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(DQ-Wk1) Week 1 Online Discussion Question Discussion (1) Applications of statistics can be divided into two broad areas: descriptive statistics and inferential statistics. Use your own words to (brie y) de“fne what descriptive statistics is and what inferential statistics is?

Discussion (2) Someone mentions to you that there is a 68% – 95% – 99% rule in statistics; use your own words (or graph) to (brie y) explain what this rule is about?

Week 1 Quiz Problem(1) (15 points) A sample (data set) contains the observations 13 12 15 10 9 11 16 ; “fnd (a) the sample size ?

(b) 7 i=1 x i (c) 7 i=1 x2 i (d) 7 i=1 (x i 1) (e) 7 i=1 (x i 1)2 (f ) ( 7 i=1 x i) 2 (g) the sample mean ¯ x (h) the sample variance s2 (i) the sample standard deviation (j) the sample median (k) Is there a (sample) mode?

Recall thats2 = n i=1 (x i ¯ x )2n 1 = n i=1 x2 i ( n i=1 x i) 2 n n 1 (HW1.1) Week 1 Homework 1 Problem (1)(10 points) At a medical center, a sample of 20 days showed the following number of cardiograms done each day. (Hint: 20 i=1 x i= 661 , 20 i=1 x2 i = 25665 .) 25 31 20 32 13 14 43 02 57 23 36 32 33 32 44 32 52 44 51 45 (a) (1 points) Find the sample mean ¯ x.

(b) (2 points) Find the sample variance s2 . (The formula for s2 has been given in previous problem.) (c) (7 points) Construct a stem-leaf plot for the data. Problem (2)(15 points) A boxplot is given on next page, answer the follow ing questions. 1 90 95 100 105 110 115 120 125 130 135 140 Values Column Number (a) Estimate the median of the sample.

(b) Estimate the 25th percentile of the sample.

(d) Estimate the interquartile range of the sample.

(e) Is (Are) there any outlier(s) in the sample. Estimate its(their) value(s).

(By default, an outlier is a value that is more than 1.5 times t he interquartile range away from the top or bottom of the box.) (f ) Disregard the outliers, estimate the maximum of the samp le.

(g) Disregard the outliers, estimate the minimum of the samp le.

(h) Disregard the outliers, estimate the range of the sample .

(i) Do you think the median and the mean of the sample are close ? Please explain. Problem (3)Consider a random experiment of throwing two fair dice toget her.

(a) (7 points) Describe the sample space Sand assign probabilities to each sample point in S .

(b) (3 points) Let Xbe the random variable that corresponds to the sum of the two d ice.

What are the values that the random variable Xtakes?

(d) (3 points) Find P(X 4).

(e) (4 points) Find P(X 3). (HW1.2) Week 1 Homework 2 Problem(1)(10 points) Inspector A visually inspected 1000 ceramic mug s and found aws in 36 of them. Inspector B inspected the same 1000 ceramic mug s and found aws in 42 of them. A total of 950 mugs were found to be good by both inspec tors. One of the 1000 mugs is selected randomly. (a) Find the probability that a a w was found in this mug by at least one of the two inspectors. (b) Find the probability tha t aws were found in this mug by both inspectors. (c) Find the probability that a aw was fo und by inspector B but not by inspector A.

Problem(2) (20 points) A lot of 20 components contains 4 that are defecti ve. Two com- ponents are drawn at random and tested. Let Abe the event that the rst component drawn is defective, and let Bbe the event that the second component drawn is defective.

(a) (2 points) Find P(A ). (b) (2 points) Find P(B |A ). (c) (2 points) Find P(A B). (d) (2 points) Find P(A C B). (e) (2 points) Find P(A BC ). (f ) (2 points) Find P(A C BC ).

(g) (4 points) Find P(B ). (h) (2 points) Find P(A B). (i) (2 points) Are events Aand B independent? Explain. (HW1.3) Week 1 Homework 3 Problem(1)(20 points) A biasedcoin is tossed, and it is assumed that the chance of getting a head, H, is 2 5 . (Thus the chance of getting a tail, T, is 3 5 ) Consider a random experiment of throwing the coin 4 times.

(a) (2 points) Let Sdenote the sample space. Describe the elements in S.

(b) (2 points) Let Xbe the random variable that corresponds to the number of the h eads coming up in the four times of toss. What are the values that th e random variableXtakes?

(d) (2 points) Find the probability that at least one of them l ands head, that is,P(X 1).

(e) (2 points) Suppose that for each toss that comes up heads w e win $4, but for each toss that comes up tails we lose $3. Clearly, a quantity of interes t in this situation is our total wining. Let Ydenote this quantity. What are the values that the random var iableYtakes?

(f ) (4 points) Compute the expectation of the random variabl eX , i.e., E(X ) =?

(g) (6 points) Compute the expectation of the random variabl eY , i.e., E(Y ) =? (HW1.4) Week 1 Homework 4 Problem(1)(5 points) Compute following “permutations” and “combinat ions”:

(a) P5 3 (b) P5 2 (c) C5 3 (d) C5 2 (e) Px 2 Problem(2) (8 points) Consider a random experiment of throwing two fair dice together.

De ne the following events: A:{ You roll a 7 }& B:{ At least one of the dice shows a 6 } (a) Identify the sample points in the events A,B ,A B,A B, and AC .

(b) Find P(A ), P(B ), P(A B), P(A B), and P(A C ).

Problem(3) (5 points) For two events, Aand B, P (A ) = 0 .2 , P (B ) = 0 .4 , and P(A |B ) = 0 .5 .

Find P(A B) and P(B |A ).

Problem(4) (2 points) A lottery has 52 numbers. To win the jackpot one nee ds to match all 6 number that are drawn by the machine. What is the probabi lity to hit the jackpot?

What is the odds to hit the jackpot? Is the probability very di erent from the odds? (Test Wk1) Week 1 Test Problem(1)(5 points) Applications of statistics can be divided into tw o broad areas:de- scriptive statistics andinferential statistics . Use your own words to (brie y) de ne what descriptive statistics is and whatinferential statistics is?

Problem(2) (15 points) At a medical center, a sample of 20 days showed the following number of cardiograms done each day. (Hint: 20 i=1 X i= 671 , 20 i=1 X2 i = 25805 .) 25 31 20 32 13 14 43 12 57 23 36 32 33 32 44 32 52 44 51 45 (a) (1 points) Find the sample mean ¯ X .

(a) (9 points) Show that the sample variance s2 = n i=1 ( x i ¯ x )2 n 1 can also be expressed as s 2 = n i=1 x 2 i ( n i=1 x i)2 n n 1 . That is, to show s2 = n i=1 ( x i ¯ x )2 n 1 = n i=1 x2 i ( n i=1 x i) 2 n n 1 .

Find the sample variance S2 for the data of 20 days of cardiograms above.

(c) (5 points) Construct a stem and leaf plot for the data. Problem(3)(10 points) 1 108 110 112 114 116 118 120 122 124 126 Values Column Number A boxplot is given below, answer the following questions.

(a) Estimate the median of the sample.

(b) Estimate the 25th percentile of the sample.

(d) Estimate the interquartile range of the sample.

(e) Is (Are) there any outlier(s) in the sample? (By default, an outlier is a value that is more than 1.5 times the interquartile range away from the top or bo ttom of the box.) (f ) Estimate the maximum of the sample.

(g) Estimate the minimum of the sample.

(h) Estimate the range of the sample.

(i) Do you think the median and the mean of the sample are close ? Please explain. Problem(4)(a) (2 points) Professor Chan has to select 5 students out of h is Math 3302 class randomly to participate a regional Statistics contes t. There are 24 students in the class.

How many ways can Prof. Chan choose his students? (Express yo ur answer in combination or permutation form; no need to a give a de nite number.) (b) (4 points) If 2 out of the 5 students selected will attend t he national contest, and the other 3 will go international, how many ways can Prof. Chan ch oose from his 25 students?

Problem(5) (6 points) A college is composed of 60% men and 40% women. It is known that 60% of the men and 40% of the women smoke cigarettes. What is the probability that a student observed smoking a cigarette is a man?

Problem(6) (8 points) Assume that cars are equally likely to be manufact ured on Monday, Tuesday, Wednesday, Thursday, or Friday. Cars made on Monda y have a 4% chance of being “lemons”; cars made on Tuesday, Wednesday or Thursday have a 1% chance of being lemons; cars made on Friday have a 2% chance of being lemons. If Jacky C han bought a car and it turned out to be a lemon, what is the probability it was manufa ctured on Monday? Problem(7)Consider a random experiment of throwing TWO dice together.

(a) (2 points) Describe the sample space.

(b) (2 points) Let Xbe the random variable that corresponds to the sum of the two d ice.

What are the values that the random variable Xtakes?

(d) (2 points) Find P(X 10).

(d) (4 points) Find the expectation of the random variable X,E (X ).

Problem(8) (10 points) Let x 1, x 2, · · · ,x n be a sample and ¯ xand s2 x be the sample mean and the sample variance of the x i’s, respectively. Let aand bbe any nonzero constants. If y 1 = a x 1+ b, y 2 = a x 2+ b, · · · ,y n = a x n+ b, and let ¯ yand s2 y be the sample mean and the sample variance of the y i’s, respectively.

What is the relationship between ¯ xand ¯ y?

What is the relationship between s2 x and s2 y ? Problem(9)(a) (3 points) An unfair coin is tossed, and it is assumed that the chance of getting a head, H, is 2 5 . (Thus the chance of getting a tail, T, is 3 5 ) Consider a random experiment of throwing the coin 4 times. Let Sdenote the sample space. Describe the elements in S.

(b) (2 point) Let Xbe the random variable that corresponds to the number of the h eads coming up in the four times of toss. What are the values that th e random variableXtakes?

(d) (3 points) Find the probability that at least one of them l ands head, that is,P(X 1).

(e) (2 points) Suppose that for each toss that comes up heads w e win $3, but for each toss that comes up tails we lose $2. Clearly, a quantity of interes t in this situation is our total wining. Let Ydenote this quantity. What are the values that the random var iableYtakes?

(f ) (3 points) Find P(Y = 0).

(g) (3 points) Find P(Y 10).

(h) (5 points) Compute the expectation of the random variabl eX , i.e., E(X ) =?

(i) (6 points) Compute the expectation of the random variabl eY , i.e., E(Y ) =?