Test 3 SOLUTIONS. x P(x) xp(x)

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1 16 1. A couple of weeks ago in class, each of you took three quizzes where you randomly guessed the answers to each question. There were eight questions on each quiz, and four possible answers to each question (A, B, C, or D). Based on the results of the quizzes, the probability distribution shown here gives the random variable x, which represents the number of questions out of eight that were guessed correctly, and the experimental probability P(x) for each value of x. Test 3 SOLUTIONS (a) Find the mean number of questions that were guessed correctly (round to one decimal place). Create a third column, x times P(x). The sum of this column equals the mean of the random variable. (b) What is the probability that a randomly selected student from this class guessed the answers to exactly 2 questions correctly? P(2) = (c) What is the probability that a randomly selected student from this class guessed the answers to less than 2 questions correctly? P(0) + P(1) = = x P(x) xp(x) Mean = 1.9 (d) Would it be unusual for a student to guess 3 or more questions correctly? Yes or no, plus briefly explain your answer here, and include the relevant probability in your answer. P(3 or more) = P(3) + P(4) + + P(8) = = NOT unusual, because the probability of 3 or more is > (a) 1.9 (b) (c) (d) Answer in space to left. Page 1 of 8

2 2. Now consider a different quiz, a True/False quiz with 10 questions on it. Randomly guessing the answer to each question is a binomial experiment. Use the binomial probability distribution to answer the following questions. Remember that the random variable x represents the total number of questions out of ten that are guessed correctly. Round off the probabilities to three significant figures. You don t have to do this, but you might want to start by identifying: S = success = what? (in words) = guess the correct answer n = 10 (total number of questions) p = 0.5 (probability of guessing any one question correctly) 16 (a) Calculate the mean of the random variable x, the number of questions that will be guessed correctly. µ = np = (10)(0.5) = 5 (b) What is the probability of guessing exactly 4 questions correctly out of 10? There is no need to do any calculations, just look up the probabilities in Table III. (see next page) (a) 5 (b) From the table, P(4) = Page 2 of 8

3 (c) What is the probability of guessing 4 or less questions correctly out of 10? P(4 or less) = P(0) + + P(4) = Just add up the probabilities from the table! (d) Would it be unusual to guess 8 or more questions correctly? Yes or no, plus briefly explain your answer here. Your answer must include the relevant probability. (c) (d) Answer to left. P(8 or more) = P(8) + P(9) + P(10) = This is > 0.05, therefore it is NOT unusual. Note: Because p = 0.5 exactly, the distribution will be bellshaped (even though the sample size is small), so you could also use the Empirical Rule for this. If you did that, you would find that the maximum usual = 8.16, and therefore 8 is not unusual because it is less than that. Page 3 of 8

4 3. According to a recent (May 18, 2017) poll done by Monmouth University *, a majority of American adults believe President Trump fired FBI Director James Comey recently in an effort to slow down or stop the investigation into the Trump campaign s ties with Russia. Specifically, 59% of Americans believe that it was very or somewhat likely that he was fired for this reason. Suppose that 65 American adults are randomly selected, and answer the following using the binomial probability distribution: 18 * Source: S = likely or somewhat likely p = probability of success in any one trial = 0.59 n = number of trials = 65 (a) Find the mean number of Americans out of 65 who believe that Trump fired Comey to stop the Russia investigation. Do not round off your answer µ = np = (65)(0.59) = (b) Find the probability that exactly 38 out of 65 Americans believe that Trump fired Comey to stop the Russia investigation. Round off your answer to three significant figures. P(38) = 65C 38(0.59) 38 (0.41) 27 = (c) Would it be unusual for 50 of the 65 Americans to believe that Trump fired Comey to stop the Russia investigation? Yes or no, plus explain your answer, including a relevant calculation. Use the Empirical Rule: σ = (65)(0. 59)(0. 41) = Max usual = µ + 2 σ = ( ) = is greater than the max usual, so YES, it would be unusual. Note: you cannot use probabilities here, because you would have to calculate P(50 or more), which is basically not possible by hand. Just quoting P(50) is not adequate. (d) If you plotted the probability histogram for this binomial distribution, what shape would you expect it to have? Explain your answer, and include a relevant calculation. It would be normal, because the sample size is large enough. Demonstrate this as follows: np(1 p) = 65(0.59)(0.41) = Since this is > 10, it will be normal. (a) (b) (c) Answer to left. (d) Answer to left. Page 4 of 8

5 4. Determine the area under the standard normal curve that lies: (a) To the left of 0.83 Area = (b) To the right of 0.09 Area = = (c) Between 1.48 and 0.33 Area below z = = Area below z = 0.33 = Area between = = (d) Obtain the z-score for which the area under the standard normal curve to its right is Therefore, the area to its left = = Closest area in the table is , corresponding z-score is (a) (b) (c) (d) (e) Determine the two z-scores that divide the area under the standard normal curve symmetrically into a middle 0.62 area and two outside 0.19 areas (the tails). The closest area in the table to 0.19 is , which corresponds to a z-score of By symmetry, the other z- score is (e) ± 0.88 Page 5 of 8

6 5. It has traditionally been taken as fact that the mean normal body temperature for humans is 98.6 F. However, more recent medical research suggests that the true mean is approximately 98.2 F *. *Source: Healthy body temperatures are known to be normally distributed. Assume a mean of 98.2 F and a standard deviation of 0.7 F for the following calculations. Note: do not round off any of the probabilities. (a) If one person is randomly selected, find the probability that their temperature is greater than 98.6 F. z = = Area below z = 0.57 = Area above = = (a) (b) If one person is randomly selected, find the probability that their temperature is between 96.5 F and 98.6 F. z 96.5 = = Area below = Already found in part (a) that the area below 98.6 = Area between = = (b) (c) Fever is generally agreed to be present if a person s temperature is over 99.9 F. What percentage of healthy people have a temperature of over 99.9 F? z = = Area above is , which you can infer from symmetry and part (b), since the area below z = is To convert into percentage, multiply by 100. (c) 0.75 % Page 6 of 8

7 (d) Find the value of the 80 th percentile. In other words, what is the cutoff temperature that separates the bottom 80% from the top 20% of the body temperatures? Round your answer to two decimal places. (d) F (e) Answer to left. The closest area in the table is , which corresponds to a z- score of Calculate the x-value: x = µ + z = (0.84)(0.7) = F (e) Would it be unusual for a healthy person to have a temperature of 96.5 F? Yes or no, plus explain briefly, and include a relevant calculation. You can answer this EITHER using probabilities or the Empirical Rule. Empirical Rule: min usual = µ 2 = (0.7) = Yes, it would be unusual, because 96.5 F is less than the min usual. Probabilities: You already calculated in part (b) that the area below = Because this probability is very small (<0.05), it would be considered unusual. OVER FOR EXTRA CREDIT!!! Page 7 of 8

8 Extra Credit #1 (4 points): Men tend to have longer feet than women. So, if you find a really long footprint at the scene of a crime, then in the absence of any other evidence, you would probably conclude that the criminal was a man. Assume that men s foot lengths are normally distributed with a mean of 25 cm and a standard deviation of 4 cm, and women s foot lengths are normally distributed with a mean of 19 cm and a standard deviation of 3 cm. Assume that a footprint belongs to a man if it is longer than 22 cm, and to a women if it is less than 22 cm (the midpoint of the means). Using this rule, what is the probability that you will mistakenly identify a woman s footprint as having come from a man? If the footprint is greater than 22 cm, then you would assume that it came from a man. However, the yellow highlighted portion of the Females curve represents the women that have feet longer than 22 cm. Therefore, that area is the probability that you need to find. z = = 1. 0 Area to the right = = Therefore, there is a 15.87% chance that it is a woman s footprint, but it is mistakenly identified as a man s. Page 8 of 8