Advanced Math Topics Chapters 6 and 7 Review

To find the mean of the probability distribution: μ = Σ x p(x) # of Heads (x)Probability /8 3/8 1/8 1/8 Find the mean of the probability distribution from the last slide. x p(x) 0 3/8 6/8 3/8 μ = 12/8 = 1.5

A bowling ball manufacturer makes bowling balls in 2 pound intervals from 8 to 18 pounds. The probability that a customer will buy a particular weighted ball is shown. Find the mean and standard deviation. x (lbs.)p(x) x p(x)x 2 x 2 p(x) (64)(0.11) = (100)(0.21) = (144)(0.28) = (196)(0.17) = (256)(0.13) = (324)(0.10) = 32.4 σ = √Σx 2 p(x) – μ 2 μ = σ = √8.6 ≈ Standard deviation of a probability distribution: σ 2 = – (12.6) 2 = 8.6

Ninety percent of graduates of Harvard University get a job in the first year after graduation. You meet a group of six young people all who graduated Harvard in the last year. Find the probability that exactly 4 of them have a job..90 x 6 C 4 = x x x x.10x 6 C 4 (.90) 4 (.10) 2 =.0984 =9.84% Binomial Distribution Formula Let p(success) = p and p(failure) = q. The experiment is performed n times, then… p(x successes) = nCxnCx (p) x (q) n-x n! x!(n-x)!

The formula for the mean of a binomial distribution, where n = the number of trials and p = p(success) is… μ = np A retail clothes store generally gets a return of 8% of its merchandise. If the store sells 100,000 items this month, about how many items will be returned? μ = np μ = (100,000)(.08) μ = 8,000 Approximately 8,000 items will be returned.

The formula for the variance and standard deviation of a binomial distribution, where n = the number of trials, p = p(success), and q = p(failure) is… σ 2 = npq A retail clothes store generally gets a return of 8% of its merchandise. Last month it sold 100,000 items. Find the standard deviation. σ =√npq σ =√npq σ =√(100,000)(.08)(.92) σ =√7,360 σ = The standard deviation is

Five cards are randomly drawn from a deck of 52 without replacement. What is the probability of drawing 3 red and 2 black cards? Hypergeometric Probability Function The probability of obtaining x successes when a sample of size n is selected without replacement from N items of which k are labeled success and N – k are labeled failure is… 26 3 ( ) 26 2 ( ) 52 5 ( ) = (2600)(325) 2,598,960 = % = kxkx ( ) N - k n - x ( ) NnNn x = 0, 1, 2, …. n

Chapter 7

2) Find the area between z = and z = 2.02 in the standard normal curve. Area between 0 and z z %.4783 =.9267

3) Find the area between z = 0.87 and z = 1.97 in the standard normal curve. Area between 0 and z z %.3078 =.1678

4) Find the probability of getting a value less than z = 0.42 in the standard normal distribution. Area between 0 and z z %.1628 =.6628

5) Find the probability of getting a value less than z = in the standard normal distribution. Area between 0 and z z %.4370 =.0630

6) Find the closest z-value that corresponds with the 11 th percentile? Area between 0 and z z z = =.39

Steps: 1) Draw a bell curve. 2) Draw your interval. 3) Find the probabilities in the back of the book. 4) Knowing that the table gives the value from the mean to the Z value and using your picture do one of the following: a) Use the probability as your answer b) Add the two probabilities c) Subtract the two probabilities d) Add the probability to e) Subtract the probability from

1)A tire company has a reputation that their tires last an average of 28,000 miles with a standard deviation of 4,000 miles. What percentage of the tires are expected to last more than 35,000 miles? Area between 0 and z Z = x - μ σ = 35,000 – 28,000 4,000 = 1.75 z % of the tires can be expected to last more than 35,000 miles =.0401

10) It is claimed the 45% of all students at Bork College smoke. What is the probability that a survey of 700 randomly selected students at this school will contain at most 300 smokers? Round to the nearest hundredth of a %. Answer: 13.57% Process: This is a binomial distribution approximation problem. Find the mean = 700(0.45) = 315 and the standard deviation = √700(0.45)(0.55) = We are looking for smokers, thus we add 0.5 to the outside of the interval, thus x = Find z = (300.5 – 315)/ = Look this up in the chart, And subtract from =

HW P. 349 #7-15 Skip 9, 10, and 12 P. 395 #1, 7, 11, 18, 19