diff --git a/source/main.ptx b/source/main.ptx index f3b346a..22c12b1 100644 --- a/source/main.ptx +++ b/source/main.ptx @@ -65,6 +65,7 @@ + diff --git a/source/review/Final-Exam-Review.ptx b/source/review/Final-Exam-Review.ptx new file mode 100644 index 0000000..e17c0ee --- /dev/null +++ b/source/review/Final-Exam-Review.ptx @@ -0,0 +1,2018 @@ + + +
+ Final Exam Review + + +

+ Use the following problems to prepare for the exam. + There will be in-class review on Thursday, April 23. + Your recitation this week will also be exam review. +

+ + + + + Allowed Materials + +

+ You will be allowed to use a scientific calculator (not a graphing calculator, not a calculator app on your phone). + You may not share a calculator with another student; you must use your own calculator. +

+ +

+ You may bring a standard 3 in x 5 in index card with prepared notes. + You may use both sides of the notecard. + You must put your full name in the top right corner of the card, and turn it in along with your exam. +

+ + + + + +

+ Consider the sets A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}, B = \{2, 4, 9, 10, 12, 14, 19\}, and C = \{9, 10, 11, 14, 16, 17, 20\}, which are all subsets of \Omega = \{1, 2, 3, \dotsc, 20\}. +

+ + + + + +

+ Find A - (B \cap C). +

+ + + +

+ \{1, 2, 3, 4, 5, 6, 7, 8\} +

+ + + + + + +

+ Find |A|, |B|, |C|, |A\cup B|, |A \cap B|, |B\cap C|, |A\cap C|, and |A\cup B\cup C|. + Is it true that the size of the union of sets is equal to the sum of the sizes of the individual sets? +

+ + + +

+ |A| = 10, |B| = 7, |C| = 7, |A \cup B| = 13, |A \cap B| = 4, |B\cap C| = 3, |A\cap C| = 2, |A\cup B\cup C| = 17. In particular, note that |A\cup B| = 13 \neq 10 + 7 = |A| + |B|, so it is not true in general that the size of the union of sets is the sum of the sizes of the individual sets. +

+ + + + + + +

+ Find A^c and (A\cup B)^c. +

+ + + +

+ A^c = \{11, 12, 13, 14, 15, 16, 17, 18, 19, 20\}, (A\cup B)^c = \{11, 13, 15, 16, 17, 18, 20\}. +

+ + + + + + +

+ Suppose we have a 6-sided die that's weighted to roll a 6 half of the time. + We roll the die two times. + List the set of all possible results. + [Note: the result (2, 4)---rolling a 2 and then a 4---is different from the result (4, 2)---rolling a 4 and then a 2.] +

+ + + +

+ + \Omega = \{\amp (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), + \amp (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), + \amp (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), + \amp (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), + \amp (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), + \amp (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\} + + Note that \Omega simply lists outcomes with no reference to the probabilities. + So the answer here is the same as in . +

+ + + + + +

+ Suppose we flip a coin two times. + List the set of all possible results. + What about flipping three times? Four times? If we flip the coin 10 times, how many possible results will there be? +

+ + + +

+ For two flips: \Omega = \{ HH, HT, TH, TT \}. +

+ +

+ For three flips: \Omega = \{ HHH, HHT, HTH, THH, HTT, THT, TTH, TTT \}. +

+ +

+ For four flips: + + \Omega = \{ \amp HHHH, HHHT, HHTH, HTHH, + \amp THHH, HHTT, HTHT, HTTH, + \amp THHT, THTH, TTHH, HTTT, + \amp THTT, TTHT, TTTH, TTTT \}. + +

+ +

+ Each additional flip doubles the number of outcomes. + So, with ten flips, we'll have |\Omega| = 2^{10} = 1024. +

+ + + + + +

+ If we roll a 6-sided die ten times, how many possible results will there be? +

+ + + +

+ Each additional roll will multiply the number of outcomes by 6. + So, with 10 rolls, we'll have |\Omega| = 6^{10}. +

+ + + + + +

+ Consider the sample space \Omega = \{1, 2, 3, 4, 5, 6, 7, 8\} with probability distribution below. + Calculate the probabilities of A = \{1, 3, 7, 8\}, B = \{2, 3, 6, 7\}, A\cup B, and A \cap B. +

+ + + + + + + x + \Pr(x) + + + + 1 + 0.1 + + + + 2 + 0.05 + + + + 3 + 0.2 + + + + 4 + 0.15 + + + + 5 + 0.15 + + + + 6 + 0.1 + + + + 7 + 0.05 + + + + 8 + 0.1 + + + + 9 + 0.1 + + +
+ + + +

+ \Pr(A) = 0.45, \Pr(B) = 0.4, \Pr(A \cup B) = 0.6, \Pr(A \cap B) = 0.25. +

+ + + + + +

+ Suppose a die has the values 1, 2, 3, 4, 5, 6 on the faces, but the die is not fair. + Instead, the probabilities scale by the same amount as the face values. + For example, a result of 4 is twice as likely as a result of 2, since 4 is twice as large as 2; a result of 6 is six times more likely than a result of 1; and so on. + Write a probability distribution table for this die. +

+ + + + + Probability Distribution for a Linearly Scaled Die + + + + x + \Pr(x) + + + + 1 + 1/21 + + + + 2 + 2/21 + + + + 3 + 3/21 + + + + 4 + 4/21 + + + + 5 + 5/21 + + + + 6 + 6/21 + + +
+ + + + + +

+ Suppose a die has the values 1, 2, 3, 4, 5, 6 on the faces, but the die is not fair. + Instead, each even value has an equal probability, each odd value has an equal probability, and the even values are each twice as likely as the odd values to appear on a roll. + Write a probability distribution table for this die. +

+ + + + + Probability Distribution for an Even-biased Die + + + + x + \Pr(x) + + + + 1 + 1/9 + + + + 2 + 2/9 + + + + 3 + 1/9 + + + + 4 + 2/9 + + + + 5 + 1/9 + + + + 6 + 2/9 + + +
+ + + + + +

+ A toxin molecule inside a cell has a 0.3 probability of leaving the cell during a 1-minute period. + For each value of n = 1, 2, 3, \dotsc, find the probability of the toxin molecule leaving the cell during the nth minute. + What is the probability of the molecule leaving the cell during the first 3 minutes? +

+ + + +

+ For short, write \Pr(n) to mean the probability of the toxin molecule leaving during the nth minute. + Then \Pr(n) = (0.7)^{n - 1} (0.3). +

+ +

+ The probability of leaving during the first 3 minutes is \Pr(1) + \Pr(2) + \Pr(3) = 0.657. +

+ + + + +

+ In each of the following scenarios with given events A and B, alculate \Pr(A), \Pr(B), \Pr(A\cap B), \Pr(A \mid B), and \Pr(B \mid A). +

+ + + + +

+ An experiment consists of rolling a fair die two times. + Let A be the event that the sum is even, and let B be the event that the second roll is higher than the first. +

+ + + +

+ + A = \{ \amp (1, 1), (1, 3), (1, 5), (2, 2), (2, 4), (2, 6), + \amp (3, 1), (3, 3), (3, 5), (4, 2), (4, 4), (4, 6), + \amp (5, 1), (5, 3), (5, 5), (6, 2), (6, 4), (6, 6)\} + B = \{ \amp (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), + \amp (2, 3), (2, 4), (2, 5), (2, 6), + \amp (3, 4), (3, 5), (3, 6), + \amp (4, 5), (4, 6), + \amp (5, 6)\} + A \cap B = \{ \amp (1, 3), (1, 5), (2, 4), (2, 6), (3, 5), (4, 6)\} + + So \Pr(A) = \frac{18}{36} = \frac{1}{2}, \Pr(B) = \frac{15}{36} = \frac{5}{12}, and \Pr(A\cap B) = \frac{6}{36} = \frac{1}{6}. + Finally: + + \Pr(A \mid B) \amp = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{6/36}{15/36} = \frac{6}{15} = \frac{2}{5} + \Pr(B \mid A) \amp = \frac{\Pr(B\cap A)}{\Pr(A)} = \frac{6/36}{18/36} = \frac{6}{18} = \frac{1}{3} + +

+ + + + + +

+ An experiment consists of flipping a fair coin three times. + Let A be the event that the first and second flips match. + Let B be the event that there are at least two heads. +

+ + + +

+ + A \amp = \{ HHH, HHT, TTH, TTT \} + B \amp = \{ HHH, HHT, HTH, THH \} + A\cap B \amp = \{HHH, HHT\} + + So \Pr(A) = \frac{4}{8} = \frac{1}{2}, \Pr(B) = \frac{4}{8} = \frac{1}{2}, and \Pr(A\cap B) = \frac{2}{8} = \frac{1}{4}. + Finally: + + \Pr(A \mid B) \amp = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{2/8}{4/8} = \frac{2}{4} = \frac{1}{2} + \Pr(B \mid A) \amp = \frac{\Pr(B\cap A)}{\Pr(A)} = \frac{2/8}{4/8} = \frac{2}{4} = \frac{1}{2} + +

+ + + + + + +

+ A diagnostic test is developed to detect a disease present in 3.2% of the population. + For a patient who has the disease, the test will accurately give a positive result 65% of the time. + When the patient does not have the disease, the test will accurately give a negative result 99.9% of the time. +

+ + + + + +

+ For a patient who receives a positive test, what is the probability they have the disease? +

+ + + +

+ Let P be the event of testing positive and D the event of having the disease. + Then the prevalence \Pr(D) is given as 3.2%, or 0.032. + The sensitivity is \Pr(P\mid D) = 0.65, and the specificity is \Pr(P^c\mid D^c) = 0.999. + So, according to Bayes' Theorem: + + \Pr(D\mid P) \amp = \frac{\Pr(P\mid D)\Pr(D)}{\Pr(P\mid D)\Pr(D) + (1 - \Pr(P^c\mid D^c))\Pr(D^c)} + \amp = \frac{(0.65)(0.032)}{(0.65)(0.032) + (1 - 0.999)(1 - 0.032)} + \amp \approx 0.96 + +

+ + + + + + +

+ For a patient who receives a negative test, what is the probability they do not have the disease? +

+ + + +

+ + \Pr(D^c\mid P^c) \amp = \frac{\Pr(P^c\mid D^c)\Pr(D^c)}{\Pr(P^c\mid D^c)\Pr(D^c) + (1 - \Pr(P\mid D))\Pr(D)} + \amp = \frac{(0.999)(1 - 0.032)}{(0.999)(1 - 0.032) + (1 - 0.65)(0.032)} + \amp \approx 0.99 + +

+ + + + + + +

+ An experiment consists of rolling a fair die two times. + Let A be the event that the sum is even, and let B be the event that the second roll is higher than the first. + Are A and B independent? +

+ + + +

+ + A = \{ \amp (1, 1), (1, 3), (1, 5), (2, 2), (2, 4), (2, 6), + \amp (3, 1), (3, 3), (3, 5), (4, 2), (4, 4), (4, 6), + \amp (5, 1), (5, 3), (5, 5), (6, 2), (6, 4), (6, 6)\} + B = \{ \amp (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), + \amp (2, 3), (2, 4), (2, 5), (2, 6), + \amp (3, 4), (3, 5), (3, 6), + \amp (4, 5), (4, 6), + \amp (5, 6)\} + A \cap B = \{ \amp (1, 3), (1, 5), (2, 4), (2, 6), (3, 5), (4, 6)\} + + So \Pr(A) = \frac{18}{36} = \frac{1}{2}, \Pr(B) = \frac{15}{36} = \frac{5}{12}, and \Pr(A\cap B) = \frac{6}{36} = \frac{1}{6}. + Finally: + + \Pr(A \mid B) \amp = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{6/36}{15/36} = \frac{6}{15} = \frac{2}{5} \neq \Pr(A), + + so A and B are not independent. +

+ + + + + +

+ An experiment consists of flipping a fair coin three times. + Let A be the event that the first and second flips match. + Let B be the event that there are at least two heads. + Are A and B independent? +

+ + + +

+ + A \amp = \{ HHH, HHT, TTH, TTT \} + B \amp = \{ HHH, HHT, HTH, THH \} + A\cap B \amp = \{HHH, HHT\} + + So \Pr(A) = \frac{4}{8} = \frac{1}{2}, \Pr(B) = \frac{4}{8} = \frac{1}{2}, and \Pr(A\cap B) = \frac{2}{8} = \frac{1}{4}. + Finally: + + \Pr(A \mid B) \amp = \frac{\Pr(A\cap B)}{\Pr(B)} = \frac{2/8}{4/8} = \frac{2}{4} = \frac{1}{2} = \Pr(A), + + so A and B are independent. +

+ + + + + +

+ Let A = \{1, 2, 3\} and B = \{3, 4, 5\} be events in the sample space \Omega = \{1, 2, 3, 4, 5, 6\}. + Create a probability distribution for \Omega so that A, B are independent. +

+ + + + + Example Distribution + + + + x + \Pr(x) + + + + 1 + 0.1 + + + + 2 + 0.2 + + + + 3 + 0.2 + + + + 4 + 0.1 + + + + 5 + 0.1 + + + + 6 + 0.3 + + +
+ +

+ Now \Pr(A) = 0.5, \Pr(B) = 0.4, and + + \Pr(A\cap B) = 0.2 = (0.5)(0.4) = \Pr(A)\Pr(B), + + so A and B are independent. +

+ + + + + +

+ An experiment consists of flipping a biased coin 20 times. + If the coin comes up heads with probability p = 0.3, find the probability of seeing 5 heads. + Find the probability of seeing up to (and including) 3 heads. +

+ + + +

+ Let S be the number of heads. + Then S \sim \Bin(20, 0.3), so: + + \Pr(S = 5) \amp = {20 \choose 5} (0.3)^5 (0.7)^{20 - 5} + \amp = \frac{20!}{(5!)(15!)} (0.3)^5 (0.7)^{15} + \amp = \frac{20 \times 19 \times 18 \times 17 \times 16}{5 \times 4 \times 3 \times 2 \times 1} (0.3)^5 (0.7)^{15} + \amp = (19 \times 3 \times 17 \times 16) (0.3)^5 (0.7)^{15} + \amp \approx 0.179 + +

+ + + + + +

+ An experiment consists of flipping a coin repeatedly until we first see heads. +

+ + + + + +

+ If the coin comes up heads with probability 0.4, what is the probability we'll see our first heads within three flips? What about precisely on the third flip? +

+ + + +

+ Let T be the number of flips until we see heads. + Then T is geometric with parameter p = 0.4, so: + + \Pr(T = k) \amp = (1-0.4)^{k-1}(0.4) = 0.6^{k-1} \cdot 0.4 + +

+ + + + + + +

+ Which flip has the highest chance of being the first flip to come up heads? +

+ + + + + + +

+ A particular store has an average of 20 customers each hour. + During a 4-hour afternoon shift, what is the probability of serving 80 customers. +

+ + + + + +

+ A continuous random variable X taking values in [1, 4] has p.d.f. + f(x) = k(x - \sqrt{x}) for some constant k. +

+ + + + + +

+ What is the value of k? +

+ + + +

+ k = \frac{6}{17}. +

+ + + + + + +

+ Find \Pr(2 \leq X \leq 3). +

+ + + +

+ \Pr(2 \leq X \leq 3) \approx 0.325 +

+ + + + + + +

+ A continuous random variable X taking values in [1, 2] has p.d.f. + \displaystyle{f(x) = \frac{1}{2}\left(\frac{1}{x^2} + x\right)}. + Find the c.d.f. + F(x). + Use your c.d.f. + to find \Pr\left(1 \leq X \leq \frac{3}{2}\right). +

+ + + +

+ \frac{1}{2}\left(\frac{x^2}{2} - \frac{1}{x}\right) + \frac{1}{4}. \Pr\left(1 \leq X \leq \frac{3}{2}\right) \approx 0.479. +

+ + + + + +

+ A continuous random variable X taking values in [2, 3] has c.d.f. + F(x) = \frac{x^3}{3} - x^2 + 4. + Find the p.d.f. + f(x). +

+ + + +

+ f(x) = x^2 - 2x. +

+ + + + + +

+ Consider X, Y with the joint distribution table below. + Are X, Y independent? +

+ + + Joint distribution for <m>X, Y</m> + + + + + X = 0 + X = 1 + + + + Y = 0 + 0.2 + 0.3 + + + + Y = 1 + 0.4 + 0.1 + + +
+ + + +

+ No. + For example, \Pr(X = 0, Y = 0) \neq \Pr(X = 0)\Pr(Y = 0). +

+ + + + + +

+ Suppose X, Y have the distributions: + + \Pr(X = 0) \amp = 0.1 \amp \Pr(Y = 0) \amp = 0.4 + \Pr(X = 1) \amp = 0.4 \amp \Pr(Y = 1) \amp = 0.6 + \Pr(X = 2) \amp = 0.5 + + Assuming X, Y are independent, write a joint distribution table. +

+ + + + + Joint Distribution + + + + + X = 0 + X = 1 + X = 2 + + + + Y = 0 + 0.04 + 0.16 + 0.2 + + + + Y = 1 + 0.06 + 0.24 + 0.3 + + +
+ + + + + +

+ Consider a random variable X with probability distribution below. + Find \E(X). +

+ + + + + + + x + \Pr(X = x) + + + + 1 + 0.1 + + + + 2 + 0.05 + + + + 3 + 0.2 + + + + 4 + 0.15 + + + + 5 + 0.15 + + + + 6 + 0.1 + + + + 7 + 0.05 + + + + 8 + 0.1 + + + + 9 + 0.1 + + +
+ + + +

+ 4.8. +

+ + + + + +

+ Suppose we flip a coin n = 100 times, and let N count the number of heads. +

+ + + + + +

+ If the coin comes up heads on a flip with probability p = 0.4, what is \E(N)? +

+ + + +

+ 40. +

+ + + + + + +

+ What if n = 80 and p = 0.6? +

+ + + +

+ 48. +

+ + + + + + +

+ What if n = 200 and p = 0.5? +

+ + + +

+ 100. +

+ + + + + + +

+ If \E(X) = 3, \E(Y) = -2, and \E(Z) = 1, what is \E(4X + 5Y - Z + 3)? +

+ + + +

+ 4. +

+ + + + + +

+ A continuous random variable X taking values in [0, 1] has p.d.f. + f(x) = 2x. + What is \E(X)? +

+ + + +

+ 2/3. +

+ + + + + +

+ A continuous random variable X taking values in [1, 4] has p.d.f. + f(x) = \frac{4}{3x^2}. + What is \E(X)? +

+ + + +

+ \frac{4}{3}\ln(4) \approx 1.85. +

+ + + + + +

+ A continuous random variable X taking values in [1, 2] has p.d.f. + \displaystyle{f(x) = \frac{1}{2}\left(\frac{1}{x^2} + x\right)}. + Find \E(X). +

+ + + +

+ \frac{1}{2}\left( \ln(2) + \frac{7}{3}\right) \approx 1.513. +

+ + + + + +

+ Consider a random variable X with probability distribution below. + Find \Var(X). +

+ + + + + + + x + \Pr(X = x) + + + + 1 + 0.1 + + + + 2 + 0.05 + + + + 3 + 0.2 + + + + 4 + 0.15 + + + + 5 + 0.15 + + + + 6 + 0.1 + + + + 7 + 0.05 + + + + 8 + 0.1 + + + + 9 + 0.1 + + +
+ + + + + +

+ Suppose we flip a coin n = 100 times, and let N count the number of heads. + If the coin comes up heads on a flip with probability p = 0.4, what is \Var(N)? What if n = 80 and p = 0.6? What if n = 200 and p = 0.5? +

+ + + + + +

+ If \E(X) = 3, \Var(X) = 2, what is \E(X^2)? +

+ + + + + +

+ A continuous random variable X taking values in [0, 1] has p.d.f. + f(x) = 2x. + What is \Var(X)? +

+ + + + + +

+ A continuous random variable X taking values in [1, 4] has p.d.f. + f(x) = \frac{4}{3x^2}. + What is \Var(X)? +

+ + + + + +

+ A continuous random variable X taking values in [1, 2] has p.d.f. + \displaystyle{f(x) = \frac{1}{2}\left(\frac{1}{x^2} + x\right)}. + Find \Var(X). +

+ + + + + +

+ Suppose a coin has an unknown probability of coming up heads. + We perform the experiment in n independent trials, during which it takes k_1, k_2, \dotsc, k_n flips to see our first heads in each trial. + Find a "common sense" MLE formula for the geometric distribution. +

+ + + + + +

+ A particular store owner wants to approximate the average hourly rate at which customers come into the store. + They observe 80 customers enter during a particular 4-hour shift. + What is the maximum likelihood estimation for the hourly customer rate? +

+ + + + + +

+ A radioactive material emits particles at an unknown probabilistic rate \lambda particles per minute. + We observe particles emitted at times 1.1, 1.7, 1.3, 2.2, 1.9, and 1.8 minutes. + Write the likelihood function \mathcal{L}(\lambda) based on this data. + What is the maximum likelihood estimation for \lambda? +

+ + + + + +

+ Suppose a parameter \theta takes values in [0, 1] with likelihood function \mathcal{L}(\theta) = \sqrt{\theta} - \theta^2. + Find the maximum likelihood estimation of \theta. +

+ + + + + +

+ Suppose a coin has probability 0.4 of coming up heads, and we flip the coin 100 times. + Let S be the number of heads. + Estimate the probability that 34 \leq S \leq 44. +

+ + + +

+ \Pr(34 \leq S \leq 44) \approx 0.8212 - 0.0918 = 0.7294. +

+ + + + + +

+ Suppose a fair die is rolled 100 times, and let m be the average value of the rolls. + Estimate the probability that 3.45 \leq m \leq 3.55. +

+ + + +

+ \Pr(3.45 \leq m \leq 3.55) \approx 0.6255 - 0.3745 = 0.2510. +

+ + + + + +

+ The heights of men in the US have a mean of 69 in and a variance of about 9 in^2, and the heights of women in the US have a mean of 63.5 in with a variance of 6.25 in^2. +

+ + + + + +

+ Suppose the heights of 30 men are sampled, and a sample mean m is taken. + Find the expected value and variance of m. +

+ + + +

+ \E(m) = 69 and \Var(m) = 0.3. +

+ + + + + + +

+ Estimate the probability that m \geq 69.5. +

+ + + +

+ \Pr(m \geq 69.5) \approx 1 - 0.8186 = 0.1814. +

+ + + + + + +

+ If the sampled group was women, estimate the probability that m \geq 64. +

+ + + +

+ \E(m) = 63.5 and \Var(m) \approx 0.208. \Pr(m \geq 64) \approx 1 - 0.8643 = 0.1357. +

+ + + + + + +

+ Suppose we flip a coin 100 times and count 60 heads. + Let p be the (unknown) probability that the coin comes up heads on a flip. + Give an approximate 95% confidence interval for the value of p. +

+ + + +

+ [0.504, 0.696]. +

+ + + + + +

+ Suppose in a sample of 100 people, 12 are left-handed. +

+ + + + + +

+ Give a 95% confidence interval for the proportion p of left-handed people. +

+ + + +

+ [0.056, 0.184]. +

+ + + + + + +

+ Give a 90% confidence interval. +

+ + + +

+ [0.066, 0.174]. +

+ + + + + + +

+ The weights of five mice are measured and recorded below. + Give a 95% confidence interval for the sample mean weight of mice. + (Pretend 5 measurements is large enough for the CLT to apply.) +

+ + + + + + + mouse i + weight W_i (g) + + + + 1 + 26 + + + + 2 + 32 + + + + 3 + 33 + + + + 4 + 20 + + + + 5 + 29 + + +
+ + + +

+ [23.39, 32.61]. +

+ + + + + +

+ The heights of five plants are measured and recorded below. + Give a 95% confidence interval around the sample mean for the heights of the plants. + (Pretend 5 measurements is large enough for the CLT to apply.) +

+ + + + + + + plant i + height H_i (in) + + + + 1 + 15 + + + + 2 + 14 + + + + 3 + 18 + + + + 4 + 21 + + + + 5 + 17 + + +
+ + + +

+ [14.61, 19.39]. +

+ + + + + +

+ In each of the following scenarios, determine whether we should use a 1-tailed test or a 2-tailed test. +

+ + + + + +

+ We have a coin which we've flipped many times, seeing an above-average number of heads. + We suspect the coin comes up heads more often than a fair coin would. +

+ + + + + + +

+ We find a coin on the street and wonder whether or not it's a fair coin. +

+ + + + + + +

+ We suspect there will be a difference in average weight of mice caught during the summer versus during the winter. +

+ + + + + + +

+ In a medical study, a group of patients are gathered and the proportion experiencing particular symptoms is measured. + A new drug intended to eliminate these symptoms is administered, after which the proportion experiencing symptoms is measured again. +

+ + + + + + +

+ Suppose we find a coin and wonder whether it's fair. + As a first test, we decide to flip the coin 200 times and count the number of heads. + If we see 112 heads, should we accept or reject the null hypothesis of a fair coin at a significance level of 0.05? +

+ + + + + +

+ Suppose we have a coin which we suspect comes up heads more often than a fair coin would. + As a first test, we decide to flip the coin 200 times and count the number of heads. + If we see 112 heads, should we accept or reject the null hypothesis of a fair coin at a significance level of 0.05? +

+ + + + + +

+ Suppose we find a six-sided die and wonder whether it's fair. + As a first test, we decide to roll the die 100 times and count the number of times it comes up 1. + If we roll 22 1's, should we accept or reject the null hypothesis of a fair die at a significance level of 0.05? +

+ + + + + +

+ Suppose a particular plant when grown outdoors has an average height of 39 in with a variance of 20 in^2. + We suspect that growing this plant in a greenhouse will increase its height. + A sample of 50 plants grown in a greenhouse has an average height of 40 in. + Is this significant enough data to reject the null hypothesis of equal means at the p = 0.05 significance level? +

+ + + + + +

+ A farmer is testing an experimental new plant fertilizer that is supposed to increase the weight of a particular apple variety. + A control sample of 25 apples grown using the usual fertilizer have a mean weight of 75 grams and a sample variance of 90 grams^2 (for an individual apple). + An experimental sample of 25 apples grown using the new fertilizer have a mean weight of 79 grams and a sample variance of 90 grams^2. +

+ + + + + +

+ We have an established factory which produces coins that are close to fair. + We're opening up a second factory, and we'd like to ensure the machines are calibrated to produce coins which behave similarly to the ones produced in the established factory. + We pick one sample coin from each factory, and flip each sample coin 100 times. + The coin from the established factory flips 52 heads in 100 flips. + The coin from the new factory flips 62 heads in 100 flips. + Is this strong enough evidence to reject the null hypothesis that the two factories produce similar coins at a 0.05 significance level? +

+ + + + + +

+ Suppose we find a coin and wonder whether it's fair. + As a first test, we decide to flip the coin 200 times and count the number of heads, S. + What values of S would be extreme enough to reject the null hypothesis of a fair coin? If the coin actually has a 0.6 probability of coming up heads, what is the power of this test? +

+ + + + + +

+ Suppose we have a coin which we suspect comes up heads more often than a fair coin would. + As a first test, we decide to flip the coin 200 times and count the number of heads, S. + What values of S would be extreme enough to reject the null hypothesis of a fair coin? If the coin actually has a 0.6 probability of coming up heads, what is the power of this test? +

+ + + + + +

+ Suppose we find a six-sided die and wonder whether it's fair. + As a first test, we decide to roll the die 100 times and count the number of times it comes up 1. + The expected number of 1's is 50/3, with a variance of 125/9. +

+ + + + + +

+ Using a normal approximation, what is the smallest number of 1's greater than 50/3 that would be extreme enough to reject the null hypothesis of a fair die? +

+ + + + + + +

+ Using a normal approximation, what is the greatest number of 1's less than 50/3 that would be extreme enough to reject the null hypothesis of a fair die? +

+ + + + + + +

+ Suppose that this die is weighted so that it rolls a 1 with probability 0.2. + What would be the power of our test? +

+ + + + + + +

+ Suppose we roll the die 100 times and see 23 1's. + Use the maximum likelihood value for the probability of rolling a 1 to calculate the power of the test. +

+ + + + + + +

+ Suppose a particular plant when grown outdoors has an average height of 39 in with a variance of 20 in^2. + We suspect that growing this plant in a greenhouse will increase its height. + We take the average height of a sample of 50 plants grown in a greenhouse. + What is the minimum average height of this sample that would be extreme enough to reject the null hypothesis of equal means at the p = 0.05 significance level? If the plants, when grown in a greenhouse, would truly have an average height of 41 in, what is the power of our test? +

+ + + + + +

+ A store owner wants to determine how much shelf space to allocate to each of the drinks that they sell. + They survey their customers about their favorite drinks. + Is the data below consistent with the null hypothesis that each type of drink will be equally preferred? +

+ + + + + + + drink type + water + soda + tea + coffee + energy drinks + + + + favorite + 28 + 17 + 15 + 26 + 14 + + +
+ + + + + +

+ A particular drug is administered in 100 independent trials. + In each trial, the drug is administered to four people, and we count how many respond to the drug. + The table below shows how many trials have each different count of people who respond to the drug. +

+ + + + + + + # who respond to drug + 0 + 1 + 2 + 3 + 4 + + + + # of trials + 3 + 11 + 31 + 34 + 21 + + +
+ + + + + +

+ Is the data consistent with a binomial distribution with parameter \theta = 0.7? +

+ + + + + + +

+ What is the total number of people who have been administered the drug? What is the total number who have responded to it? What is the maximum likelihood estimation \widehat{\theta} for the probability that a person will respond to the drug? +

+ + + + + + +

+ Is the data consistent with a binomial distribution with the MLE value of \widehat{\theta}? +

+ + + + + + +

+ Calculate the covariance of X and Y: +

+ + + + + + + + X = 1 + X = 2 + + + + Y = 0 + 0.12 + 0.24 + + + + Y = 1 + 0.3 + 34 + + +
+ + + + + +

+ Calculate the covariance of X and Y: +

+ + + + + + + + X = 0 + X = 1 + X = 2 + + + + Y = 0 + 0.08 + 0.16 + 0.1 + + + + Y = 1 + 0.14 + 0.2 + 0.32 + + +
+ + + + + +

+ Suppose we roll a fair, 4-sided die two times. + Let X be the sum of the rolls, and let Y be the product of the rolls. + Find the covariance of X and Y. +

+ +

+ [Note: Somewhat tedious.] +

+ + + + + +

+ Calculate the correlation of X and Y: +

+ + + + + + + + X = 1 + X = 2 + + + + Y = 0 + 0.12 + 0.24 + + + + Y = 1 + 0.3 + 34 + + +
+ + + + + +

+ Calculate the correlation of X and Y: +

+ + + + + + + + X = 0 + X = 1 + X = 2 + + + + Y = 0 + 0.08 + 0.16 + 0.1 + + + + Y = 1 + 0.14 + 0.2 + 0.32 + + +
+ + + + + +

+ Suppose we roll a fair, 4-sided die two times. + Let X be the sum of the rolls, and let Y be the product of the rolls. + Find the correlation of X and Y. +

+ +

+ [Note: Somewhat tedious.] +

+ + + + + +

+ A survey of local companies collects information about marketing budgets X and revenue Y (each measures in thousands of dollars), shown below. + A linear regression gives the best linear fit as Y = 18.28 X + 29.69. + What is the coefficient of determination r^2? +

+ + + + + + + x_i + y_i + (y_i - \text{avg})^2 + \text{pred } y_i + \text{res}^2 + + + + 200 + 4300 + 2073600 + 3686 + 377377 + + + + 420 + 7700 + 3841600 + 7707 + 53 + + + + 270 + 4500 + 1537600 + 4965 + 216495 + + + + 380 + 7000 + 1587600 + 6976 + 572 + + + + 300 + 5200 + 291600 + 5514 + 98401 + + + + sum: + 28700 + 9332000 + 28848 + 692898 + + +
+ + + + + +

+ A sample of 100 measurements are taken and a best fit line is calculated, resulting in the data below (the final line shows the sums for each column). + Find the coefficient of determination. +

+ + + + + + + x_i + y_i + \text{pred } y_i + (y - \text{avg})^2 + \text{res}^2 + + + + 38.00 + 121.00 + 93.17 + 11.83 + 774.33 + + + + 87.00 + 241.00 + 238.95 + 13586.23 + 4.22 + + + + 30.00 + 61.00 + 69.37 + 4024.63 + 70.12 + + + + 35.00 + 85.00 + 84.25 + 1555.51 + 0.57 + + + + ⋮ + ⋮ + ⋮ + ⋮ + ⋮ + + + + 33.00 + 108.00 + 78.30 + 270.27 + 882.19 + + + + 26.00 + 32.00 + 57.47 + 8545.15 + 648.91 + + + + sum: + 12444.00 + 12444.00 + 515206.64 + 29789.80 + + +
+ + + +
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