STAT Exam 2
For any particular interval we state that we are __ confident that the true parameter value lies between the lower and upper bounds of the confidence interval.
(1-a)*100%
the confidence level is denoted by
(1-a)/100% -> most of the time is 95% (a=0.05)
if the sampling distribution of p^ is normal, wee can use the z score formula :
(p^-p)/ square root(p(1-p)/n)
Example: Now suppose the economist didn't have this previous estimate. What size sample should be obtained if he wishes the estimate to be within 0.02 with 90% confidence?
.25(1.645/0.02)^2= 1691.27= 1692
find p(z<=-2.43)
0.0075
IQ scores have a bell shaped distribution with a mean of 100 and a standard deviation of 15. what is the probability that a randomly selected person has an IQ score between 75 and 90
0.2039
if there is no estimate for p^(1-p^) in the sample size equation use
0.25
in 2004 baseball season, inchiro Suzuki of the Seattle mariners set the record for the most hits in a season with a Toal of 262 hits. in the following probability distribution, the random variable X represents the number of hits inchiro obtained in a game. x 0. 1. 2. 3. 4. 5 p(x). 0.17. 0.33. 0.29 0.15 0.04 0.02 what is the probability that in a randomly selected game inchiro got more than 1 hit
0.50
for a normal distribution 95% of the observations fall between __ deviations from the mean
1.96
binomial distributions only have __ possible outcomes
2
the sample proportion will almost certainly fall between
3 standard deviations of the mean
a random variable with a standard normal distribution is typically denoted by
Z
normal probability plot
a graph that plots observed data versus normal scores
think of a population distribution as
a parameter that selects a subject at random from a population
The sampling distribution of 𝑝𝑝̂ is approximately normal for large samples. Thus, in confidence intervals for the population proportion 𝑝𝑝, the standard normal distribution is used to find the appropriate critical values. Critical values from a normal distribution are denoted 𝑧𝑧𝛼𝛼 2⁄ , which indicates they are the z-value with an area of 𝛼𝛼 2⁄ to the right. (a) Illustrate why the critical value for a 95% confidence interval for a population proportion p is 1.96. (b) Find the critical value for a 90% confidence interval for a population proportion p. (c) Find the critical value for a 99% confidence interval for a population proportion p.
a) 1-a=0.95 a=0.5 a/2= 0.025 -> in table is -1.96 and 1.96 and Z.025= 1.96 b) 1-a= 0.90 a=0.10 a/2= 0.05 ->-1.645 but Z.05 is 1.645 bc its to the right c) 1-a=0.99 a=0.01 a/2=0.005 ->2.575
A hot dog manufacturer claims that its new lean beef hot dogs have an average of 3.5g of fat per hot dog. The fat content of these hot dogs is known to be approximately normally distributed with a standard deviation of 0.5g. (a) What is the probability that one of these lean beef hot dogs has more than 4g of fat? (b) What is the probability that a randomly selected package of 8 lean beef hot dogs will have an average fat content greater than 4g?
a) 4-3.5/0.5= 1.00 -> 1-0.8413= 0.1587 b) 4-3.5/(.5/8) = 2.83 -> 1-0.9977 = 0.0023
the mean credit card debit for a US household is $7115 with a standard deviation of $2160. the mean is such a large value because of deeply indebted households. consider a random sample of 50 US households and let x bar represent sample mean and credit card debt a) describe the distribution of x bar b) what is the approximate probability that the mean credit card debt for a sample of 50 households is less than $6500
a) approximately normal with a mean of $7115 and a standard deviation of $305.47 (theta/ square root n = 2160/square root 50) b) x-mu/(theta/square root n)= 6500-7115/305.47 = -2.01 -> 0.0222
Example: According to the American Humane Society, 47% of American households own a dog. Let 𝑝𝑝̂ represent the proportion of households that own a dog in a random sample of 100 American households. (a) Describe the sampling distribution of 𝑝𝑝̂. (b) Draw a sketch of the sampling distribution of 𝑝𝑝̂. Label the horizontal axis with the mean and with the values that are one, two, and three standard deviations from the mean rounded to two decimal places. (c) What is the probability that 35 or fewer households in a random sample of 100 American households own a dog? (d) If you actually obtain a sample in which 35 of 100 randomly selected American households own a dog, would this cause you to suspect that the Humane Society's claim that 47% of American households own a dog is too high? Explain.
a) center= p= 0.47 shape: np(1-p)= 100*.47(1-0.47)= 24.91 >=10 so p is normal spread: square root (p(1-p)/n = .47(1-.47)/100)= 0.04991 b) draw normal distribution of values c) p^=35/100 = .35 z= p^-p/square root(p(1-p)/n) = .35-.47/0.04991 = -2.40 find z score in table-> 0.0082 d) yes because the probability calculated is a rare event, and having an observation out so far int eh tail makes us think its less than 47%
Example: In a random sample of 500 US adults, 255 answered "yes" to the question "Do you approve of the President's job performance?" The margin of error is reported to be 4%. (a) Identify the target parameter in this study. (b) Find the point estimate of the target parameter. (c) Construct the interval estimate for the target parameter. (d) Is this evidence that a majority of Americans approve of the President's performance?
a) p= true proportion of US adults that approve of presidents performance b) p^= x/n= 255/500= 0.51 c) p^ + E = .51+- 0.4 = (.47, .55) d) is p>=.5 -> no, the entire interval would need to be greater than .5 to claim majority
in a poll, 874 of 1016 candidates said they would vote for the president. a)find the point estimate of the proportion of all adults americans that would vote for the president b) find the 99% confidence interval
a) p^ = x/n = 874/1016= 0.86024 b) p^ +- 2.575(standard error) = 0.86024 +- 2.575(.08024*0.139/1016)
Example: Suppose that a particular candidate for public office (in a race with only two candidates) is in fact favored by 48% of the voting population in the district. (a) A polling organization takes a random sample of 625 voters and will use 𝑝𝑝̂, the sample proportion, to estimate p. What is the approximate probability that 𝑝𝑝̂ will be greater than 0.5, causing the polling organization to incorrectly predict the result of the upcoming election? (b) How would this probability change if the polling organization instead took samples of 2,500 voters? (c) Comment on how the sample size affects the likelihood of this polling organization incorrectly predicting the election results.
a) shape: np(1-p)= 156 >=10 center: p=.48 spread: 0.01998 z= .5-.48/0.01988= 1.00 z score in table is 0.8413 1-0.8413= 0.1587 b) theta = square root(0.48*.52/2500)= 0.01 z=0.50-0.48/0.01= 2.00 -> 1-0.9772= 0.0228 c) the distribution gets smaller so they are less likely to be incorrect
if sample data are taken from a populate that is normally distributed, a normal probability plot of the observed values versus the expected scores will be __
approximately linear
a bottling company uses a filling machine to fill plastic bottles with cola. the bottles are supposed to contain 300ml of cola. the contents vary according to normal distribution with mean=302 and standard deviation of 3. let x bar represent the contents of the bottles in a 6 pack. describe the sampling distribution
approximately normal with a mean of 302 and standard deviation of 1.22 -> 3/square root 6 = 1.22
the time required to get an oil change at a particular oil change facility is skewed right. records indicate that the mean time for an oil change at this facility is 11.4 minutes with a standard deviation of 3.2 minutes. let x bar represent the mean time for an oil change at this facility for a random sample of 50 cars. describe the sampling distribution of x bar -> normal or skewed, mean, standard deviation
approximately normal with mu xbar of 11,4 and theta xbar of 0.45 ->theta x bar = theta/square root n = 3.2/square root 50= 0.45 -> normal bc the sample size is greater than 30
point estimate
best guess for target parameter
For a binomial random variable X with n trials and probability p of success in each trial, the sampling distribution of the sample proportion of successes 𝑝𝑝̂ has the following characteristics:
center: the mean of the sampling distribution of 𝑝̂ is the probability of success p spread: theta=square root (p(1-p)/n) (standard error: as the sample size increases the spread decreases) shape: a sample size is large if np(1-p)>=10
every confidence interval has an associated __
confidence level
spread is described by the word
consistent
as sample size increases, the denominator increases so the standard error __
decreases
increasing sample size __ the width of confidence interval
decreases
round __ up even if its a small value
determining sample size for confidence intervals
a or alpha is the __ rate
error
confidence level contains the __ of confidence intervals that will contain __
expected proportion; the value of a target parameter if there is a large number of samples
if the population isn't normal, the sample size must be __ for x bar to be normal
greater than or equal to 30
the standard error of x bar describes
how much the sample mean varies from sample to sample
if a random sample if taken from a large population, the shape shows
if the population is normal, the sampling distribution of the mean will be normal regardless of size
increasing the confidence level __ the width of the confidence interval
increases
interval estimate
interval of numbers where the parameter is believed to fall
an. interval estimate is more useful than point estimate because
more likely to contain true parameter and lets us see the accuracy of the point estimate
population parameter mean symbol
mu
if a random sample if taken from a large population, the center shows
mu x = mu the mean of the sampling distribution is equal to the mean of the population
the margin of error is a __ of the sampling distribution
multiple of standard
The sample size required to obtain a (1 −𝛼)100% confidence interval for a population proportion p with a specified margin of error E is:
n= p^(1-p^)(Za/2 /E)^2
population parameter binomial symbol
p
(1-a)/100% confidence interval for p is:
p+-Za/2 * square root(p(1-p)/n)
sample statistic binomial symbol
p^
interval estimate formula
p^ +- E (E=margin of error)
a 95% confidence interval for p is :
p^ +-1.96*square root(p(1-p)/n) or the standard error formula point estimate +- critical value*standard error
An urban economist wishes to estimate the proportion of Americans who own their homes. A prior study found that 64.8% of Americans own their homes. What size sample should be obtained if he wishes the estimate to be within 0.02 with 90% confidence?
p^= 0.648 E=0.02 Za/2= 1.645 n= .648(0.352)(1.645/0.02)^2= 1543.08 = 1544
Example: A recent Gallup poll found that 493 of 1000 randomly selected U.S. adults believe it is the responsibility of the federal government to make sure all Americans have healthcare coverage. Based on these results, find a 95% confidence interval for the proportion of all U.S adults who believe it is the responsibility of the federal government to make sure all Americans have healthcare coverage.
p^= 493/1000 p= true proportion of all adults Americans that believe the gov is responsible for making sure all Americans have healthcare conditions: stand confidence interval: p^ +- Za/2 *(square root p^(1-p^)/n) = .493 +- 1.96(.493*.507/1000) = (.46, .52) we're 95% confident that the true proportion of adult Americans who believe its the responsibility of the government is between .46 and .52->about m95% of possible samples will result in a confidence interval that contains the true proportion
point estimate formula
p^= x/n= number of successes/sample size
suppose a random sample is taken where an individual either does or does not have a charcterisitic, the sample proportion p^ is:
p^=x/n= number of successes/ sample size
using x bar to estimate mu or using p^ to estimate p is an example of
point estiamtes
confidence interval formula
point estimate +/- margin of error
for quantitative data the target parameter is
population mean (mu)
the sampling distribution of the sample statistic is a
probability distribution for all possible values of the statistic computed from a sample of size
distiribution of the sample proportions __ data
qualitative
sample statistic standard deviation symbol
s
we verify the independence assumption by checking if
sample size n is no more than 5% of the population size N (n<=0.05N)
independence assumption violation
sample values must be independent of one another, so sampling is done without replacement
as the sample size increases what happens to the sampling distribution (shape, center, spread)
shape: normal center: mean stays consistent spread: decreases
we use normal probability plots for __ because __
small data sets; the shape of the histogram doesn't accurately represent that shape of the population
standard error formula
square root(p(1-p)/n)
the standard deviation of the sampling distribution of x bar is called
standard error
in a sample proportion, x is the number of
successes in n trials
for large data sets, a random variable x is approximately normally distributed provided the histogram is __ and __
symmetric and bell shaped
the margin of error measures
the accuracy y of the point estimate in estimating the parameter
the critical value is determined by
the desired confidence interval
normal score
the expected z-score of the data value, assuming that the distribution of the random variable is normal
the means of the samples tend to vary less than
the individual observations
central limit theorem
the larger the sample (30 or above), the the sampling distribution mean will be normal and close to the populate mean
For a (1 −a)100% confidence interval, the critical value corresponds to the number of standard errors within which __ of possible sample statistics are contained
the middle (1 −a)100%
confidence interval is an interval containing
the most plausible values for a parameter
critical value
the number of standard errors we add and subtract from the point estimate to achieve the desired confidence level (1.96 or Za/2)
for qualitative data the target parameter is
the population proportion of successes p
the sample proportion p^ is a statistic that estimates the
the population proportion p
target parameter
the unknown population parameter we want to estimate p
population parameter standard deviation symbol
theta
if you are asked to find the probability of an individual observation, divide __ in a z score formula
theta (x-mu/theta)
if a random sample if taken from a large population, the spread shows the formula
theta x = theta/ square root n or standard deviation of the population/square root sample size of population
if you are asked to find the probability of a sample mean, divide by __ in a z score formula
theta/square root n ((x bar - mu/theta)/square root n)
center is described by the word
unbiased
the larger the sample3r, the lower the __ of the sampling distribution
variability
statistical inferences allow us to make decisions and predictions about populations even if
we have data for few subjects
sample statistic mean symbol
x bar
IQ scores have a bell shaped distribution with a mean of 100 and a standard deviation of 15. what IQ score represents the 90th percentile
x= 100+ 1.28*15 = 119.2
conditions of a confidence interval
• The sample data come from a random sample or a randomized experiment. • The sample size is small relative to the population size (𝑛≤0.05𝑁). • The sample size n is large enough such that 𝑛𝑝̂(1 −𝑝̂) ≥10.
the mean of a probability distribution is denoted by __ and its standard deviation by__
𝜇 ; 𝜎
for a random variable of X, its mean is denoted by __ and its standard deviation by __
𝜇x ; 𝜎x
standard deviation of a discrete random variable formula
𝜎^2 x = sigma (x-𝜇x)^2*p(x) so the deviation is the square root of it
IQ scores have a bell shaped distribution with a mean of 100 and a standard deviation of 15. what proportion of the population has an IQ higher than 140
(z<=2.67) = 0.9962 1-0.9962= 0.0038 -> key word higher, to the right
any observation that lies __ may be considered unusual
+ or - 2 standard deviations away from the mean
find p(z<1.56)
0.9406 (table)
the standard normal distribution has a mean of __ and standard deviation of __
0;1
Find Z0.025
1- 0.025= 0.9750 find this value in the table tan which numbers it corresponds to 1.96
find the z scores that separate the middle 99% of the distribution from the areas in the tails
1-0.99= 0.01 0.01/2= 0.0050 0.0050 is the average of the 2 z scores its in between, so average to the 2 z scores -z.005= -2.575 z.005= 2.575
properties of probability distributions
1. areas under the curve correspond to probabilities 2. the probability for one particular outcome is always 0 (the probability that X=x is 0) 3. the interval containing all possible values has probability equal to 1, so the total area under the curve equals 1
ex: determine whether each is continuous or discrete and give possible values 1. the number of heads observed when flipping 3 coins 2. the height of a randomly selected college student 3. the amount of snowfall in clemson during the month of January 4. the number of football games clemson will win next season
1. discrete (x=0,1,2,3) 2. continuous (x>0) 3. continuous (x>=0) 4. discrete (x=0,1,2...15)
standard deviation of a discrete random variable x measures
the spread of a distribution and describes how far the random variable falls from the mean
formula converting z to x
x = mu + z*theta
find Z.7950
z= -0.82
random variable
a variable that assumes numerical values associated with the outcomes of a probability experiment, where only one numerical value is assigned to each outcome
continuous random variable X can take on
any value within a given interval
find p(IzI>1.67)
area to the left of -1.67 and the right of 1.67 are the same value p(z<-1.67) + p(z>1.67) = 0.0475x2= 0.0950
for a fixed p, as the number of trials n in a binomial experiment increases, the probability distribution of a random variable becomes __
bell shaped
find the z score such that the area to its left is 0.15
between 2 values so pick the one its closer to -1.04
a continuous random variable __ take on every possible value in an interval of numbers
can
a discrete random variable __ take on every possible value in an interval of numbers
cannot
mean and standard deviation are most common to describe __ and __
center and spread
if you measure to get the value of a random variable its __
continuous
expected value of x can be denoted as
e(x)
the mean of a discrete value of x is also called the
expected value of x
a continuous random variable
has an infinite number of possible values that are not countable
a discrete random variable
has either a finite number of values or at least a countable number of values
a continuous random variable is approximately normally distributed if
its relative frequency histogram has the shape of a normal curve
table v in appendix a gives the area to the __ of a specific z score
left
expected values are what we expect in the __ of observations
long run
Example: Find the mean 𝜇𝑋 of the random variable X = number of cars a randomly selected American household owns. X 0 1 2 3 4 5 P(X) 0.09 0.36 0.35 0.13 0.06 0.01
mean = 0(.09)+1(.36)+2(.35)+4(.06)+5(.01)= 1.74
to describe characteristics of a discrete probability distribution, we can use
mean and standard deviation
nCx=
n!/x!(n-x)!
is the expected value the most likely value
no
if ___>=10 then the binomial distribution is bell shaped
np(1-p)
binomial probabilities formula
p(x)=p(X-x)= nCx p^x (1-p)^n-x
summaries of populations are called
parameters
binomial distribution
specific type of discrete probability distribution that can be presented using a formula
a probability distribution can be in the form of
table, graph, or mathematical formula
the notation Za is the z score that
the area under the standard normal curve to the right is a
inclusive means
the first and last values are included
ex: consider the sample of flipping 3 coins S: {HHH,HHT,HTH,THH,THT,TTH,TTT} let x= the number of heads observed when flipping 3 coins. then x is a random variable. what are the possible values of x and what is the probability of each value?
S:{HHH,HHT,HTH,THH,THT,TTH,TTT} X: 3 2 2 1 2 1 1 0 x: 0. 1. 2. 3 p(x):1/8 3/8 3/8 1/8 ->this table represents the probability of the distribution of x
Example: Choose an American household at random and let the random variable X be the number of cars they own. Below is the probability distribution of X in table form. Note that while it is possible for a household to own more than five cars, this probability is so small that it is negligible. X 0 1 2 3 4 5 P(X) 0.09 0.36 0.35 0.13 0.06 0.01 (a) Verify that this is a legitimate discrete probability distribution. (b) What is the probability that a randomly selected American household owns 2 cars? (c) What is the probability that a randomly selected American household owns at least 1 car? (d) Your company builds houses with two-car garages. What percent of households have more cars than the garage can hold?
a) 0<=p(x)<=1 and sigma p(x) =1 b) p(X=2) = .35 c) p(x>=1)= 1-p(x=0)= 1-0.09= .91 d) p(X>2)= p(x>=3)= .13+.06+.01= 0.20
ex: drive through times have mean of 138.5 and a standard deviation of 29. a) what time would you recommend Wendy's advertise the maximum wait time before a free meal is awarded to 1% of patrons? b) the fastest 5% of cars to make it through the drive through will spend what amount of time waiting?
a) 1- 0.0100= 0.9900 -> z score is 2.33 x = 138.5 + (2.33)(29) = 206.07 b) x = 138.5 + (-1.645)(29) = 90.8
steps for finding a probability corresponding to a normal random variable
1) draw a normal curve and shade the desired area 2) convert the values of x to a z score using the z score formula 3)use the standard normal table to find the area to the left of the z score
ex: mean driver through time is 138.5 and the standard deviation is 29 seconds a) what is the probability that a randomly selected customer will get through the drive through in less than 100 seconds b) what is the proportion of cars that spend between 120 seconds and 180 seconds in the drive through
a) let x = the amount of time spent in the drive through z= (x-mu)/theta = 100-138.5/29= -1.33 -> table probability of -1.33 is 0.0918 b) find the z score for both values z= 180-138.5/29= 1.43 -> 0.2611 z= 120-138.5/29 = -0.64 ->0.9236 0.9236-0.2611= 0.6625
Example: You operate a restaurant. You read that a sample survey by the National Restaurant Association shows that 40% of adults are committed to eating healthy food when dining out. To help plan your menu, you decide to conduct a sample survey in your own area. You will use random digit dialing to contact a random sample of 20 households by telephone. If the national result holds in your area... (a) What is the expected number of respondents who seek healthy food when dining out? (b) What is the probability at most 2 respondents seek healthy food when dining out? (c) What is the probability at least 3 respondents seek healthy food when dining out?
a) let x= #wanting healthy food when dining out then x is binomial with n=20 p=0.4 b) p(x<=2)= p(x=0)+p(x=1)+p(x=2)= 20C0 (.4)^0 (0.6)^20+ 20C1(.4)^1(0.6)^19+20C2(0.4)^2(0.6)^18 = 0.0036 c) p(x>=3)= 1-p(x<=2)= 1-0.0036= 0.9964
__ can be negative but __ cannot be negative
values of x; probabilities
steps for finding the value of a normal random variable corresponding to a give probability
1) draw the normal curve and shade the given area 2)use the standard normal table to find the z score that corresponds to the shaded area 3)substitute the values of z, mu, and theta into the z score formula solved
find p(z>0.78)
1-p(z<0.78)= 1-0.7823= 0.2177
let p(x) denote the probability that a discrete random variable X equals x, or p(X=x). then:
1. 0<=p(x)<=1 2. sigma p(x)=1
characteristics of a binomial distribution
1. there are only 2 possible outcomes in each trial, we denote one by S(success) and the other by F(failure) 2. the experiment consists of a fixed number (n) of identical trials 3. the n trials are independent 4. the probability of S remains the same from trial to trial. this is denoted by p, and the probability of F is 1-p
IQ scores have a bell shaped distribution with a mean of 100 and a standard deviation of 15. consider a person with an IQ of 140, find the z score and interpret the value
2.67
how to do distributions on calculator
2nd, vars, binomial pdf or cdf
the area under the normal curve for any interval of values of the random variable represents __ or __
the proportion of the population with the characteristic; the probability that a randomly selected individual from the population will have the characteristic
normal tables give probabilities for
the standard normal distribution
in the graph of a discrete probability distribution, the horizontal axis corresponds to __ and the vertical axis represents __
the values of the random variable; the probability of each value
binomial standard deviaton formula
theta= square root of np(1-p)
in the graph of a discrete probability distribution, vertical lines are drawn to
to each values height that is the probability of that value (you can describe the shape of it)
Example: Determine which of the following probability experiments qualify as a binomial experiment. For those that are binomial experiments, define the binomial random variable X, identify the number of trials (n), identify the probability of success (p), and list the possible values of X. (a) You roll a 6-sided die 50 times. The number of ones observed is recorded. (b) You continue rolling a 6-sided die until you get a one. The total number of rolls is recorded. (c) You want to know what percent of married people believe that mothers of young children should not be employed outside the home. You plan to interview 50 people, and for the sake of convenience you decide to interview both the husband and the wife in 25 married couples. You record the number among the 50 persons interviewed that think mothers should not be employed. (d) You buy a ticket in your state's "Pick 3" lottery game every week and record the number of times in a year that you won a prize.
a) n=50 p=1/6 b) not binomial, the number of trials isn't fixed c) not binomial, the trials aren't independent bc the opinions of the husbands and wives are likely related d) n=52 p=0.01
consider a standardized test where the multiple choice questions have 5 answer choices, and 1 point is awarded for a correct answer but 1/4 of a point is deducted for an incorrect answer. a) give the probability distribution. b) what is the expected number of points when a student guesses the answer to a multiple choice question on a test
a) x: 1. -1/4 p(x): 1/5. 4/5 b) e(x)= 1(1/5)-1/4(4/5)=0
__ are terms that describe continuous variables
amount, much, and less
for greater than or equal to binomial outcomes, use __ on calculator
binomial cdf (at least __)
for individual binonmial outcomes, use __ on calculator
binomial pdf
is the distance a baseball travels in the air after being hit a discrete or a continuous random variable
continuous
to apply the standard normal table to a random variable x, we must first
convert the value of x to a z score
probability distribution of a discrete random variable x lists
the possible values of the radon variable and their corresponding probabilities
IQ scores have a bell shaped distribution with a mean of 100 and a standard deviation of 15. let x=IQ score. which of the following expressions is equivalent to the probability that a randomly selected person has an IQ score between 75 and 90 a) p(75<x<90) b) p(-1.67<x<-0.67) c) p(-1.67<=x<=-0.67) d) all of the above
d) all of the above the equal signs don't change anything
if you count to get the value of a random variable its __
discrete
is the number of runs scored in a baseball game discrete or continuous
discrete
to find p(a<=x<=b) ___
find the area under the curve between a and b
Example: According to a study done by Nick Wilson of Otago University Wellington, the probability a randomly selected individual will cover his or her mouth when sneezing is 0.73. Suppose you sit on a bench in a mall and observe people's habits as they sneeze. What is the probability that among 5 randomly observed individuals exactly 3 of them cover their mouth when sneezing?
let x= # of people covering their mouth then x is binomial with n=5 and p=0.73 p(x=3)= 5C3(.73)^3(.27)^2 = .28
the mean of the discrete random variable can be thought as the
mean outcome of the probability experiment if we repeated it multiple times
Example: Find the standard deviation of X = ideal number of children. X 0 1 2 3 4 P(X) 0.05 0.13 0.60 0.15 0.07
mean= 2.06 =(0-2.06)^2(.05)+(1-2.06)^2(.13)+(2-2.06)^2(.6)+(3-2.06)^2(.15)+(4-2.06)^2(0.7)= 0.75460 square root of 0.75460= 0.87
binomial mean formula
mu = np
a __ is symmetric, bell shaped, and characterized by mean and standard deviation
normal curve
__ are terms that describe discrete variables
number, fewer, and many
the cumulative probabilities on jmp shows
p(x<=_)
the p(x) column on jmp shows
p(x=_)
ex: when using jmp to dine the probability of 10 or more the formula is
p(x>=10)= 1-p(x<=9)
the formula for probability distribution curve is called a
probability density function
histogram curves portray __ of continuous random variables
probability distributions
the normal distribution is an important type of continuous distribution bc it provides a
reasonable approximation to the distribution of many variables
as the width of the intervals decreases, the histogram gradually approaches a
smooth curve
__ if you need the area to the right
subtract from 1
find p(-1.20<=z<=0.58)
subtract the area to the left of the upper value- the area to the left of lower value 0.7190-0.1151= 0.6039
__ if you need the area between two x values
subtract the smaller area from the larger area
binomial random variable is the number of
successes in n trials, denoted by X
Example: In the "Pick 3" lottery, a player chooses one of the 1000 three-digit numbers between 000 and 999. The lottery chooses a three-digit winning number at random. With a single bet of $1, the player wins $500 if their number is selected. Otherwise, the lottery takes the player's $1. (a) Construct the probability distribution for X = profit from one ticket. (b) What is the expected profit for a single ticket? (c) If you played the game 10 times, how much would you expect to lose?
win. lose a) x $499 -1 p(x) 1/1000. 999/10000 B) e(x)= 499*(1/1000) - 1(999/1000)= -$0.50 c) -$0.50(10) = -$5
According to a study done by Nick Wilson of Otago University Wellington, the probability a randomly selected individual will cover his or her mouth when sneezing is 0.73. Suppose you sit on a bench in a mall and observe people's habits as they sneeze. What is the mean and standard deviation that among 5 randomly observed individuals exactly 3 of them cover their mouth when sneezing?
x is binomial with n=5 and p=0.73 mu=np= 5(.73)= 3.65 theta= square root 5(.73)(.27)= .9927
Example: The possible values of a random variable X are 0, 1, 2, or 3. The probability distribution of X is defined by P(x) = 0.1x + 0.1 for x = 0, 1, 2, 3. Find the probability of each possible value of X, and verify that it is a legitimate discrete probability distribution.
x. 0. 1. 2. 3 p(x)= .1x + 1. .1 .2. .3. .4 ( .1(0) + .1 =.1) 0<=p(x)<=1 sigma p(x) = .1+.2+.3+.4 = 1
a normal random variable can be converted into a standard normal random variable by taking its
z score
formula for converting x to z
z= (x-mu)/theta
mean of a discrete random value formula
𝜇x= sigma (x * p(x)) ->x is the random variable and p(x) is the probability of observing the value x
