MATH exam 2

calculating the SD of a discrete random variable

find the variance first

how to tell if two events are independent

multiply the probabilities of A and B and then calculate the probability of A and B. if the two answers are the same, then the events are independent

a student takes a multiple choice test that has 8 questions. each question has five choices. the student guesses randomly at each answer. a. find p(3)

n=8 p=0.2 because there is one correct answer out of 5 choices so you divide 1 by 5 solving for a 1. use the binomial prob table

continuous random variable

a random variable that can take up any value in an interval - height - amount of light

how to tell if two events are mutually exclusive

calculate the probability of A and B and see if it equals 0. If it does, they are mutually exclusive

finding the areas under the normal curve

consult the cumulative distribution table 1. if the shaded area is to the right of the mean, the answer is positive and if the shaded area is to the left, then it is negative

An insurance company sells a 1-year term life insurance policy to an 83-year-old man. The man pays a premium of $1200. If he dies within 1 year, the company will pay $32,000 to his beneficiary. According to the U.S. Centers for Disease Control and Prevention, the probability that an 80-year-old man will be alive 1 year later is 0.9645. Let X be the profit made by the insurance company.

*calculating the expected value* 1. you're building four values in two different categories: two dollar values (success and failure) and two probability values (success and failure) 2. subtract the premium from the payout (32,000-1,200 = 30,800) 3. the "success" (that the event of death occurs) dollar value is -30,800 because the company is paying out that amount and the failure dollar value is 1200 because the company is keeping his premium 4. the success probability is 0.0355 because it's the complement of the failure probability of 0.9645 4. multiply the success values together and the loss values together 5. add the products of the previous step

discrete random variable

random variables that can be listed - the number that comes up on a die - the number of siblings a person has

According to a recent report, 66% of Internet searches in a particular month used the Google search engine. Assume that a sample of 22 searches is studied. Round the answers to four decimal places.

solve like a binomial

Big chickens: A report from a poultry industry news website stated that the weights of broilers (commercially raised chickens) are approximately normally distributed with mean 1340 grams and standard deviation 200 grams. Use the Cumulative Normal Distribution Table to answer the following. (a) Find the 20th percentile of the weights. (b) Find the 92nd percentile of the weights. (c) Find the first quartile of the weights. (d) A chicken farmer wants to provide a money-back guarantee that his broilers will weigh at least a certain amount. What weight should he guarantee so that he will have to give his customers' money back only 3% of the time?

solving for a a. the percentile given is an area to the left of a yet to be found z score b. use the zscore table to find the score with the given area to the left (in this case it's the 20th percentile so 0.2) c. calculate the percentile using the raw score equation solving for b. 1. solve the same as a solving for c. 1. remember quartiles (1=.25 2=.5 3=.75) 2. find the zscore corresponding the area of the quartile 3. use the raw score equation solving for d. 1. turn the percentage into a decimal 2. find the zscore with the area closest to the decimal 3. calculate the raw score

A sample of 23 notebook computers was selected. Find the complements of the following events. a. Exactly 8 of the notebook computers weigh less than 5 pounds. b. at least 8 of the notebook computers weigh less than 5 pounds c. more than 8 of the notebook computers weigh less than 5 pounds d. fewer than 8 of the notebook computers weigh less than 5 pounds

solving for a. - the compliment is: the number of computers weighing less than 5 pounds is not equal to 8 solving for b. - the compliment is: at most 7 computers weigh less than 5 pounds solving for c. - the compliment is: fewer than 9 weigh less than 5 pounds solving for d. - the compliment is: more than 7 weigh less than 5 pounds

A company audit showed that of 853 bills that were sent out, 472 were paid on time, 133 were paid up to 30 days late, 79 were paid between 31 and 90 days late, and 169 remained unpaid after 90 days. One bill is selected at random. part a. What is the probability that the bill was paid on time? part b. What is the probability that the bill was not paid on time?

solving for a. 1. divide 472 by 853 solving for b. 2. use the rule of compliments and subtract 472 from 853 to get 381 2. divide 381 by 853

Tree heights: Cherry trees in a certain orchard have heights that are normally distributed with mean=μ118 inches and standard deviation=σ12 (a) What proportion of trees are more than 122 inches tall? (b) What proportion of trees are less than 100 inches tall? (c) What is the probability that a randomly chosen tree is between 84 and 93 inches tall? Round the answers to four decimal places.

solving for a. 1. find the z score from the raw score 2. match the zscore to the chart and find the area to the left 3. because the question is asking for what proportion is greater than, take the compliment for the answer

How are your grades? In a recent semester at a local university, 600 students enrolled in both Statistics I and Psychology I. Of these students, 83 got an A in statistics, 75 got an A in psychology, and 39 got an A in both statistics and psychology. Round the answers to four decimal places, as needed. a. Find the probability that a randomly chosen student got an A in statistics or psychology or both. The probability that a randomly chosen student got an A in statistics or psychology or both is .b. The probability that a randomly chosen student did not get an A in statistics is

solving for a. 1. use the general addition rule 2. calculate the probability of event A (divide 83 by 600) 3. calculate the probability of event B (divide 75 by 600) 4. calculate the probability of event A and B (divide 39 by 600) 5. add the result of step 2 with the result of step three 6. subtract the result of step 4 from the sum of step 5 solving for b. 1. use the rule of compliments 2. subtract 83 from 600 3. divide the result of step 2 by 600

You are trying to answer a multiple choice question on a standardized test. There are four choices. If you get the question right, you gain one point, and if you get it wrong, you lose 1/3 point. Assume you have no idea what the right answer is, so you pick one of the choices at random. What is the expected value of the number of points you get?

*calculating the expected value* 1. you're building four values in two different categories: two point values (success and loss) and two probability values (success and loss) 2. the success point value is 1 because you gain one point if you get it right and the loss point value is -1/3 3. the success probability value is 1/5 because one out of five questions is correct and the loss probability is 4/5 4. multiply the success values together and the loss values together 5. add the products of the previous step

how to tell if it is a binomial distribution

1. a fixed number of trials are conducted 2. there are two possible outcomes for each trial (success and failure) 3. the trials are independent 4. the random variable x represents the number of successes 5. success and failure are equal in every trial 6. n means the number of trials and p means the probability of a success

how to tell if something is a probability distribution

1. all the probabilities must be between 0 and 1 2. all the probabilities must add up to 1

calculating the variance of a probability distribution

1. calculate the mean 2. subtract the mean from each number x 3. raise each of the previous to the second 4. multiply each of the previous by its corresponding probability 5. add the previous products

finding the shaded areas underneath a normal curve

1. consult the cumulative normal distribution table and find the row with the numbers in the 1s and 10ths places and then the column with the digit in the 100ths place 2. if the shaded area is to the right of the mean, then take the complement of the answer on the table

how to make a probability distribution

1. find all the possible values for x - ex. if you're doing a dice problem, there are going to be 6 possible values for x 2. find the probability of each value of x

a mineral economist estimated that a particular venture had a probability of 0.4 of a 30 million dollar loss, probability 0.5 of a 20 million dollar profit, and probability 0.1 of a 40 million dollar profit. let x represent the profit. find the probability distribution of the profit and the expected value of the profit. does this venture represent an expected gain or an expected loss?

1. make sure that the profits are positive numbers and the losses are negatives 2. make the probability distribution 3. to find the expected value, multiply all of the values by their probabilities and sum the products

finding the mean of a discrete random variable

1. multiply each possible value by its probability 2. add all of the previous probabilities

calculating the mean of a probability distribution

1. multiply each x by its probability 2. add them all together

raw score equation

1. multiply the zscore by the SD 2. add mew

z score from the raw score

1. subtract mew from the raw score 2. divide by the SD

finding the z scores on either side of a known area

1. take the compliment of percentage given 2. divide the compliment by two 3. find the zscore with the area closest to the half of the compliment 3. plug in that zscore and it's positive or negative equal

how to solve a GIVEN probability ex. what is the probability that gene 2 is recessive given that gene 1 is recessive

1. use the general method for computing probabilities 2. determine the p of a and b (genes 1 and 2 being recessive) 3. determine p of a (gene 1 is recessive) 4. divide step 2 by step 3 *****remember the GIVEN event is on the far or right side of the line

According to a survey by Nickelodeon TV, 89% of children under 13 in Germany recognized a picture of the cartoon character SpongeBob SquarePants. Assume that among those children, 67% also recognized SpongeBob's cranky neighbor Squidward Tentacles. What is the probability that a German child recognized both SpongeBob and Squidward? Write your answer as a fraction or a decimal, rounded to four decimal places.

1. use the general multiplication rule 2. multiply .67 by .89

how to compute AT LEAST probabilities ex. The probability that a certain make of car will need repairs in the first four months is 0.1. A dealer sells three such cars. What is the probability that at least one of them will require repairs in the first four months? Round your final answer to four decimal places.

1. use the rule of compliments (find the probability that none of the cars will need repaired and then subtract from 1) 2. use the multiplication rule for independent events 3. multiply the compliments of each event together (1-0.1*1-0.1.... up until all three events OR raise 1-0.1 to the third power) 4. subtract the results of step 3 from 1 4.

independent events

The outcome of one event does not affect the outcome of the second event

Big chickens: According to a poultry industry news website, the weights of broilers (commercially raised chickens) are approximately normally distributed with mean 1423 grams and standard deviation 169 grams. Use the Cumulative Normal Distribution Table to answer the following: (a) What proportion of broilers weigh between 1180 and 1280 grams? (b) What is the probability that a randomly selected broiler weighs more than 1500 grams? (c) Is it unusual for a broiler to weigh more than 1750 grams?

answering for a. 1. use the zscore finding equation to find the individual zscores for the measurements given (z = x - mew / SD) 2. find each of the zscores on the table *be sure to round up 3. subtract the lesser area from the greater area (the one further to the right)

four patients have made an appointment to have their blood pressure checked at a clinic. let x be the number of them that have high blood pressure. the probability distribution of x is a. find the probability of two or three of the patients having high blood pressure b. find the probability that more than one of the patients have high blood pressure c. find the probability that at least 1 patient has high blood pressure

solving for a. - the events can not happen at the same time - the probability of 2 or 3 is equal to the probability of 2 added to the probability of 3 solving for b. - more than 1 means the probability of 2 or 3 or 4 - it's an OR problem so add all of those probabilities solving for c. - at least one is the same as 1 - the probability that none of them have high blood pressure - so subtract the probability of 0 from 1

a pew research center reported that about 30% of americans own a tablet. suppose a simple random sample of 15 people is taken. use the binomial probability distribution to find the following probabilities a. find the probability that exactly 4 of the sampled people own a tablet b. find the probability that more than one person owns a tablet

solving for a. 1. n = 15 (the number of trials) 2. p = 0.3 (the probability of success) 3. plug in the numbers and solve with the binomial equation solving for b 1. use the rule of compliments you can also use the binomial probabilities table

In horse racing, one can make a trifecta bet by specifying which horse will come in first, which will come in second, and which will come in third, in the correct order. One can make a box trifecta bet by specifying which three horses will come in first, second, and third, without specifying the order. Part 1 of 2. Your Answer is correct a. In an eight-horse field, how many different ways can one make a trifecta bet? b. In an eight-horse field, how many different ways can one make a box trifecta bet?

solving for a. permutation solving for b. combination

probability distribution for a discrete random variable

specifies the probability for each possible value of the random variable - the probability of x is always between 1 and 0 - the sum of all of the probabilities must equal 1

the expected value

the mean of a random variable

random variable

the outcome of a random process that has a numerical value - ex. rolling a die

binomial distribution

the probability of a random variable that represents the number of successes in a series of trials

binomials

use binomial prob calculator or the binomial prob table *for binomials that say "or fewer" or "or more" just add all the probabilities that fit, the mean, SD, and the variance don't change; if worse comes to worse, do it the hard way and use the formulas *for binomials that don't give you a success number, just plug in the sample again

Tree heights: Cherry trees in a certain orchard have heights that are normally distributed with mean μ=110 inches and standard deviation σ=12 inches. Use the Cumulative Normal Distribution Tableto answer the following. (a) Find the 24th percentile of the tree heights. (b) Find the 82nd percentile of the tree heights. (c) Find the third quartile of the tree heights. (d) An agricultural scientist wants to study the tallest 1% of the trees to determine whether they have a certain gene that allows them to grow taller. To do this, she needs to study all the trees above a certain height. What height is this? Round the answers to at least two decimal places.

use https://mathcracker.com/standard-deviation-percentile-calculator#results for one 1% look for the 99th percentile