction for the approximate probability that a randomly selected driver is not drunk, given that he or she got a positive result. You are given that the probability that a randomly selected driver will test positive is 500055. (4 pts)
Answer: a) False, you need more information, e.g. the base rate, b) 4995 · 0 .01 = 49. 95 ≈ 50, c) 4995 5000 ,^ d)^ D^ = drunk,^ P os^ = tested positive. By Bayes Theorem:^ P^ ( not^ D|P os) =^
P (P os| not D)P ( not D) P (P os) =
- (^001 ) 500055 ≈^5055 =^1011.
Sample Exam Question 4
The proportion of scores for 1,000 students in a class are well approximated by a normal distribution, with mean 50 and standard deviation 10. Approximately how many students scored 70 and above? Show your work. Illustrations are allowed. (2 pts)
Answer: using the 68-95 heuristic/rule from the textbook, approximately 95 % of students scored between 30 and 70. So 2.5 % scored 70 and above. Then, approximately 1000 · 0 .025 = 25 is the answer.
Sample Exam Question 5
The weights for 1000 bars of gold are well approximated by a normal distribution, with mean 100 grams and standard deviation 20 grams. You may round your final answers to the nearest integer.
a) Approximately how many bars of gold weigh between 100 grams and 130 grams inclusive? (2 pts)
b) Approximately how many bars of gold weigh less than 80 grams?
Answer: Let X be the weight of a randomly selected bar of gold. P (X ≤ k) = P (Z ≤ k− 20100 ), where Z is standard normal (z-tables).
a) So, P (X ≤ 100) = P (Z ≤ 10020 −^100 = 0) = 0.50, and P (X ≤ 130) = P (Z ≤ 13020 −^100 ) = P (Z ≤ 1 .5) = 0 .9332. So, P (X ≤ 130) − P (X ≤ 100) = 0. 9332 − 0 .50 = 0.4332. Ans: approximately 0. 4332 · 1000 ≈ 433 bars.
b) Also, P (X ≤ 80) = P (Z ≤ 80 − 20100 ) = P (Z ≤ −1) = 0.1587. Ans: 0. 1587 · 1000 ≈ 159.
hoping to find out if the scale is correct on average, or systematically giving you lower/higher than the actual weight.
If the scale is correct on average, it would produce results drawn from a normal distribution with mean 500 grams and unknown variance σ^2. (E.g. real life scales will tell you they are intended to be accurate within X grams)
The sample mean of your n = 25 data points is M = 490 grams.
- Can you conclude that the electronic weighing scale is faulty and is systematically giving you results that are lower than the actual weight, on average? Since M = 490 < 500 true weight? Explain. (2 pts)
- Using this set of sample data, you calculated an estimate of the standard deviation s = 20 grams. What sampling distribution of the mean would you use? State all the parameters of this sampling distribution (using μ for the real mean). Why do you use this sampling distribution? (3 pts)
- Using the statistics produced by your sample, construct a 95 % confidence interval for the mean. (4 pts)
Answers
No. Even if the scale is correct on average, you could have gotten 490 by chance.
No population variance, have to use t distribution. Mean μ, standard error √^2025 = 4, degrees of freedom 24.
Degrees of freedom = 25 − 1 = 24. Find the t-value, t = 2.06. Formula: [M − t · SE, M + t · SE], where SE is the standard error given.
Lecture 1 - Class Exercises
Summation Notation Practice
X 1 = 1, X 2 = 2, X 3 = 3, X 4 = 4, X 5 = 5
- Calculate
∑^5
i=
Xi.
- Calculate
∑^3
i=
Xi.
- Calculate
∑^3
i=
X i^2.
- Calculate
i=
Xi
- Calculate
i=
X i^2
Y 1 = 1, Y 2 = 2
- Calculate
∑^2
i=
XiYi.
- Calculate
∑^2
i=
X i^2 Yi.
- Is Ui = log 4 (Wi) a linear transformation of the Wis?
Answers
- No
- 4
- 4
- 3
- No
- 1, 2, 3, 4
- Yes
- No
Lecture 2 - Class Exercises
Central Tendency and Variability Practice
Our favourite dataset : X 1 = 1, X 2 = 2, X 3 = 3, X 4 = 4, X 5 = 5
- Calculate the mean of this set of data.
- Calculate (^15)
∑^5
i=
Xi.
- Calculate the median of this set of data.
- What is the mode of this set of data? Trick question: it is possible for there to be no mode.
- Compare the median and the mean for this set of data. Are they the same? Why or why not?
New dataset : Y 1 = 1, Y 2 = 3, Y 3 = 7, Y 4 = 1
- Calculate the mean of this new set of data.
- Calculate the median of this new set of data.
- Compare the median and the mean for this set of data. Are they the same? Why or why not?
- What is the mode of this set of data? Not a trick question this time.
- Which one “balances the scale”? The mean or the median?
- Which one minimizes the sum of squared deviations? The mean or the median?
- Which one minimizes the sum of absolute deviations? The mean or the median?
- Consider our favourite dataset : X 1 = 1, X 2 = 2, X 3 = 3, X 4 = 4, X 5 = 5. Let us apply a linear transformation Zi = 5Xi − 2 to each one of them. Without calculating each Zi, can you deduce what their mean would be?
- If I tell you the variance of our favourite dataset in question 13 is 2. Can you deduce what the variance of the Zis will be?
Answers
your answer. (2 points)
Answers
- mean = 100, median = 70.
- positively skewed. mean is affected more by large values and outliers. can also give the balance scale argument.
- a = 2.
- r = 1. Perfect linear relationship.
- r will probably be lower. Less/no correlation if randomized.
- Will not be 1. Points will not have a perfect linear relationship.
- New r = 0.65. r is not affected by change in units of measurement of one of the variables.
Lecture 5 - Class Exercises
- Given 4 dice rolls, write down an expression for the probability that at least two rolls have the same outcome. You do not have to simplify your answer.
- Given 7 dice rolls, what is the probability that none of the rolls had the same outcome? (in mathematics, this is known as the pigeonhole principle)
Recall that for the Binomial Distribution, the probability of getting k successes in n trials is nCk pk(1−p)n−k, where nCk = (^) k!(nn−!k)!.
- Suppose the probability of being admitted to a college is 0.50 for every student who applies. If 3 students applied, what is the probability that 0 or 2 students got admitted? Give a simplified numerical answer and show your work.
- If instead, the probability of being admitted is 0.30 and 10 students applied, write down a numerical expression for the probability that between 1 and 3 students got admitted. You do not have to simplify your answer.
- Suppose there are two candidates running for president, Mr T and Mrs H. There are only 1000 voters. 400 of them voted for Mrs H, while 600 of them voted for Mr T.
Suppose you took a simple random sample of size 100 from these 1000 voters. Write down a numerical expression of the probability that 35 in your sample voted for Mr T. You do not have to simplify your answer.
Answers
1 − 56 · 46 · 36.
6 possible outcomes, 7 dice. So, the probability is zero.
P ( 0 success in 3 trials ) + P ( 2 success in 3 trials ) = 18 + 38 = 12.
10 C 1 (0.3)^1 (0.7)^9 +^10 C 2 (0.3)^2 (0.7)^8 +^10 C 3 (0.3)^3 (0.7)^7
100 C 35 (0.6)^35 (0.4)^65
Lecture 6 - Class Exercise
The scores of a class of 10, 000 students are approximately normally distributed with mean 50 and standard deviation 10.
Hint: you can do this entire question using just the 68-95-99.7 rule, and I recommend you do.
- Approximately how many students scored between 40 and 60 inclusive? (1 pt)
- Approximately how many students scored between 50 and 70 inclusive? (2 pts)
- Approximately how many students, in total, scored ≥ 70 or scored ≤ 20? (4 pts)
Answers
- 6800
- 4750
- 250 + 15 = 265
Lecture 7 - Class Exercise
Suppose that the mean annual salary for a particular job is $60, 000 with a standard deviation of $10, 000. The distribution of salaries for this job is heavily positively skewed (long tail to the right).
- What is the approximate probability that the mean of a simple random sample of 100 salaries lies in the interval [58360, 61640]? (4 pts)
