EM521
SAMPLE MIDTERM EXAM QUESTIONS
1. In a questionnaire, respondents are asked to mark their gender as male or female. Gender is an example of the __________ scale. a) Ordinal b) Nominal c) Ratio d) Interval Answer: b 2. The Central Limit Theorem is important in Statistics because: a) It tells us that large samples do not need to be selected. b) It guarantees that, when it applies, the samples that are drawn are always randomly selected. c) It enables reasonably accurate probabilities to be determined for events involving the sample average when sample size is large regardless of the distribution of the variable. d) It tells us that if several samples have produced sample averages, which seem to be different than expected, the next sample average will likely to be close to its expected value. e) It is the basis for much of the theory that has been developed in the area of discrete random variables and their probability distributions. Answer: c 3. A factory produces items with p=0.10 defective probability. Items are distributed in boxes of n=20. Hence the number of defective items, X, in a box has a binomial distribution with mean np and variance np(1-p). Using normal approximation (with the correction factor) to binomial find the approximate probability that a box contains more than 4 defectives. a) 0.01 b) 0.03 c) 0.04 d) 0.07 e) 0.10 Answer: b
4. Work hours lost due to accidents over a year were sampled from 20 companies. Their stem and leaf plot was obtained using MINITAB as shown below. Stem-and-leaf of lost hours Leaf Unit = 10 2 4 6 7 9 (3) 8 7
0 0 0 0 1 1 1 1
N
= 20
23 45 67 9 11 222 4 6677777
a) Reconstruct the data from the above plot. 20, 30, 40, 50, 60, 70, 90, 110, 110, 120, 120, 120, 140, 160, 160,170, 170, 170, 170, 170 b) What is the median of these data? (120+120)/2=120 c) What is the mode of these data? 170 (the most frequently observed data) 5. The average annual amount of American households spend for daily transportation is $6312 (Money, August 2001). Assume that the amount spent is normally distributed. a) Suppose you learn that 5% of American households spend less than $1000 for daily transportation. What is the standard deviation of the amount spent? b) What is the range of spending for the 3% of households with the highest daily transportation cost? Solution: = 6312 a) P(X<1000) = 0.05 P( Z
1000 6312
) 0.05
z = -1.645 cuts off .05 in the lower tail. 1.645
1000 6312
So,
1000 6312 3229 1.645
b)
P(X>x) = 0.03
z = 1.88 cuts off approximately .03 in the upper tail x = 6312 + 1.88(3229) = 12,383 The households with the highest 3% of expenditures spent more than $12,383. 6. Measurement error that is continuous and uniformly distributed from -3 to+3 millivolts is added to the true voltage of a circuit. Then the measurement is rounded to the nearest millivolt so that it becomes discrete. Suppose that the true voltage is 250 millivolts. a) What is the probability mass function of the measured voltage? b) What is the mean and variance of the measured voltage? Solution: (a) Let X denote the measured voltage. 1 So the probability mass function is P( X x) , 7 (b) E(X)=250
for
x 247,..., 253
Var(X)=E(X2)- (E(X))2= (1/7)(2472+…+2532)-2502=4 7. In the Let’s make a deal show (from seventies), a contestant is given a choice of three doors of which one contained a prize. The other two doors contain gag gifts like a chicken or a donkey. After the contestant chose an initial door, the host of the show then reveals an empty door among the two doors not chosen, and asks the contestant if he or she would like to switch to the remaining door. The question is should the contestant switch. Do the odds of winning increase by switching to the remaining door? What if there are 100 doors and only one contains a prize? Solution: W= the event that the contestant wins A= the event that the contestant has chosen the right door initially B= the event that the contestant has chosen a wrong door initially
Case 1: not switch p(W)=p(W|A).p(A)+p(W|B).p(B) =(1)(1/3)+(0)(2/3) =1/3 Case 2: switch p(W)=p(W|A).p(A)+p(W|B).p(B) =(0)(1/3)+(1)(2/3) =2/3 The contestant has a higher probability of winning if he switches. The same thing holds for the case with 100 doors. Note that after the contestant chose an initial door, the host will reveal 98 empty doors. In this case the probability of winning if the contestant does not switch is (1/100) which will increase to (99/100) in case that he does switch (since he has a 0.99 chance of choosing the wrong door initially). 8. The Department of Economic and Community Development (DECD) reported that in 2009 the average number of new jobs created per county was 450. The department also provided the following information regarding a sample of 5 counties in 2010. County Bradley Rhea Marion Grundy Sequatchie
New Jobs Created In 2010 410 480 407 428 400
a) Compute the sample average and the standard deviation for 2010. = 425 and s = 32.44 (rounded)
b) We want to determine whether there has been a significant decrease in the average number of jobs created. Provide the null and the alternative hypotheses. Ho: > 450 H1: < 450
c) Compute the 95% confidence interval for the mean and the hypotheses. P( x t /2,n 1 ( s / n ) x t /2,n 1 ( s / n )) P(425 5.571(32.44 / 5) 425 5.571(32.44 / 5)) P(344.18 505.82) 0.95
Since the interval covers 450, we cannot reject H0.
