test2reviewanswers

Exam 2 Review Answers MAT 167 1. Given the following data, x -4 -3 0 2 y 10 8 5 0 (a) Create a scatter diagram of the data. Does it look like there’s a linear correlation? If yes, is it positive or negative? It looks like there’s a negative linear correlation. (b) Calculate the correlation coefficient. Is there a strong linear correlation? Is it positive or negative? r ≈ - 0 . 98. This is a strong negative correlation since | - 0 . 98 | > 0 . 950, the critical value for the correlation coefficient. (c) Find the regression line for the data above and use it to predict a y-value for x = 1 ˆ y = - 1 . 55 x + 3 . 81; for x = 1, ˆ y = 2 . 26 (d) Interpret the slope and y-intercept of the least-squares regression line (if they make sense). The slope is - 1 . 55. This indicates that for each increase of 1 for x , y is predicted to decrease by 1.55. The y-intercept is (0 , 3 . 81). This means the regression line predicts a value of 3.81 for x = 0 (although we have an actual data point that shows that when x is 0, y is 5).

2. In a factory, it was observed that employees who were well-liked by at least 60% of their colleagues usually performed well on their annual performance evaluation. The correlation coefficient for being well-liked and performing well was 0.97. Do you think that being well-liked causes a high score on the performance evaluation? Explain why or why not. If you do not think so, give some examples of other factors that could influence performing well on the evaluation. Correlation does not imply causation, so probably being well-liked does not cause a good evaluation. Other factors that might influence performing well (answers may vary): a good night’s sleep before the evaluation, enjoying one’s job, high skill level, etc. 3. A probability experiment is performed in which an 8-sided die is rolled and then a card is chosen at random from a standard deck of cards. (a) What is the probability of rolling a number greater than 5 and choosing an even-numbered card? 3 8 · 20 52 ≈ 0 . 144 (b) Is rolling a number greater than 5 and choosing an even-numbered card a simple event? Why or why not? No, the event has more than one possible outcome, e.g., roll a 6 and choose the 2 of spades, roll a 7 and choose the 4 of clubs, roll an 8 and choose the 2 of diamonds, etc. 4. You go to the store to buy milk, eggs, and bread. For milk you have a choice of whole milk, 2%, 1%, and skim; for eggs you have a choice of white or brown eggs; for bread you have a choice of white or whole-wheat. How many possible ways could you choose the three items? What is the sample space? 4 choices for milk, 2 choices for eggs, 2 choices for bread; so there are 4 · 2 · 2 = 16 ways to choose the three items. The sample space is: { (whole milk, white eggs, white bread),(whole milk, white eggs, whole-wheat bread),(whole milk, brown eggs, white bread),(whole milk, brown eggs, whole-wheat bread),(2%, white eggs, white bread),(2%, white eggs, whole-wheat bread),(2%, brown eggs, white bread),(2%, brown eggs, whole-wheat bread),(1%, white eggs, white bread),(1%, white eggs, whole- wheat bread),(1%, brown eggs, white bread),(1%, brown eggs, whole-wheat bread),(skim, white eggs, white bread),(skim, white eggs, whole-wheat bread),(skim, brown eggs, white bread),(skim, brown eggs, whole-wheat bread) } 5. A prison rehabilitation program has a success rate of 68%. Jerry, Jamal, and Jos´ e have just completed the program. (a) What is the probability that all three of them will not re-offend after release from prison? 0 . 68 · 0 . 68 · 0 . 68 ≈ 0 . 314 (b) What is the probability that all three of them will re-offend after release from prison? The probability that a person will re-offend is 1 - 0 . 68 ≈ 0 . 32, so the probability that all three will re-offend is 0 . 32 · 0 . 32 · 0 . 32 ≈ 0 . 033 (c) What is the probability that at least one of them will not re-offend after release from prison? At least one re-offending is the complement of none of them re-offending, so the probability that at least one will re-offend is 1 - 0 . 314 = 0 . 686 6. You choose two cards from a standard deck of cards without replacing the first card. What is the probability that the first card you draw is a king and the second card you draw is not a face card (king, queen, jack)? P(K and not a face card)=P(K) · P(not a face card | K)= 4 52 · 40 51 = 160 2652 ≈ 0 . 06 since P(not a face card | K)=1-P(face card | K)=1 - 11 51 = 40 51

15. A gambling game costs 2 to play. It’s possible to lose the game (and pay an additional 50), to break even (and pay no additional money), or to win big (and gain 10,000). Let X represent the payout to the player. x P(x) - $52 0.399 - $2 0.600 $9 , 998 0.001 (a) Determine whether the table describing the payout is a discrete probability distribution. Explain your choice. This is a discrete probability distribution because there are finitely many outcomes (discrete), each of the probabilities is between 0 and 1, and 0 . 399 + 0 . 600 + 0 . 001 = 1. (b) Use the table to find the expected value of the game (to the player). Round the answer to two decimal places. Based on the expected value, would you want to play the game? Why or why not? The expected value is - $11 . 95. I would not want to play the game because the expected value is negative (not likely that I would win money). (c) Find the standard deviation of the discrete random variable, X . Round the answer to two decimal places. The standard deviation is 317.65.