SCM 202: Module 2

binomial probability distribution examples

- 10 coin flips: heads or tails each flip - random sample of n = 20 items: each item either defective or OK

what is p(x) if x = 2, pi = .5, and n = 3

0.375

if x is a normal random variable with a mean of 50 and a standard deviation of 8, how many standard deviations away from the mean is x = 66?

2

random variable

a variable whose values occur at random, following a probability distribution

failure

event does not occur

determine the mean, variance, and standard deviation of x = the number of defective parts when n = 3,000 and pi = .01

mean = 30 variance = 29.7 standard deviation = 5.45

expected value is...

not what we expect to happen next time, its a long run average

z

number of standard deviations away from the mean

the probability of success, pi, which represents the proportion of items in the population or process that we would like to know is...

often unknown

n and pi are the ______ of the binomial distribution

parameters

we sample in order to estimate...

pi

systematic

randomly select starting point and everyone will have a chance of being picked

trials

repetitions of a random experiment under identical conditions

simple

sample selected so that each number of the population has the same chance of being selected

true or false? a weighted die is thrown. success is defined as getting an even number. the probability of success is .55. the probability of getting two even numbers when throwing the die four times is approximately .3675

true

characteristics of the normal distribution

- bell-shaped, unimodal and symmetrical - mean, median, and mode all lie at the center - parameters are mean and standard deviation - two tails extend indefinitely - x is continuous and ranges from negative to positive infinity - infinite number of normal curves

reasons for sampling (versus taking a census)

- cost savings - time savings - destruction testing - accuracy

MC use the standard normal curve to determine the probability that the random variable z will fall between -0.04 and 0.44. that is, find the following probability: P(-0.04 < z < 0.44): a. .4000 b. .4800 c. .1860 d. .1540

c. .1860

MC find the standard deviation of x for the following probability distribution: x: 20, 30, 60 P(x): .50, .10, .40 a. 4.29 b. 18.4 c. 19 d. 361

c. 19

MC the number of times a machine breaks down during any given month is given by the probability distribution below: breakdowns (x): 0, 1, 2 probability P(x): .4, .4, .2 the expected value of the random variable x is equal to 0.8. this number is the: a. most likely value of x b. median of the distribution c. mean number of breakdowns per month d. median of the distribution AND mean number of breakdowns per month

c. mean number of breakdowns per month

self-selected

call-in, internet, research studies. let people know participants are needed but people make the decision to participate

discrete

consists of whole numbers or values that have distance between them and are countable; does not mean finite - can have an infinite number of possible outcomes (but not within a given range); the probability is the height of the bar when the distribution is in graph form

normal distributions are...

continuous

MC The lifetime of tires is normally distributed with a mean of 50,000 miles and a standard deviation of 3,000 miles. The warranty is for 46,000 miles. What proportion of the tires will fail after the warranty but before 52,000 miles? Indicate the interval below that contains this probability. a) .0000 to .2000 b) .2001 to .4000 c) .4001 to .6000 d) .6001 to .8000 e) .8001 to 1.000

d) .6001 to .8000

MC find the variance of x for the following probability distribution: x: 1, 3, 5 P(x): .4, .2, .4 a. 1.6 b. 4 c. 8/3 d. 3.2

d. 3.2

binomial distributions are...

discrete

convenience

ease of access, often physical access, to a lot of people, e.g. street corner or hub

success

event occurs

continuous

theoretically an infinite number of outcomes within a given range; probabilities are assigned to a range of continuous values rather than to distinct individual values

true or false? if x is a normal random variable with a mean of 10 and a standard deviation of 1/2, then x = 14 is 8 standard deviations away from the mean

true

true or false? non-random samples involve unequal probabilities

true

true or false? the standard deviation of the distribution in the previous problem is 1.37

true

non-random samples (lead to bias)

- self-selected - judgement - convenience

random (equal probabilities) samples

- simple - systematic - stratified

types of samples

- simple - systematic - stratified - self-selected - judgement - convenience

conditions of a binomial distribution

- the random variable of interest is x = number of successes in n trials - the probability of success is the same for every trial 1. denoted by pi 2. probability of failure is denoted by 1 - pi

if x is a normal random variable with a mean of 50 and a standard deviation of 8, how many standard deviations away from the mean is x = 44?

-0.75

to find the probabilities, proportions, or percentages for a normal curve:

1. convert x to z 2. find z in the table 3. look in the body of the table for the appropriate probability

types of probability distributions (theoretical frequency distributions)

1. discrete 2. continuous

a random variable has...

1. numerical values 2. probabilities associated with those values (the probabilities need not be equal)

qualities of a continuous probability distribution

1. probability of any specific value is zero, the total area under the curve is 1 2. probabilities are calculated using probability density function (PDF) 3. the probability associated with a range of values is equal to the area under the curve

requirements for a discrete probability distribution

1. the probability of each event or combination of events must range from 0 to 1 2. the sum of the probabilities of all possible events must equal 1 (these requirements are about the P(x) column; a discrete probability distribution can be a table, formula, or graph that lists all possible values and their associated probabilities)

summary of a binomial probability distribution

1. the two possible outcomes of each trial are success and failure 2. the experiment has n identical trials 3. the n trials are independent of each other 4. the probability of success (pi) and the probability of failure (1 - pi) remain the same from trial to trial 5. the random variable x represents the number of successes

MC samantha calls on 10 houses per day selling girl scout cookies. historically, 3 out of 4 customers buy cookies. assuming the binomial distribution applies here, the variance of this distribution is: a. 1.875 b. 1.37 c. 7.5 d. 2.5

a. 1.875

MC for a random variable x to have a binomial distribution, it is necessary that: a. x represents the number of successes b. the outcome of each trial is a success c. the n trials are statistically dependent on each other d. x represents the number of successes AND the outcome of each trial is a success

a. x represents the number of successes

MC a quiz with 10 questions was given to 10 students. each question was worth one point and the results are summarized below: x = number of correct answers = 10, 9, 8, 7, 6 with frequencies f(x) = 3, 2, 1, 2, 2 respectively. the quiz score mean (expected value) is: a. 8 b. 8.2 c. 9 d. 7.8

b. 8.2

stratified

separate the population into non-overlapping subpopulations or strata and then draw simple random samples from each stratum

find the expected value, variance and standard deviation of the following probability distribution x: 20, 40, 60 p(x): 1/2, 1/4, 1/4

expected value = 35 variance = 275 standard deviation = 16.58

true or false? a success in a binomial distribution always indicates that something good has occurred

false

true or false? in the standard normal z-distribution, the probability between z = -1 and z = +1 is the same as the probability between z = -0.5 and z = +1.5

false

true or false? probabilities associated with random variables must all be equal

false

true or false? the distribution of people's heights is an example of a discrete probability distribution

false

true or false? the standard normal distribution has a mean of 1 and a standard deviation of 0

false

true or false? the sum of the probabilities in a discrete probability distribution could total 1.2

false

judgement

handpick target cities or individuals to sample