MAT 120 Unit 2 Review
Probability rules: 1) each event P(E) must have a probability that is some value between blank and blank -This implies that probabilities cannot ever be blank or (greater/lesser) than 1. 2) the sun or probabilities for al possible outcomes in the sample space must sum to be blank. 3) if an event is impossible it's probability is blank. P(E)= blank 4) if an event is certain to occur P(E)= blank 5) in general any event with a probability of less than 0.05 would be deemed blank
1) 0 and 1 -negative or greater than 2) 1 3)0,0 4) 1 5)unusual
Example: find an area of z score (TO THE LEFT) 1) P(Z< -0.42) 2) P(Z<1.18) SC
1) 0.3372 2)0.8810 stat/calc/normal/ select standard or between/flip arrow for sign needed/change mean sd and data value/compute
Example: find an between two z scores (TO LEFT) (large number minus small number) 1) P(-0.86 < Z < 0.53) 2) P(-1.19< Z < 2.05)
1) 0.7019-0.1949= 0.5070 2)0.9798- 0.1170= 0.8628 IN SC: stat/calc/normal/ select standard or between/flip arrow for sign needed/change mean sd and data value/compute
Example: find area of z score (TO THE RIGHT) 1) P(Z> -0.13) 2) P(Z> 1.77) SC
1) 1-0.4483= 0.5517 2) 1-0.9616= 0.0384 stat/calc/normal/ select standard or between/flip arrow for sign needed/change mean sd and data value/compute
Example: if a worker were selected at random what is the probability they did not drive alone to work? Method. Frequency. Relative frequency Drive alone. 153. .765 Carpool. 22. .110 Public transport. 10. .05 Walk. 5. .025 Other. 3. .015 Work at home. 7. .035
1) add everything but drive alone(rel frequency)
Example: a sample of homes in briar acres revealed the following information about household pets Type. # of pet type Cat. 87 Dog. 55 Guinea pig 8 Goldfish. 12 None. 37 If a home were selected at random what is the probability the home had a cat or dog as a pet?
1) add number of pet type= 199 2) divide 87/199 & 55/199 3) add together= 0.7135 or 71.4%
Find the z scores that separate the middle 13% distribution from the area in the tails of the stand normal distribution. SC
1) divide 13% into two halves 6.5 or .065 and .065 2).5-.065= .435 3) use table to find area closest to .435 to the left = -.16 and .16
Example: a survey found that women spend on average $146.21 on beauty products during the summer months. The standard deviation is $29.44. Find the percentage of women who spend less than $160. Assume the variable is normally distributed. SC
1) draw picture and shade area to find of u= 146.21 o= 29.44 x= 160 2) u= 0 o= 1 Z= x-u/o 160-146.21/29.44 = 0.47 3) P(Z < 0.47) (left because asking for less than) Use table =0.6808 4) 69.08% of women spend less than $160 on cosmetics during the summer stat/calc/normal/ select standard or between/flip arrow for sign needed/change mean sd and data value/compute
An experiment is considered a binomial experiment if: 1) the experiment is performed a blank number of times (trials) 2) the trials are blank. This means the occurrence of one trial does not affect the occurrence of other trials. 3) for each trial there are two mutually exclusive outcomes. What are these? 4) the probability of blank is the same from trial to trial
1) fixed 2) independent 3) success or failure 4) success
Find the value of Za a= 0.27
1) in SC: area to the right 2) stat calc normal standard arrow to right/ p(x > blank)= .27 3)0.61
Example: to help students improve their math skills a new math program is instituted by a local school district and aimed at students in the lowest 5% based on an achievement test. If the average score for students on the test is 122.6 find the cutoff score for admittance into the program: the standard deviation is 18 and scores are normally distributed
1) u= 122.6 o= 18 2) Z= x-u/o concert to x=zo+u 3)u=0 o=1 4) use table to find area closest to .05(because lowest 5%)(use negative side because lowest/left) =-1.64(z-score) 5)x=zo+u x= -1.64(18)+122.6 =93.08 round to nearest whole number =93% or lower is what's needed to make acceptance for program IN SC: stat/calc/normal/standard/arrow towards left </ u=0 o=1/ P(x< blank)= .05/ compute and gives you -1.64 as z score/ do math from there
Example: to qualify for a police academy applicants must score in the top 10% of a general abilities test. The test has a mean of 200 and standard deviation of 20. The scores are normally distributed. Find the lowest possible score to assure admittance you the academy round to the nearest whole number.
1) u= 200 o= 20 be in top 10% or .10 2) Z= x-u/o flip this to x=z•o + u 3) u= 0 o=1 we know the area what's the z score? Want the area to right. The Area to the left is .90 because .10 is somewhere to the right 4) use table find area closest to .90 =.8997 Which is 1.28 Z scores= 1.28 5) x= z•o+u x= 1.28(20)+200 = 25.6+200 =225.6 round to nearest whole number = 226 score needs to be made in order to qualify on exam IN SC: Stat/calc/normal/standard/u=0 sd=1/ p(x > blank)= 0.10/compute you get 1.28/do math from there
Example: an automobile dealer find that the average price of a previously owned vehicle is $8256. He decides to sell cars aimed at the middle 60% of the market in terms of price. Find the minimum and maximum prices of the cars the dealer will sell. Assume the standard deviation is $1150 and that the variable is normally distributed.
1) u= 8256 o= 1150 middle 60% 2) u= 0 o=1 Two z scores 60%= .30 and .30 in the middle Area to left= .20 Area to right= .20 3) use table to find area closest to right of .20 and left of .80(.30+ .30 + .20) 4) to the left(negative)= 0.2005 Z score= -0.84 To the right(positive)= 0.7995 Z score= 0.84 5) x1= z1•o + u = -0.84(1150)+ 8256= $7290 X2= z2•o+u = 0.84(1150)+ 8256= $9222 6) the dealer will sell his cars for a minimum or $7290 to $9222 IN SC: Stat/calc/normal/u=0 o=1/ p(x< blank)= .20/ compute/same just change sign= .20/compute/ do math from there(steps 5&6)
Producer her for solving problems involving applications of the normal distribution 1) identify the blank blank (from population or sample) and blank 2) find the blank for the data value listed in the problem 3) use the standard normal table( or technology) To find blank under curve either to left or right or those z scores. Area will represent proportion and therefore probability being calculated
1)Data value, mean, standard deviation 2) z score 3) area
Example:describe the sampling distribution of p hat. Assume the size of the population is 30,000 n= 200 p= 0.6
1-p= 0.4 p=0.6 N=30000 n=200 -Is 200< 5% of 30000? (Yes) -np(1-p) > 10 200(.6)(.4) > 10? =48 (Yes) Up hat= P= 0.6 Op hat= square root p(1-p)/n = square root .6(.4)/200 =0.035
Example: What is the probability of getting an even number when you throw the die one time? What happens if the number of die increases to two? Does this still qualify as a classical probability experiment?
1/2 or 0.50 No. Rolling two dice instead of just one results in a sample space where the outcomes are not all equally likely
In the following the random variable x represents the number of activities a parent of a 6th to 8th grade student is involved in. X. P(x) 0. 0.288 1. 0.343 2. 0.243 3. 0.061 4. 0.065 This is a discrete probability distribution because the sum of the probabilities is blank and each probability is blank Compute the mean and standard deviation and interpret What is the probability that a randomly selected student has a parent involved in three activities? What is the loan nil ty a student has parent involved in 3 or 4 activities?
1; between 0 and 1 Mean= 1.272; as the number of experiments increases the mean of the observations will approach the mean of the random variable Standard deviation= 1.1 Three activities= .061 Three or four activities= 0.061+ 0.065 = .126
example: the number of chocolate chips in a big of cookies is approximately normally distributed with a mean of 1261 chips and standard deviation of 118 chips. A) determine the 30th percentile for number of chocolate chips in bag B) determine the number of chocolate chips in a bag that make up the middle 96% of bags C) what is the interquartile range of the number of chocolate chips in a bag of cookies?
A) 1) u=1261 o=-118 30th percentile= 30% at or below (to the left) of .30 2) x= zo+u Use table to find closest area to .30 = 0.3015 Z score= -0.52 In SC: stat calc normal p(x < blank)=.30 = -.52 3) x=zo+u = -0.52(118)+ 1261 =1199.64 or 1200 B) 1) 96/2 = .48 & .48 so we need .02 on each side In SC: stat calc normal p(< blank)= .02 and > = -2.05 and 2.05 2)x1=zo+u =(-2.05)(118)+1261 =1019.1 is 1019 X2= zo+u =(2.05)(118)+1261 =1502.9 is 1503 Round to nearest whole number 3) so of all chocolate chips in bag 96% will have between 1019 and 1503 chips C) 1) Q3-Q1= (75th percentile and 25th percentile) 2) in sc: stat calc normal p< blank= .75 = 0.67 P> blank= .25 =-0.67 3) -0.67(118)+1261= 1181.94 is 1182 (Q1) .67(118)+ 1261= 1340 (Q3) IQR=1340- 1182= 158
Example: each month an American household generates on average 28 pounds of newspaper or garbage for recycling. Assume the standard deviation is 2 pounds and that the variable is normally distributed. If a household were selected at random... A) P( the household generates more than 30.2 pounds of newspaper or garbage each month) B) P(the household generates between 27 and 31 pounds of newspaper or garbage each month)
A) 1: u= 28 o=2 x= 30.2 P(x> 30.2) 2) u= 0 o= 1 P(x>30.2) X-U/o 30.2-28/2 =1.1 3) P(Z > 1.10) = 1-0.8643 =0.1357 4) 13.6% a household generates more than 30.2 pounds of garbage each month B) 1: x1= 27 x2= 31 u= 28 o= 2 P(27<x<31) 2) x=0 o=1 X-u/o (two times) Z1= 27-28/2 = -.50 Z2= 31-28/2 = 1.50 P(-0.50<x<1.50) 3)subtract(large minus small) 0.9332-0.3085= 0.6247 4) 62.5% a household produces between 27 & 31 each month of garbage IN STATCRUNCH!: stat/calc/normal/ select standard or between/flip arrow for sign needed/change mean sd and data value/compute
Example: Identify these pairs of events as disjoint or not A) a student is a male and female B) a student is a female and a sophomore C) a pet is a cat and a guinea pig
A) Disjoint B)not disjoint C) disjoint
Example: A) suppose that the mean time for an oil change at a ten minute shop is 11.4 minutes with a standard deviation of 3.2 minutes if a random sample of n=35 oil changes is selected describe the sampling distribution of the sample mean B) if a random sample of n=35 oil changes is selected what is the probability the mean oil change time is less than 11 minutes?
A) X barr is approximately normal So normal= same mean: With a mean of 11.4 And stand deviation: o/(square root)n: 3.2/ (square root) 35= 0.5409 B) Z=11-11.4/0.5409 = -0.74 P(Z < -0.74) =0.23 About 23% chance less than 11 minutes
According to the National Vehicle Association of America, the average age of cars in the USA was 96 months with a standard deviation of 15 months. Suppose a random sample of 35 vehicles in the parking garage on church street in Greenville was taken. A) describe the sampling distribution B) if an individual vehicle was chosen at random what is the probability its age was more than 98 months. Round to four decimals C) suppose a sample of 35 cars was taken. What is the probability that the sample mean was more than 98 months? Round to four decimals.
A) bell shaped and symmetric ux barr= 96 Ox barr= 0.2148 B) 0.4483 C) 0.2148
According to the FAA 87% of domestic us flights depart airports on time(leaving mo more than 15 'minutes beyond time) suppose a random sample of 9 US flights is taken and the number of flights that departed on counted were counted A) Explain why this is a binomial experiment B) why is the probability that exactly 7 flights departed on time? C) what is the probability that at least 7 flights departed on time? D) what is the probability that between 6 and 8 flights departed on time? E) calculate the mean and standard deviation
A) independence, 87%, fixed number of trials, success or fails of flights B) n=9, c=7, p=.87, 1-p=.13/ 9C7 • .87^7 • .13^2 =36(.3773)(.0169) = .2295 C 0.8991 (use sc) D)0.6936 (use sc) E) u= np= 9(.87)= Mean: 7.83; if repeated samples of size 9 were taken a great many times we would expect on average almost 8 flights would leave on time Standard deviation: O=square root(np(1-p) = square root(7.83(.13) = 1.0
Example: a researcher studying public opinion obtains a simple random sample of 25 adult Americans and asks them whether or not they support the proposed changes. To say that the distribution of adults who responded yes is normal. How many more adult Americans does the researcher need to sample in the following cases? A) 10% of all adults Americans support changes B) 15% of all adult Americans support the changes
A) n= 25 p= .1 1-p= 0.9 np(1-p)> 10 25(.1)(.9)> 10 NO! Solve for n np(1-p)/p(1-p) > 10/p(1-p) n=10/p(1-p) = 10/(.1)(.9) = 10/.09 = 111.11 = 112 112-25= 87 more people B) p= .15 n= 25 1-p= .85 n= 10/.15(.85) = 10/. 1275 = 78.43 = 79 79-25 = 54 more people
A sample random sample of size n=49 is obtained from a population with u= 70 and o= 14 A) describe sampling distribution of c barr B) p(x > 73.4) C) what is p(x < 65.8)? D) what is p(67.2 < x< 73.9)?
A) n=49 u= 70 o=14 U= 70 Ox= o/ square n = 14/ square 49 = 2 Distribution shape is normal B) p(x > 73.4) Z= x barr-ux/ox = 73.4-70/2 =1.70 P(Z> 1.70) Use table 1-0.9554 =0.446 C) p(x < 65.8) Z= barr-ux/ox = 65.8-70/2 = -2.10 Use table =0.0179 D: p(67.2 < x < 73.9) Z= barr-ux/ox = 67.2-70/2 = -1.40 Z= 73.9-70/2 =1.95 Use table(1.95 and -1.40) 0.9744-0.0808= 0.8936
According to county record of the 84,116 registered homeowners in Burris county 76.2% had more than one motor vehicle registered. Suppose a random sample of 100 homeowners in Burris county was selected and it was determined that 82 had more than one motor vehicle registered A) what is the point estimate for P? B) verify that the sampling distribution of motor vehicles registered to homeowners in Burris county C) describe the sampling distribution of motor vehicles registered to homeowners in Burris county D) what is the probability that a random sample of 100 homeowners in Burris county will feature at least 82 that have more than one motor vehicle registered?
A) p= 82/100=0.82 B) large sample assures a bell shaped and symmetric distribution 100(.82)(.18)= 14.76> or equal to 10 The sample size is less than 5% of the population size C) bell shapes and symmetric distribution Up= p= 0.762 Op= square root (.762)(.238)/100= .0426
Example: according to a study the proportion of people who are satisfied with things going in their lives is 0.72 suppose that a random sample of 100 people is obtained complete the questions A) suppose the random sample of 100 people is asked "are you satisfied with life) qualitative or quantitative? B) explain why sample proportion is a random variable. What is the source? C) describe the sampling distribution of p hat the proportion of people who are satisfied with the way things are going in their life D) in the sample what is the probability that the proportion who are satisfied exceeds 0.75? E) would is be unusual for a survey of 100 people to reveal that 66 or fewer people in the sample are satisfied with their lives?
A) qualitative because no number B) varies from sample to sample. people are the source C) p= 0.72 1-p= 0.28 n= 100 n<.05N yes np(1-p) > 10 =100(.72)(.28) > 10 = 20.16 > 10 yes Up hat= P= 0.72 Op hat= square root p(1-p)/n =square root .72(.28)/100 =0.449 Since sample size is NO MORE than 5% of population size and np(1-0)= 20.16 > 10 the distribution of p hat is APPROXIMATELY MORMAL with up hat= .72 and op hat= .28 D) p( p hat > 0.75) Z= p hat-up hat/ o hat = 0.75-0.72/ 0.0449 = 0.67 P(Z>0.67) use table =1-p(z < 0.67) = 1-0.7486 = 0.2514 E) Z=.66-.72/.0449 =-1.34 Use table (to the left because fewer) = 0.0901 Is not unusual because probability is not less than 5%
Example: determine the area under the standard normal curve that lies to the LEFT of A)z=-0.84 B) z=0.82 C) z=0.91 D) z= -0.32
A) stat/calc/normal/standard/ </ -0.84 = .2005 B) .7939 C) .8186 D) .3745
Example: suppose a probability experiment involved throwing a die one time A) identify the outcomes B) determine the sample space C) list the outcomes for E= "roll an even number"
A){ 1, 2, 3, 4, 5, 6} B){ 1, 2, 3, 4, 5, 6} C){2, 4, 6}
With the addition rules we blank probabilities of events together How we do this depends on whether the events in question are blank Two events are disjoint if they (can/cannot) occur again the same time
Add Disjoint Cannot
The z distribution is blank shaped and has a mean of blank and standard deviation of blank
Bell shapes and symmetric 0 1
How many two letter permutations can be constructed from the word cat?
CA AC CT TC AT TA 6 permutations 3P2= 3!/(3/2)= 3•2•1/1=6/1= 6
How many two letter combinations can be constructed from the word cat?
CA CT
What probability is this? Example: tossing a coin, rolling a die, drawing a card from a deck
Classical
an experiment is a blank probability or experiment if each outcomes is equally likely
Classical
A blank is a selection of items from a group in which the ordering of the items is not considered important
Combination
??? A blank random variable is a
Continuous
A blank random variable is a random variable that can assume an infinite number of balues
Continuous
Example: according to a poll conducted in June 42% of voters believed gay couples should be allowed to marry. Describe the sampling distribution of the sample proportion for samples of n= 10, 50, 100
-As the size n increases the shape becomes normal -The mean of the sample proportion equals the population p -the standard deviation of the sample proportion decreases as the size n increases
Discrete or continuous random variables: - the number of As Essenes on Test 1 for math - the number of cars that go through drive thru at McDonald in the next hiur -the speed of the next car that passes a state trooper
-Discrete -discrete -continuous
Complete the questions with the info: U= 300 SD= 10 -If the sample size n= 16 what is likely true about the shape of the population? -???Jf the sample size is n=16 what is the standard deviation of the population?
-The shape is approximately normal (because bell shaped) ???Ox= o/square root n = 10/ square 16 = 2.5
Example: according to the us census bureau 84% of households have either cable television or satellite service. Suppose 6 home are selected at random. -list the reasons this qualifies as a binomial experiment -what is the probability that exactly 4 home in the sample will have either cable or satellite? -what is the probability that at least 5 homes will have cable or satellite? -what is the mean and standard deviation?
-fixes number of trials(6)/ outcomes are independent - n=6 X=4 P= .84 1-p= 1-.84= .16 P(x=4)=6 C 4 • .84^4 • .16^2 In calc: 6/prb/ 2(nCr)/ 4 =15(.4979)(.0256) = .1912 -p(x greater than or equal to 5)= p(=5) + p(x=6) 6C5 •.84^5 • .16^1 + 6 C 6 • .84^6 • .16^0 = 6(.4182)(.16) + 1(.3513)(1) =.4015 + .3513 = .7528 -Mean u=np= 6(.84) = 5.04; if we took repeated sample of size 6 from population we would expect about 5 homes on average to have cable or satellite tv -Standard deviation O= (square root)np(1-p) = square root 5.04(.16) = 0.9
Example: a recent survey revealed that 58% of Mat 120 students experiences extreme stress while taking the course. suppose four students taking math were selected at random. What is the probability that all four will admit to suffering extreme stress?
.58 .58. .58. .58 P(all 4 suffering extreme stress)= (.58)(.58)(.58(.58)=.113 or 11%
Example: an insurance agent currently insures 192 teenage drivers. Last year, 24 of those drivers had to file a claim for an accident. Given this info what is the probability that in a given year a teenager will have to file an insurance claim?
0.125
Determine the missing probability X. P(x) 3. 0.23 4. ? 5. 0.30 6. 0.25
0.22
If we toss a coin one time that is a classical experiment. If we toss a coin 25 times and count the number of heads, then we have blank evidence about the pro imita a head will be observed using this coin. In the long run the probability will approach the theoretical value of 0.500 by the blank
Empirical law of large numbers
Probabilities based on evidence (data) that have been observed are called blank. Can be though of relative frequencies.
Empirical probabilities
Blank is a collection of outcomes from a probability experiment. It could be a single outcome or multiple outcomes. In probability we call this P(E)
Event
Blank is any process that can be repeated over and over. We often assign probabilities to these
Experiment
???An experiment is considered a binomial experiment if: 1) the experiment is performed a blank number of times(trials) 2) the trials are blank. This means the occurrence of one trial does not affect the occurrence of other trials.
Fixed
A binomial probably experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n=9 p= 0.8 x less than or equal to 3 The probability of x less or equal to 3 successes is=?
In sc: n=9 p=0.8 p(x) less than what to 3 =.0031
If a binomial experiment were repeated a large amount of times the blank serves as the expected value as the number of successes in the trials
Mean
The general blank rule can be extended to more than two events happening one right after each other
Multiplication
Intervals of values to the left would have a blank z score and to the right would be a blank z score
Negative Positive
Example: a simple random sample of size n=61 is obtained from a population with u=85 and o=6 does the population need to be normally distributed for the sampling distribution of x barr to be approximately normally distributed? Why? What is the sampling distribution of x barr?
No because the central limit theorem states that regardless of the shape of the underlying population the sampling distribution of x barr becomes approximately normal as the sample size n increases -Sampling distribution Ux barr= U O x barr= O/ (square root) n U= 85 O= 6 U x barr= U= 85 O x barr= o/ (square root) n = 6/ square root 61 = 0.768 The samp dis of x barr is normal with ux=85 and ox=0.768
Determine if this is binomial: a random sample of 25 high school seniors is obtained and the individuals selected are asked to state their heights.
No because variable is continuous and not two mutually exclusive outcomes (not a success or fail)
A blank plots observed data versus normal scores
Normal probability plot
A blank is the expected z score if the data value if the distribution of the random variable is normal. The expected. Score of an observed value will depend upon the number of observations in the data set
Normal score
Multiplication rules When events occur blank another (in sequence) Assumes that two events are blank T/F: Two events are independent if the first event has no effect on the probability the second events takes place
One right after Independent True
Blank is the result of an experiment
Outcome
Example: a velvety bag contains 4 red marbles 3 blue marbles 2 black marbles and a gold marble. Suppose a marble is drawn from the bad it's color notes the marble is put back in the bag then a second marble is drawn. What is the probability that a red marble was drawn first followed by a black marble? Are these events independent?
P( red marble)= 4/10 P( black marble)= 2/10 P( red followed by black)= 4/10 • 2/10= 8/10= 0.8 or 8% chance of drawing red marble by black marble These two events are independent
P(A or B)= blank
P(A) + P(B)
Example: In a university poll conducted in May, 1745 voters were asked whether they approved of bush handling the economy. 349 responded yes. Obtain a pint estimated for the proportion of voters who approve.
P(hat)= 349/1745 = 0.2
A blank is an arrangement of items selected from a group in which the order in which the items are arranged is important
Permutation
Blank is the measure of likelihood of a random phenomenon or chance behavior occurring
Probability
A blank for a discrete random variable lists all values that a random variable can assume along with their probabilities
Probability distribution
A blank variable is a numerical measure of the outcome of a probability experiment so it's value is determined by chance
Random
A blank variable is a numerical measure of the outcome of a probability experiment, so it's value is determined by chance
Random
***Find the relative probabilities for the following data answer choices on a SAT exam: Choice. Frequency. Relative frequency A. 19 B. 38 C. 74 D. 69 If a question were selected at random what is the probability the correct answer choice was a C? What is the probability it was an A or a B?
Relative frequency: A .095 B .190 C .370 D .345
Example: according to a poll conducted in June 42% of voters believed gay couples should be allowed to marry. Suppose that we obtain a simple random sample of 50 voters and determine which voters believe that gay couples should Marry. Describe the sampling distribution of the sampling proportion for voters who believe they should be allowed to marry.
Sample of n=50 is smaller than 5% of population size (all voters in u.s) np(1-p)= 50(0.42)(0.58)=12.18> 10 The sampling distribution of sample proportion is normal with mean of 0.42 and sd of All Square root 0.42(1-0.42)/50 =0.0698
Blank is a listing of all possible outcomes of a probability experiment
Sample space
Standard deviation for a discrete random variable Multiply the blank of each value x can assume by its probability in the distribution and add up those products. Then blank out the square of the mean. Finally take the blank to the find the standard deviation
Square Subtract Square root
A binomial probably experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n= 10 p= 0.75 x=9 P(9)=?
Stat/calc/binomial/n=10 p=.75 p(x)=9 = 0.1877
Properties of the normal density curve 1) the normal curve is blank about its mean 2) because mean=median= mode the normal curve has a single blank and the highest point occurs at x=u 3) the normal curve has blank point at u- o and u + o 4) the area under the normal curve is blank 5) the area under the normal curve to the right of u equals the area under the curve to the left of u which equals blank
Symmetric Peak Inflection 1 1/2
Example: The following table summarizes staffing in the emergency room of a hospital on a certain day. Doctors. Nurses. Total Male. 2. 1 Female. 3. 8 Total If a stag member were selected at random what is the probability they were a male or a nurse? First...are these events disjoint? Can they occur at the same times
Table: Doctors. Nurses. Total Male. 2. 1. 3 Female. 3. 8. 11 Total. 5. 9. 14 P(male or nurse)= P(male)+ P(nurse)- P(male nurse) = 3/14 + 9/14-1/14= 11/14 = .786 or 79% chance will selects male or nurse Yes they can occur at the same time; not disjoint
Central limit theorem T/F: Regardless of the shape of the underlying population the sampling distribution of x barr becomes approximately normal as the sample size n increases
True
Example: t/f The mean of the sampling distribution of p hat is p.
True
Example: determine the mean(ux)and standard deviation(ox) from the this: U= 72 O= 18 n= 81
U= 72 Ox= o/ square root n = 18/ square 81 =2
Mean of a discrete random variable -Multiply each value of blank by its probability then add up all the products -the mean of a blank random variable can be thought of as the mean outcome of the probability experiment ire repeated it many times
X Discrete
Example: the weights of pennies minted after 1982 are approximately normally distributed with mean 2.46 and standard deviation 0.02 grams Approximate the sampling distribution of the sample mean by obtaining 200 simple random samples of size n=5 from this population Sample size: 2.433 2.466 2.423 2.442 2.456 What is sample mean? ??What is the probability that is a simple random sample of 10 pennies minted after 1982 we obtain a sample mean of at least 2.465 grams?
X barr=2.444
Suppose a basketball player were asked to shoot 3 FT at a time over a summer camp. This experiment was repeated over and over again. Coach will count the number of successful free throws in each group of the three. Like 0/3 or 1/3 or 2/3 or 3/3. Let x= the number of fr made out of three attempts. What values can x assume? The probability distribution for this random variable is given below. X. P( x) 0. 0.01 1. 0.10 2. 0.38 3. 0.51 Calculate the mean and interpret Find the standard deviation and interpret
X can assume 0, 1,2, 3 U=€ x• p(x) =0(.01) + 1(.10)+ 2(.38)+ 3(.51) =0 + .10 + .76+ 1.53 =2.39 = 2.4; over many repetitions of shooting 3 free throws at a time we would expect this basketball player to make on average 2.4 out of 3 O=^square root €x^2 • p(x) -U2 =square root)0^2(.01)+ 1^2(1.0)+ 2^2(.38)+ 3^2(.51)-2.4^2 = (square root) 0+ .10+ 1.52+ 4.59+ 5.76 = (square root).45 = 0.67 -> 0.7 Standard deviation=0.7
A die is rolled and a "4" is observed Is this binomial?
Yes
Example: Students passing math class is this binomial?
Yes
Example: households are surveyed and the number with a dog is recorded Is this binomial?
Yes
Determine whether the distribution is a discrete probability distribution(are all numbers countable? Do they equal 1? X. P(x) 0. 0.08 1. 0.17 2. 0.32 3. 0.19 4. 0.24
Yes because sun equals 1
The binomial distribution The outcome of each probability experiment are mutually exclusive outcomes(success or failures) or can be adjusted so there are two mutually exclusive outcomes. Then the experiment qualifies as a blank probability experiment.
binomial
Example: according to the centers of disease control and prevention 18.8% of school aged children aged 6-11 years were overweight in 2004. A) in a random sample of 90 school aged children aged 6-11 years what is the probability that at least 19% are overweight? B) suppose a random sample of 90 school aged children aged 6-11 years results in 24 overweight children. What might we conclude?
n=90 is less than 5% of population size np(1-p)= 90(.188)(1-.188) = 13.7 which is greater than 10 P hat is normal with mean 0.188 and sd of ALL square root (0.188)(1-0.188)/90 =0.0412 A) Z= 0.19-0.188/0.0412 = 0.0485 P(Z> 0.05) use table(to the right) =1-0.5199 = 0.4801 Or about 48% of the time B) p hat= 24/90 = 0.2667 Z= 0.2666-0.188/0.0412 =1.91 We would only expect to see about 3 samples in 100 resulting in a sample proportion of 0.2666 or more. This is ínsula sample
Counting rules Sometimes it is necessary to count the number of ways an event can occur. Two important counting methods in probability are blank and their cousin,blank
permutations combinations
Example: list the sample space if a letter (A B C D) were selected and a spinner contained the number 1, 2, and 3.
{A1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3}
A blank random variable is a random variable whose values are always just countable values
Discrete
A blank random variable is a random variable whose values are what's just countable values: give example
Discrete Amount of children someone has
Discrete or continuous ? Number of As on a math test Number of cars in McDonald's drive thru for an hour The speed of the next car that passes a state trooper
Discrete Discrete Continuous
If two events are not blank then that means the events have some elements in common. Those common elements need to be removed from the sun of the probabilities to avoid "double counting"
Disjoint
To find probability, blank the number of outcomes which satisfy the event by the number of outcomes in the sample space.
Divide
The complement rule: If an events occurs, the probability that it (does/ does not) occur is 1-P(E) This can be useful when finding probabilities for events that have multiple outcome
Does not