Probability Questions and Answers

For a monthly subscription fee a video download site allows people to download and watch up to five movies per month Based on past download histories the following table gives the estim probabilities that a randomly selected subscriber will download 0 1 2 3 4 or 5 movies in a particular month Number of downloads b at most three movies c four or more movies d zero or one movie 0 e more than one movie 1 0 03 0 44 0 24 3 4 Estimated probability If a subscriber is selected at random what is the estimated probability that this subscriber downloads the following amount of movies a three or fewer movies 0 10 0 10 5 0 09

The article Scrambled Statistics What Are the Chances of Finding Multi Yolk Eggs gives the probability of a double yolk egg as 0 001 a Give a relative frequency interpretation of this probability In the long run about b If 9 000 eggs were randomly selected about how many double yolk eggs would you expect to find eggs Need Help of eggs are double yolk Read It

According to a Pew Research Center study in May 2011 33 of all American adults had a smart phone one which the user can use to read email and surf the Internet A communications professor at a university believes this percentage is lower among community college students She selects 358 community college students at random and finds that 130 of them have a smart phone In testing the hypotheses Ho p 0 33 versus Ha p 0 33 Do the hypothesis test Use a level of significance of a 0 0 0 05 Use the unrounded values in Excel find the answers for 3 and 4 hypothesis test 1 This is an example of a 2 Find the standard error 3 Find the z value 4 Find the p value C 5 The communication professor should the null hypothesis Fill in the blanks by selecting one option from each menu aby Part 1 3 points ab Part 2 3 points aby Part 3 3 points ab Part 4 3 points aby Part 5 3 points 15 points

Use the theoretical probability formula to solve the problem Express the probability as a fraction reduced to lowest terms A fair coin is tossed two times in succession The set of equally likely outcomes is HH HT TH TT Find the probability of getting the same outcome on each toss 1 OA OB 34 O C 1 OD 1 1 4

Use the spinner shown It is equally probable that the pointer will land on any one of the six regions If the pointer lands on a borderline spin again If the pointer is spun twice find the probability that it will land on purple and then grey Find the probability that the spinner will land on purple and then grey Type an integer or a simplified fraction MICHERCH purple she purple grey bhe

Use the theoretical probability formula to solve the problem Express the probability as a fraction reduced to lowest terms A single die is rolled twice The set of 36 equally likely outcomes is 1 1 1 2 1 3 1 4 1 5 1 6 2 1 2 2 2 3 2 4 2 5 2 6 3 1 3 2 3 3 3 4 3 5 3 6 4 1 4 2 4 3 4 4 4 5 4 6 5 1 5 2 5 3 5 4 5 5 5 6 6 1 6 2 6 3 6 4 6 5 6 6 Find the probability of getting two numbers whose sum is greater than 10 O A OB O C 1 18 5 18 12 OD 3

Find the median for the group of data items 97 97 94 48 71 97 A 97 B 95 5 OC 94 D 48

According to flightstats com American Airlines flights from Dallas to Chicago are on time 80 of the time Suppose 20 flights are randomly selected and the number of on time flights is recorded a Explain why this is a binomial experiment b Determine the values of n and p c Find and interpret the probability that exactly 14 flights are on time d Find and interpret the probability that fewer than 14 flights are on time e Find and interpret the probability that at least 14 flights are on time f Find and interpret the probability that between 12 and 14 flights inclusive are on time is Round to four decimal places as needed Interpret the probability In 100 trials of this experiment it is expected that about flights being on time Round to the nearest whole number as needed e Using the binomial distribution the probability that at least 14 flights are on time is will result in fewer than 14 Round to four decimal places as needed Interpret the probability In 100 trials of this experiment it is expected that about will result in at least 14 flights being on time Round to the nearest whole number as needed f Using the binomial distribution the probability that between 12 and 14 flights inclusive are on time is Round to four decimal places as needed Interpret the probability In 100 trials of this experiment it is expected that about will result in between 12 and 14 flights inclusive being on time Round to the nearest whole number as needed

possible According to flightstats com American Airlines flights from Dallas to Chicago are on time 80 of the time Suppose 20 flights are randomly selected and the number of on time flights is recorded a Explain why this is a binomial experiment b Determine the values of n and p c Find and interpret the probability that exactly 14 flights are on time d Find and interpret the probability that fewer than 14 flights are on time e Find and interpret the probability that at least 14 flights are on time f Find and interpret the probability that between 12 and 14 flights inclusive are on time C Using the pinomial distribution the probability that exactly 14 Tights are on time is Round to four decimal places as needed Interpret the probability In 100 trials of this experiment it is expected that about will result in exactly 14 flights being on time Round to the nearest whole number as needed d Using the binomial distribution the probability that fewer than 14 flights are on time is Round to four decimal places as needed Interpret the probability In 100 trials of this experiment it is expected that about will result in fewer than 14 flights being on time Round to the nearest whole number as needed e Using the binomial distribution the probability that at least 14 flights are on time is Round to four decimal places as needed nterpret the probability n 100 trials of this experiment it is expected that about will result in at least 14

In the probability distribution to the right the random variable X represents the number of hits a baseball player obtained in a game over the course of a season Complete parts a through f below c Compute and interpret the mean of the random variable X Hx hits Type an integer or a decimal Do not round Which of the following interpretations of the mean is correct possible X 0 1 2 Tyne an integer or a decimal Do not round 3 4 5 Type an integer or a decimal Do not round f What is the probability that in a randomly selected game the player got more than 1 hit P x 0 1667 0 3356 0 2847 OA In any number of games one would expect the mean number of hits per game to be the mean of the random variable 0 1505 0 0371 0 0254 OB Over the course of many games one would expect the mean number of hits per game to be the mean of the random variable OC The observed number of hits per game will be equal to the mean number of hits per game for most games OD The observe number of hits per game will be less than the mean number of hits per game for most games d Compute the standard deviation of the random variable X x hits Round to three decimal places as needed e What is the probability that in a randomly selected game the player got 2 hits

The graph to the right is the uniform probability density function for a friend who is x minutes late a Find the probability that the friend is between 25 and 30 minutes late b It is 10 A M There is a 20 probability the friend will arrive within how many minutes Density 1 30 0 0 10 20 30X Time min a The probability that the friend is between 25 and 30 minutes late is

Write the equation of this line in slope intercept form 8 7 6 5 4 3 2 1 8 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 1 2 3 4 5 6 7 7 8 X

An experiment is performed where a 3 color spinner is spun and then a 6 sided die is rolled The possible outcomes for each event are red R blue B and yellow Y for the 3 color spinner and 1 2 3 4 5 and 6 for the 6 sided die Identify the sample space for this experiment Answer Separate the elements of the sample space with commas Keypad Keyboard Shortcu

Question 1 35 Points A soil profile is provided below The estimated stresses from a spread footing overlying a 10 meter thick normally consolidated clay layer are 180 0 kN sq m top of the clay layer 64 3 kN sq m bottom of the clay layer 97 2 kN sq m middle of the clay layer Estimate the primary settlement in meters of the clay layer assuming Cc Cc log 1 eo 0 44 and Cs Cs log 1 eo 0 08 taken from consolidation curves Assume the pre consolidation pressure is 100 kN sq m Assume no settlement from the overlying sand layer dan 17 3 V YSAT 19 81 KNV m FN m SAND CLAY Depth 0 GS l 3 m 1 00

Chapter 5 6 100 pts Name Continuous Random Variables and The Normal Curve Probability Statistics Multiple Choice Write a brief explanation for your choice Which of the following can be used to describe the distribution for a continuous random variable a Binomial Distribution c Poisson Distribution b Normal Distribution d Hypergeometric Distribution When working with a continuous random variable the outcomes are b estimated d None of these a counted c measured Which of the following is not a characteristic of the normal distribution a It is symmetric b It is bell shaped c Mean Median Mode d None of these The standard normal curve is a distribution with a 1 and 0 0 c u larger than b 0 and 0 1 d u usually larger than O 5 The area under the curve to the right of z 0 is b 1 0000 a 0 6816 c 0 5000 d 0

6 If z is a standard normal random variable the area between z 0 and z 1 2 compared to the area between z 1 2 and z 2 4 will be b smaller c the same a larger d Not Enough Info 7 The probability density function f x of a random variable X that is normally distributed is completely determined once the a mean and standard deviation of X are specified b mean and mode of X are specified c median and mode of X are specified d mean and median of X are specified 8 The exponential distribution is often concerned with a the number of failures in an experiment b the amount of time until an event occurs c the number of successes in an experiment d events that are equally likely to happen

20 A normal random variable x has an unknown standard deviation and mean 54 If the probability that x is less than 58 is 0 8944 find the standard deviation

19 The mayor of the city of Mayville was informed that household garbage pickup was a normally distributed random variable with 65 pounds and 0 9 pounds per week a Find the probability that a randomly chosen household has more than 55 pounds of garbage per week h Find the probability that a randomly chosen household has between 55 and 70

18 Suppose that the longevity of a light bulb is exponential with a mean lifetime 8 years a Find the probability that a light bulb lasts less than 1 year b Find the prob that a bulb lasts between 6 and 10 years

If scores for a school team were normally distributed with a mean 68 and a eviation 3 Determine the z score for a golfer who scored 65 ind the probability that a randomly selected golfer scored less than 65 and the probability that a randomly selected golfer scored between 60 and 70 10 Su parar

1 To assess the effectiveness of the artificial marks the team selected 14 cattle herds in Botswana Within each herd adult cattle were randomly assigned into one of three treatment groups artificial eyespots cross marked or unmarked The researchers have decided to use a 6 sided die in order to randomly assign the treatments to a herd of 50 cows One such die is shown here for you to use as a reference Describe a procedure which uses the die to assign each cow in the herd to one of the three treatments You can also write on the die rolling area to show your thinking

tiven the point and slope write the equation in slope intercept form 3 3 slope N M 2 3

Given the graph write the equation in slope intercept form 2 6 5 4 3 10 1 2 3 4 5 6

13 Consider a small collection of four letters A B C D Think of this collection as a small population How many samples of size two are possible if sampling is done with replacement Answer samples How many samples of size two are possible if sampling is done without replacement Answer samples Previous Answers 10 6

Fill out the table below 1 2 3 4 5 SUM x 2 2 4 6 Y 25 23 19 17 11 xy x

3 A warehouse employs 22 workers on first shift and 16 workers on second shift Eight workers are chosen at random to be interviewed about the work environment Complete parts a through d a Find the probability of choosing all first shift workers Round to four decimal places as needed b Find the probability of choosing all second shift workers Round to four decimal places as needed c Find the probability of choosing exactly six first shift workers Round to four decimal places as needed d Find the probability of choosing exactly four second shift workers Round to four decimal places as needed

The pie chart to the right shows how adults rate their financial shape Suppose 18 people are chosen at random from one of the 18 people would rate their financial shape as fair Make the assumption that the 500 people are represented by the pie chart group of 500 What is the probability that 4 Excellent 4 Good 42 Fair 38 Poor 14 Other 2

For the given data set find the a mean b median c mode or state that there is no mode and d midrange The ages of teachers in the mathematics department of a certain high school are shown below 26 54 38 57 26 54 26 50 50 41 OA a 31 6 b 45 5 c 26 d 41 5 B a 26 6 b 45 5 c 26 d 41 5 IN OC a 42 2 b 45 5 c 50 d 41 5 OD a 42 2 b 45 5 c 26 d 41 5

7 A company that has 200 employees chooses a committee of 13 to represent employee retirement issues When the committee is formed none of the 56 minority employees are selected a Use technology to find the number of ways 13 employees can be chosen from 200 b Use technology to find the number of ways 13 employees can be chosen from 144 nonminorities c What is the probability that the committee contains no minorities when the committee is chosen randomly without bias d Does your answer to c indicate that the committee selection is biased Explain your reasoning a There are ways to choose 13 employees from 200 employees Use scientific notation Round to two decimal places as needed Use the multiplication symbol in the math palette as needed b There are ways to choose 13 employees from 144 nonminorities Use scientific notation Round to two decimal places as needed Use the multiplication symbol in the math palette as needed c The probability is Do not round until the final answer Then round to three decimal places as needed d 1 1 Yes O No 2 the committee selection is most likely 2 O biased 3 O large O unbiased small because the probability that was found in part c is 3

A professor had students keep track of their social interactions for a week The number of social interactions over the week is shown in the following grouped frequency distribution How many students had at least 60 social interactions for the week The number of students was Type a whole number www Number of Social Interactions 30 34 35 39 40 44 45 49 50 54 55 59 60 64 65 69 70 74 75 79 Frequency 5 17 16 18 18 10 6 6 1 1

Provide a real world example in which the Z score would be a useful number to have

If we know that the mean of a given data set is 20 and its standard deviation is 3 What is the Z score for a data point with value 15 What of the data is smaller than that given data point

ecorded at the end of the experiment Z Y 4 5 Plant Height cm To start the class discussion about the results of the experiment the teacher randomly chooses two of the plants Preview Questions How many plants are there The total number of plants is 9 What do we know about each plant X 8 4 W 12 1 V 11 U 9 A B T 3 8 S 5 2 R 6 GED Test Question What is the probability of drawing two plants that ar taller than 10 cm

In your own words what is the difference between a discrete and continuous variable

Promoted Not Promoted Total Men 288 672 960 Women 36 204 240 Total 324 876 1200 Based on the table above what is the probability that someone randomly selected from a group of men was not promoted

Use the given data to find the 95 confidence interval estimate of the population mean Assume that the population has a normal distribution IQ scores of professional athletes Sample size n 15 Mean 104 Standard deviation s 14

The scores of students on an exam are normally distributed with a mean of 298 and a standard deviation of 72 a What is the first quartile score for this exam Answer b What is the third quartile score for this exam Answer

5 The offices of president vice president secretary and treasurer for an environmental club will be filled from a pool of 19 candidates Eight of the candidates are members of the debate team a What is the probability that all of the offices are filled by members of the debate team b What is the probability that none of the offices are filled by members of the debate team a P all offices filled by debate team members Round to three decimal places as needed b P no offices filled by debate team members Round to three decimal places as needed 6 The pie chart to the right shows how adults rate their financial shape Suppose 18 people are chosen at random from a group of 500 What is the probability that none of the 18 people would rate their financial shape as fair Make the assumption that the 500 people are represented by the pie chart The probability that none of the 18 people would rate their financial shane as fair in M Excellent 4 Good 42 Fair 38 Poor 14 Other 2

Paper size Print 2 A golf course architect has four linden trees three white birch trees and three bald cypress trees to plant in a row along a fairway In how many ways can the landscaper plant the trees in a row assuming that the trees are evenly spaced The trees can be planted in different ways There are Type a whole number 3 A class has 27 students In how many different ways can five students form a group for an activity Assume the order of the students is not important different ways that the five students can form a group for an activity eller 4 You look over the songs in a jukebox and determine that you like 16 of the 58 songs a What is the probability that you like the next four songs that are played Assume a song cannot be repeated b What is the probability that you do not like the any of the next four songs that are played Assume a song cannot be repeated a The probability that you like the next four songs that are played is Round to three decimal places as needed b The probability that you do not like any of the next four songs that are played is Round to three decimal places as needed

7 The disease status and exposure of a small population is summarized in the given table below An individual is randomly selected from this population Define A to be the event that the selected individual has been exposed and define event B to be the event that the selected individual has the disease Find the probability that the individual has the disease given they have not been exposed i e find P BIA Exposure Disease No Disease 56 No Exposure 24 Answer 384 Round your answer to the nearest thousandths Previous Answer 18 668

It is assumed that the duration of human pregnancies can be described by a Normal model with mean 266 days and standard deviation 16 days The results shown in the accompanying analysis of pregnancies were based on this Normal model The duration may not actually follow the Normal model however Complete parts a and b below Click the icon to view the pregnancy analysis results based on the Normal model Does the result in statement i change O A No because the Normal model can still be used according to the Central Limit Theorem even if the corre O B Yes because while the Normal model can still be used the standard deviation will potentially change OC Yes because the Normal model cannot be used if the correct model is very skewed O D Yes because while the Normal model can still be used the mean will potentially change Does the result in statement ii change OA Yes because the Normal model cannot be used if the correct model is very skewed OB Yes because while the Normal model can still be used the standard deviation will potentially change Pregnancy Analysis Results i Assuming the Normal model is true 21 1 of all pregnancies last between 270 and 280 days ii Assuming the Normal model is true the longest 25 of all pregnancies will last 276 8 days or more iii Assuming the Normal model is true a sample of 60 pregnant patients will have a sampling distribution with a mean of 266 days and a standard deviation of 2 07 days OC Yes because while the Normal model can still be used the mean will potentially change OD No because the Normal model can still be used according to the Central Limit Theorem even if the correct model is very skewed Does the result in statement iii change Print Done

8 State the z scores that most closely approximates the following probabilities hint similar to question 7 but you have to first convert the percentage into a decimal Similar to question 7 but you do not have to solve the entire problem a A top 10 of scores b Bottom 20 of scores c Top 50 of scores 9 A physiological psychologist studied the time it took a new recreational drug to begin to influence interpersonal perception among 145 senior adults The mean time for this population was 140 seconds with a standard deviation of 25 seconds What proportion of the participants in this study took between 135 and 155 seconds before the drug started to influence their interpersonal perception 25 135 14

Complete the statement below As the sample size increases the mean of the sampling distribution stays the same gets larger gets smaller

According to MarylandMatters org 77 of Marylanders are fully vaccinated for COVID 19 A college administrator in Maryland wants to test if a higher percentage of students at their institution are fully vaccinated for COVID 19 The administor randomly selects 500 students and inquires about their vaccination status 395 of the 500 surveyed indicated that they were fully vaccinated Does this provide sufficient evidence to declare that the student body has a significantly higher percentage vaccinated than the state level Ho HA A

3 Assuming each outcome is equally likely in the sample space find P AUB A COB Answer Previous Answer 8

The Centers for Disease Control estimates indicate 20 percent of the U S population had a sexually transmitted infection STI on any given day in 2018 A public health researcher thinks the percent of adutls with a STI is different in their state They randomly select 850 adults and discover that 146 have a STI Is this significant evidence that the percent of adults in this state with a STI is different than 20 Let a 0 05 z stat round to 2 decimal places 2 19 p value do not round 0 0286 What is your decision Retain Ho or Reject Ho Reject Ho

The age when smokers first start from previous studies is normally distributed with a mean of 13 years old with a population standard deviation of 2 1 years old A survey of smokers of this generation was done to estimate if the mean age has changed The sample of 28 smokers found that their mean starting age was 14 8 years old Find the 99 confidence interval of the mean Select an answer Do not round in between steps Round answers to at least 3 decimal places

8 13 Number Skills of Eighth Graders Suppose that scores on the mathemat ics part of the National Assessment of Educational Progress NAEP test for eighth grade students follow a Normal distribution with standard deviation 110 You want to estimate the mean score within 10 with 90 confi dence How large an SRS of scores must you choose

4 14 pts The life expectancy of a microwave oven is normally distributed with an average of 16 7 years and a standard deviation of 3 1 years Round all probabilities to 4 decimal places a Sketch and label a complete normal distribution for this situation include values of 30 30 16 7 3 3 1 16 7 9 3 7 4 years years H 20 16 7 2 3 1 16 7 6 2 10 2 H 0 16 7 3 1 13 6 years H 16 7 years H 0 16 7 3 1 19 8 8 years H 20 16 7 2 3 1 16 7 6 2 22 9 years H 30 16 7 3 3 1 16 7 9 3 26 0 years 7 4 center of distribution 10 5 13 6 16 7 22 9 b Find the probability of buying a microwave that lasts for more than 24 years Label this value shade in this area in the distribution above to show the connection to the probability you calculate P x 24 P z 2 35 1 P Z 2 35 0 0094 c Find the probability of buying a microwave that lasts for less than 13 years

4 14 pts The life expectancy of a microwave oven is normally distributed with an average of 16 7 and a standard deviation of 3 1 years Round all probabilities to 4 decimal places a Sketch and label a complete normal distribution for this situation include values of 30 I 7 4 years 10 2 years I center of distribution H 30 16 7 3 3 1 16 7 9 3 H 20 16 7 2 3 1 16 7 6 2 H o 16 7 3 1 13 6 years H 16 7 years H 0 16 7 3 1 19 8 years 1 H 20 16 7 2 3 1 16 7 6 2 22 9 years 1 H 30 16 7 3 3 1 16 7 9 3 26 0 17 4 10 5 13 6 years 16 7 b Find the probability of buying a microwave that lasts for more than 24 years Label this value shade in this area in the distribution above to show the connection to the probability you calculate years 19 8 22 9