stats_exam1_practice_problems_solutions

4 7. The mean of X = 10 and the standard deviation of X = 5. The variable X is linearly transformed such that X new = a + bX . The mean of X new = 50 and the standard deviation of X new = 20. Suppose b is a positive number. What does a equal? 8. The mean of X = 10 and the standard deviation of X = 2. The variable X is linearly transformed such that X new = a + bX . The mean of X new = 70 and the variance of X new = 36. Suppose b is a positive number. What does a equal? 9. The mean of X = 20 and the IQR of X = 2. The variable X is linearly transformed such that X new = a + bx . The mean of X new = 700 and the IQR of X new = 8. Suppose b is a negative number. What does a equal? 10. The mean of X = 10 and the standard deviation of X = 2. The variable X is linearly transformed such that X new = a + bx . The mean of X new = 500 and the variance of X new = 64. Suppose b is a negative number. What does a equal?

2 3. The covariance between X and Y is calculated and recorded as s XY (and s XY is a positive number). A linear transformation is performed on X . The linearly transformed variable is called W , where W i = (5) X i . The covariance between W and Y is calculated and recorded as s WY . Which statement is correct? A) In this example, s XY would be greater than s WY . B) In this example, s XY would be less than s WY . C) In this example, s XY would be equal to s WY . 4. The covariance between X and Y is calculated and recorded as s XY (and s XY is a negative number). A linear transformation is performed on X . The linearly transformed variable is called W , where W i = 6 + X i . The covariance between W and Y is calculated and recorded as s WY . Which statement is correct? A) In this example, s WY would be greater than s XY . B) In this example, s WY would be less than s XY . C) In this example, s WY would be equal to s XY . 5. The correlation between X and Y is calculated and recorded as r XY (and r XY is positive). A linear transformation is performed on X . The linearly transformed variable is called W , where W i = (-7) X i . The correlation between W and Y is calculated and recorded as r WY . In this example, _____. A) r XY would definitely be the same as r WY B) r XY would definitely not be the same as r WY C) r XY might be the same as r WY , but might not be

2 3. The probability that a randomly selected American has disease Z is 0.02. If 85 people are randomly selected from the population, what is the probability that at least 1 of them has disease Z ? (R ound your answer to four decimal places. Remember to express your answer as a proportion.) 4. Twenty-two percent of the American adult population supports candidate Green. An SRS of 13 adults asNs if they aJree with the statement ³, support candidate Green .´ What is the probability that at least 2 of those surveyed would agree with that statement? (R ound your answer to four decimal places. Remember to express your answer as a proportion.)

3. Suppose that A and B are two events with P( A ) = 0.3, P( B ) = 0.4, and P( B | A )=0.5. What is P( A or B )? 4. Suppose that F and G are two events with P( F ) = 0.4, P( G ) = 0.5, and P( F | G )=0.6. What is P( F or G )? 5. This problem involves two distinct sets of events, which we label A 1 and A 2 and B 1 and B 2 . The events A 1 and A 2 are mutually exclusive and collectively exhaustive within their set. The events B 1 and B 2 are mutually exclusive and collectively exhaustive within their set. Intersections can occur between all events from the two sets. Given P ( A 1 ) = 0.8, P ( B 1 | A 1 ) = 0.2, and P ( B 1 | A 2 ) = 0.6, what is P ( A 1 | B 1 )? (Round your answer to four decimal places.) 6. This problem involves two distinct sets of events, which we label A 1 and A 2 and B 1 and B 2 . The events A 1 and A 2 are mutually exclusive and collectively exhaustive within their set. The events B 1 and B 2 are mutually exclusive and collectively exhaustive within their set. Intersections can occur between all events from the two sets. Given P ( A 1 ) = 0.3, P ( B 1 | A 1 ) = 0.8, and P ( B 1 | A 2 ) = 0.4, what is P ( A 2 | B 2 )? (Round your answer to four decimal places.)

7 Problem 7 Round all of your final answers to four decimal places if necessary. The probability distribution of random variable X is given below: X 2 4 P ( x ) 0.2 0.8 A second random variable is defined as follows: Y = 2 X + 1. A third random variable is defined as follows: Z = X 2 – 1. a) What is μ X ? b) What is ࠵? ௒ ଶ ? c) What is ࠵? ௓ ଶ ?

8 Problem 8 Round all of your final answers to four decimal places if necessary. For parts a and b : This problem involves two distinct sets of events, which we label A 1 and A 2 and B 1 and B 2 . The events A 1 and A 2 are mutually exclusive and collectively exhaustive within their set. The events B 1 and B 2 are mutually exclusive and collectively exhaustive within their set. Intersections can occur between all events from the two sets. Consider the probabilities below: P ( A 1 ) = 0.6 P ( B 1 | A 1 ) = 0.2 P ( B 1 | A 2 ) = 0.1 a) What is P ( A 1 and B 1 )? b) What is P ( A 1 | B 1 )? c) Suppose that X and Y are two events with P ( X ) = 0.4, P ( Y ) = 0.3, and P ( Y | X )=0.2. What is ܲሺܺ ׫ ܻሻ ?

10 Problem 10 Round all of your final answers to four decimal places if necessary. a) Natalie converts 70% of her free throws. (Her free throws are independent.) She goes to the line to VKRRW D ³WZR - VKRW IRXO±´ ²:KHWKHU RU QRW WKH ILUVW VKRW LV PDGHµ VKH LV DOORZHG D VHFRQG VKRW± (DFK successful shot is worth one point.) The random variable X is the number of points resulting from 1DWDOLH¶V WULS WR WKH OLQH IRU D ³WZR - VKRW IRXO±´ What is E( X )? b) The Cubs are to play a series of 3 games with the Mets. For any one game it is estimated that the pr REDELOLW\ RI D &XEV¶ ZLQ LV ·±¸ . The outcomes of the 3 games are independent. Let X be the number of games that the Cubs win in the series. What is ࠵? ௑ ଶ ? c) The sample space contains 4 As and 5 Bs. What is the probability that a randomly selected set of 3 will include 1 A and 2 Bs?

11 Problem 11 Round all of your final answers to four decimal places if necessary. a) The mean of X = 10 and the standard deviation of X = 2. The variable X is linearly transformed such that X new = a + bx . The mean of X new = 500 and the standard deviation of X new = 8. Suppose b is a positive number. What does a equal? b) The mean of X = 5 and the standard deviation of X = 2. The variable X is linearly transformed such that X new = a + bx . The mean of X new = 800 and the variance of X new = 64. Suppose b is a negative number. What does a equal? c) The probability that a randomly selected American has disease Z is 0.04. If 30 people are randomly selected from the population, what is the probability that at least 1 of them has disease Z ?

13 Problem 13 Round all of your final answers to four decimal places if necessary. a) Consider a data set that includes every whole number from 10 to 80, and then every whole number from 200 to 410: 10, 11, 12, …78, 79, 80, 200, 201, 202, … 408, 409, 410 What is n in this data set? What is Q 3 in this data set? b) In a city election, 30% of the voters are white and 70% of the voters are non-white. Candidate Jones wins 40% of the white vote and 80% of the non-white vote. Event W is the event that a voter is white. Event NW is the event that a voter is non-white. Event F is the event that a voter votes for candidate Jones. Event A is the event that a voter votes against candidate Jones. What is P ( NW or A )? c) A shipment of 20 parts contains 6 defective parts. Two parts are chosen at random from the shipment and checked. What is the probability that exactly one defective part is found?