University:
Stony Brook University
Course:
AMS 310 | Survey of Probability and Statistics
Academic year:

2024
Views:

415

Pages:

16
Author:

study_genius

1. The following are weights of 14 young goats in pounds: 56 40 60 62 80 65 44 51 58 51 66 49 75 40 44 45 49 51 51 56 58 60 62 65 66 75 80 a) Find the sample mean. Round to one decimal place Xbar = 57.3 b) Find the median and the quartiles Q1 and Q3. Median = 57 Q1 = 49 Q3 = 65 c) Create a box plot for this data. Include a scale and label the key 5 numbers. 45 Box plot: 5 number summary 40, 49, 57, 65, 80 Boxplot 80 C1 70 60 50 40 2. Below are weights (in pounds) of males athletes in a high school. Weight Frequency 130 – 150 150 – 170 13 170 – 190 23 190 – 210 210 – 230 32 18 11 a) Draw a histogram Histogram 35 30 25 20 Frequency 15 10 5 0 130 to 150 150 to 170 170 to 190 Weigt 190 to 210 210 to 230 b) Draw a Pie Chart Pie Chart 11% 13% 19% 33% 24% 130 to 150 150 to 170 170 to 190 190 to 210 210 to 230 c) Estimate the mean for this data First note that n = 97 We use the midpoint times the frequency, sum these, then divide by n Mean = [(140)13 + (160)18 + (180)23 + (200)32 + (220)11]/97 = 182.1 3. The following stem-and leaf plot shows scores on a statistics final exam. 2|9 3|3 8 4|3 6 8 8 5|1 1 6 6|1 4 6 7 8 7|7 8 8 9 8|2 3 4 5 6 6 6 9 9 9|0 0 1 3 4 7 10|0 a) Find the median for this data. Explain your answer There are 35 data points, so use the 18th value. Median = 78 b) Compute the 66th percentile 33(.66) = 23.1, round up, use the 24th value, P66 = 86 c) How many students scored above the 80th percentile? Explain your answer 35(.80) = 28, use the 29th value, P80 = 90, students •4. A company in New Jersey is hiring 8 people for a position. Of the applicants, 10 are from New Jersey, 8 are from New York, and 7 are from Pennsylvania. What is the probability that of the 8 people hired: a) Exactly 2 are from New Jersey?  0.21 = 21% b) At least one is from New Jersey? 1 - = 99.99% •5. A and B are events such that P(A) = 0.4 and P(A  B) = 0.6. Find P(B) in each of the following cases. a) A and B are mutually exclusive. Using PIE, P(A  B) = P(A) + P(B)  P(A  B), and here P(A  B) = 0 So 0.6 = 0.4 + P(B), which implies that P(B) = 0.2 b) A and B are independent Now P(A  B) = P(A)P(B) So 0.6 = 0.4 + P(B) – 0.4P(B), which implies that 0.2 = 0.6P(B). Therefore, P(B) = 0.2/0.6 = 0.33 c) P(A|B) = 0.2 0.2 = P(A|B) = , which implies that P(A  B) = .2P(B) So .6 = 0.4 + P(B) – 0.2P(B), of simplifying we get 0.2 = 0.8P(B) Therefore, P(B) = 0.2/0.8 = 0.25 d) P(A  B) = 0.3 0.6 = 0.4 + P(B) – 0.3, so P(B) = 0.5 •6. Seventy percent of clay pots are produced by Machine 1 and 30% by Machine 2. Among all pots produced by Machine 1, 4% are defective, and of those produced by Machine 2, 8% are defective. a) What percentage of the total produced are defective? P(Defective) = (0.70)(0.04) + (.30)(0.08) = 0.052 or 5.2% b) If a pot is found to be defective, what is the probability that it was produced by Machine 2? Explain your answer. P(Machine 2 Defective) = = 0.4615 or 46.2% 7. • It is reported that 80% of workers aged 18 and over drive to work alone in New York. Suppose that Choose 10 workers at random. a) Use a formula to find the probability that exactly 6 of the 10 workers drive to work alone. Find an answer rounded off to 2 decimal places. Confirm your answer with the binomial table. = 0.088 b) Suppose that it is reported that 75% of workers aged 18 and over drive to work. Use a formula to find the probability that more than 2 of the 10 workers chosen drive to work alone. Find an answer rounded off to 2 decimal places. 1 – P(X = 0) – P(X = 1) = P(X = 2) =1- - 1 c) If we repeatedly choose samples of 10, what is the average number of workers that drive to work alone, assuming that that 75% of workers aged 18 and over drive to work ? Round to one decimal place.  = np = 10(.75) = 7.5 •8. Suppose that a website receives a hit every 2 minutes on average. a) Find the probability the website receives 3 hits in 2 minutes. Show all steps and find an answer rounded off to 2 decimal places. X is Poisson with  = 1 P(X= 3) = e-1 = 0.06 b) Find the probability the website receives 8 or 9 hits in 10 minutes. Show all steps and find an answer rounded off to 2 decimal places. First note that  = 5 P(X= 8) + P(X=9) = e-5 + e-5 = 0.065 + 0.036 = 0.10 •8c) Find the probability the website receives more than 4 hits in 6 minutes. You do not to find the final number just indicate how you would get it. First note that  = 3, P(X > 4) = 1 – P(X ≤ 4) =1 – P(X=0) – P(X=1) – P(X=2) – P(X=3) – P(X=4) =1- •9. a) Suppose components A1, A2, B1, B2 and C operate independently in the system shown below. Assume that all the components in the system start operating at the same time and a component does not work again once it fails. The probability of functioning for each component is 0.9. The entire system works if A1 or A2 works, and B1 or B2 , and C works. Find the probability that the entire system works. P(A1  A2)  P(B1  B2)  P(C) = [0.9 + 0.9 – 0.81] [0.9 + 0.9 – 0.81][0.9] = 0.88 Use PIE for P(A1  A2) and P(B1  B2) b) Suppose components A1, A2, B1, B2 and C operate independently in the system shown. Assume that all the components in the system start operating at the same time and a component does not work again once it fails. The probability of functioning for each component is 0.9. The entire system works if A1 or A2 or C, works, and B1 or B2 works. Find the probability that the entire system works. P(A1  A2  C)  P(B1  B2) = = 0.989 10. When a fair die is flipped 72,000 times, find the lower bound of the probability that the number 5 occurs between 11,500 and 12,500 times. 𝜇 = 72,000(1/6) = 12,000 𝜇 = SQRT(72,000(1/6)(5/6) ) = 100 11,500 = 𝜇 − 500 12,500 = 𝜇 + 500 𝜇𝜇 = 500 ⇒ 𝜇 = 500/100 = 5 2 = 24/25 = 0.96 𝜇|− (|𝜇−|−5 𝜇| < 500) = 𝜇(|𝜇− 𝜇| < 5 𝜇) ≥ 1 − 1/5 There is at least a 96% chance that the number of 5s is between 11,500 and 12,500