Probability Questions and Answers

Use the Student s t distribution to find the t value for each of the given scenarios Round t values to four decimal places Find the value of t such that the area in the right tail of the t distribution is 0 01 if the sample size is 110 t Find the value of t such that the area in the left tail of the t distribution is 0 05 if the sample size is 144 Find the value of t such that the area in the right tail of the t distribution is 0 0005 if the sample size is 59 Find the two values of t such that 80 of the area under the t distribution is centered around the mean if the sample size is 66 Enter the solutions using a comma separated list

An economist needs to estimate the proportion of residents of Okeechobee that have earned a college degree Determine the most conservative estimate of the sample size required to limit the margin of error to within 0 065 of the population proportion for a 90 confidence interval Round the solution up to the nearest whole number n

An agronomist would like to estimate the average weight of all Indian River County grapefruits The mean weight of a random sample of 190 Indian River County grapefruits was found to be 217 grams The standard deviation of the weights of all Indian River County grapefruit is known to be 19 grams Find three different confidence intervals one with a 96 confidence level one with a 95 confidence level and one with a 90 confidence level for the mean weight of all Indian River County grapefruits Notice how the confidence level affects the margin of error and the width of the interval Report confidence interval solutions using interval notation Round solutions to two decimal places if necessary The margin of error for a 96 confidence interval is given by E A 96 confidence interval is given by

A city official would like to estimate the mean age of all residents of Jensen Beach The standard deviation of the ages of all residents of Jensen Beach is known to be 18 years Determine the sample size necessary such that the margin of error of the estimate for a 96 confidence interval for the mean age of all residents is at most 4 8 years Round the solution up to the nearest whole number n

A researcher would like to estimate the mean weight of all adult female American alligators in the Everglades The mean weight of a random sample of 36 adult female American alligators from the Everglades was found to be 250 pounds The standard deviation of the weights of all adult female American alligators in the Everglades is known to be 15 pounds Using a 99 confidence level determine the margin of error E and a confidence interval for the mean weight of all adult female American alligators in the Everglades Report the confidence interval using interval notation Round solutions to two decimal places if necessary The margin of error is given by E A 99 confidence interval is given by

Suppose one would like to construct a 99 confidence interval for a sample mean when the population standard deviation is known Determine the z value that would be used to construct such a confidence interval Round the solution to two decimal places N

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The probability that teresa will go on vacation in vigan is 0 45 and the probability that she will go to Bohol for vacation is 0 38 Find the probability that Tevera will go to Vigan or Bohol

Sixty percent of the sixth grade students at Jackson Middle School play a sport If 114 sixth grade students play a sport how many sixth grade students are there in the school You are given the part 114 students and the percent 60 You need to find the whole In other words 60 of what number is 114 You can use bar diagrams ratio tables double number lines and equivalent ratios to find the whole Method 1 Use a bar diagram Sixty is a multiple of 10 and 10 is a factor of 100 Draw a bar diagram with 10 equal size sections of 10 each because 10 x 10 100 Shade 6 sections to represent 60 Label the shaded sections as 114 students because 60 of the whole is 114 Percent 0 114 students Percent 0 60 Of Each section represents the same number of students There are 6 shaded sections Divide 114 by 6 to find the number of students represented by each section ni qata 114 6 19 Divide Each section represents 19 students 1 114 students 19 19 19 19 19 19 19 19 19 19 d 60 100 100 19 x 10 190 Multiply The whole is 190 students C000100 Because each section represents 19 students and there are 10 total OUT sections multiply 19 by 10 to find the total number of sixth grade students no in 114 There are 190 sixth grade students MA 6 AR 3 5 e Talk About It How can you use bar diagram to fin number of sixth g students who do play a sport

Use a scatter plot to display the data shown in the table available below The data represent the numbers of hours worked and the hourly wages in dollars of 12 production workers Describe any patterns Click the icon to view the hours worked and the hourly wages data Which scatter plot below displays the data OA Hourly Wage in dollar OC Hours Worked and Hourly Wage A 20 18 16 14 12 10 20 Hourly Wage in dollam 20 18 16 14 12 10 U 20 30 PEE Hours Worked and Hourly Wage A T Hours 30 40 T B Hours What best describes the data 40 50 Q Q 50 Q Q 5 OA The data closely follow along a line that rises from left to right OB The data loosely follow along a curve that rises and then falls from left to right OC The data shows no pattern OD The data closely follow along a line that falls from left to right CEP OB OD Hourly Wage in dollars 16640 Hours Worked and Hourly Wage A 20 18 12 10 20 20 Hourly Wage in dollar NON FOTO 18 14 16 12 8 10 20 8 30 Hours Worked and Hourly Wage I ww T 30 T Hours Hours 40 40 50 50 Q 5 Q Q 5

The soft drink cups at a certain fast food restaurant have tickets attached to them Customers peel off the tickets to see whether they win a prize The proportion of tickets that are winners is p 0 30 A total of n 85 people purchase soft drinks between noon and 1 00 PM on a certain day Let be the proportion that win a prize Find the mean and standard deviation of sample proportion P Formula p 1 The mean of the sampling distribution of population proportion 0 1 is denoted by Hp p 2 The standard deviation of the sampling distribution of is denoted by p 1 P He and equals the Op is

Determine whether each experiment is a binomial experiment or can be reduced to a binomial experiment If so describe a trial determine the random variable and state n p and q You randomly remove one card from a deck to see if it is a heart You place the card back in the deck and repeat the process five times O This experiment can be reduced to a binomial experiment Success is a heart failure is not a heart a trial is removing a card and the random variable is the number of hearts n 5 p 0 25 q 0 75 This experiment can be reduced to a binomial experiment Success is a heart failure is not a heart a trial is removing a card and the random variable is the number of hearts n 5 p 0 75 g 0 25 This experiment can be reduced to a binomial experiment Success is a heart failure is not a heart a trial is removing a card and the random variable is the number of hearts n 5 p 0 5 q 0 5 This experiment cannot be reduced to a binomial experiment because the events are not independent The probability of choosing a heart changes after each selection

2 points A firm that refurbishes personal computers PCs recently received 24 faulty computers of a similar make and model 21 PCs have been fixed but an intern mistakenly packaged all the PCs to storage without labeling The repair engineer will need to test the PCs one after another until he identifies the 3 PCs that are not yet fixed Find the probability that he tested 15 computers to complete the task

County Health and Summertir Sports Center are two local youth basketball leagues Last season the total points scored per player in United County Health had a population mean of 342 points and a standard deviation of 67 points and the total points scored per player in Summertime Sports Center had a population mean of 429 points and a standard deviation of 53 points For each league the distribution of total points scored last season per player was clearly skewed not approximately symmetric Last season Chau scored a total of 488 points in United County Health and Milan scored a total of 574 points in Summertime Sports Center a Find the z scores of Chau s total points scored last season as a player in United County Health and Milan s total points scored last season as a player in Summertime Sports Center Round your answers to two decimal places z score of Chau s total points scored last season z score of Milan s total points scored last season b Relative to his population who was the lower scorer Choose the best answer based on the z scores of the two players Chau O Milan O It is unclear who was the lower scorer relative to his population X

A TV executive is interested in the popularity of a particular streaming TV show She has been told that a whopping 68 of American households would be interested in tuning in to a new network version of the show If this is correct what is the probability that all 6 of the households in her city being monitored by the TV industry would tune in to the new show Assume that the 6 households constitute a random sample of American households Round your response to at least three decimal places If necessary consult a list of formulas

At a recent meeting at the American College of Asthma Allergy and Immunology researchers reported that allergy vaccinations are effective in treating sinusitis inflammation of the membrane lining the facial sinuses in individuals predisposed to allergies Specifically the researchers reported that 98 of individuals predisposed to allergies find relief from their sinusitis after allergy vaccinations Suppose that this number is accurate and that 130 individuals predisposed to allergies are chosen at random and given allergy vaccinations Answer the following If necessary consult a list of formulas a Estimate the number of individuals in the random sample who do not find relief from their sinusitis by giving the mean of the relevant distribution that is the expectation of the relevant random variable Do not round your response 0 b Quantify the uncertainty of your estimate by giving the standard deviation of the distribution Round your response to at least three decimal places Aa

For admission to graduate school Jina is taking the French subject matter test and Lucy is taking the physics subject matter test Jina scored 531 on the French test and Lucy scored 653 on the physics test For each of these subject matter tests the distribution of scores is clearly bell shaped Scores for the French test have a population mean of 625 points with a standard deviation of 29 points Scores for the physics test have a population mean of 566 points with a standard deviation of 41 points a Find the z scores of Jina s performance as a score on the French test and Lucy s performance as a score on the physics test Round your answers to two decimal places z score of Jina s score z score of Lucy s score X b Relative to her population who scored higher Choose the best answer based on the z scores of the two test scores Jina O Lucy X B K

13 The Lumberjack Hardware store found that 7 out of every 10 hammer drills they recently received were not properly assembled If they received 325 hammer drills about how manyjammer drills can they expect to be faulty O 46 9 hammer drills

Caucasian Hispanic D Other Expected count 535 92 Deviation 15 92 Chi square 0 47294 Observed count 55 Expected count 49 91 Deviation 5 09 Chi square 0 51922 98 90 61 Deviation 7 39 Chi square Observed count Expected count 0 60234 417 50 27 50 1 81114 30 38 88 8 88 2 02866 60 70 59 10 59 1 58880 Select all of the true statements regarding the chi square contingency table 113 86 38 71 12 86 1 29 1 45336 0 04273 15 3 10 60 3 61 4 40 0 61 1 82243 0 10164 25 4 19 25 6 55 5 75 2 55 1 71624 0 99002 Except for rounding errors the sum of the expected counts is equal to the total number of observations The frequency distribution and cell counts meet the requirements of a chi square test The relatively high number of Caucasian donors of blood type A and the relatively low number of Caucasian donors of blood type O contribute the most to the chi square statistic The number of Hispanic donors of blood type AB contributes the least to the chi square statistic The chi square test requirements are not met because 3 cells have observed counts of less than 5 The relatively low number of African American donors of blood type A and the relatively low number of Hispanic donors of blood type A contribute the most to the chi square statistic The number of Caucasian donors of blood type AB contributes the least to the chi square statistic

Determine the critical value 2 2 that corresponds to the given level of confidence rounding to two decimal pla 86 OA 1 16 B 1 08 OC 0 81 OD 1 48

How much money does the average professional hockey fan spend on food at a single hockey game That question was posed to 10 randomly selected hockey fans The sampled results show that sample mean and standard deviation were 16 00 and 3 30 respectively Use this information to create a 98 confidence interval for the mean Express the answer in the form x 1 2 s n OA 16 2 764 3 30 10 OB 16 2 821 3 30 10 OC 16 2 262 3 30 10 OD 16 2 718 3 30 10

It is thought that not as many Americans buy presents to celebrate Valentine s Day anymore A random sample of 4000 Americans yielded 2200 who bought their significant other a present and celebrated Valentine s Day Estimate the true proportion of all Americans who celebrate Valentine s Day using a 90 confidence interval Express the answer in the form p E and round to four decimal places OA 0 5500 0 0129 OB 0 4375 0 0129 OC 0 5500 0 4048 OD 0 4375 0 4048

A computer package was used to generate the following printout for estimating the sale price of condominiums in a particular neighborhood A friend suggests that the mean sale price of homes in this neighborhood is 44 000 Comment on your friend s suggestion Click the icon to view the printout A Your friend is correct and you are 100 certain O B Your friend is correct and you are 90 certain C Based on this printout all you can say is that the mean sale price mig D Your friend is wrong and you are 90 certain Printout X sale price SAMPLE MEAN OF X 46 400 SAMPLE STANDARD DEV 1 SAMPLE SIZE OF X 15 CONFIDENCE 90 UPPER BOUND 52 650 6 SAMPLE MEAN OF X 46 400 LOWER BOUND 40 149 4

e If you concluded that marriage status and employment status in men are associated does that imply potential partners preferentially seek out employed men If you failed to conclude they were associated does that imply potential partners are indifferent to the employment status of men Explain your answer to receive full credit

16 Given that the degrees of freedom for the appropriate t procedure is approximately 12 what is a 95 confidence interval for High Low hint 20 6 21 1 7 7 11 146 a 1 9 to 50 5 grams b 15 1 to 37 3 grams c 20 6 to 21 1 grams d 101 3 to 127 5 grams

1 A used car dealer says that the mean price of used cars sold in the last 12 months is at least 21 000 You suspect this claim is incorrect and find that a random sample of 14 used cars sold in the last 12 months has a mean price of 19 189 and a standard deviation of 2 950 Is there enough evidence to reject the dealer s claim at a 0 05 Assume the population is normally distributed State in words the null and alternative hypotheses Null Alternative

12 6 points Find the absolute extrema value on 0 00 for f x x 6x 8

A binomial experiment with probability of success p 0 45 and n 5 trials is conducted What is the probability that the experiment results in exactly 4 successes Do not round your intermediate computations and round your answer to three decimal places If necessary consult a list of formulas

Maya runs a large animal shelter At Maya s shelter the length of stay for puppies has a mean of 85 days and a standard deviation of 11 days The Amador family adopted a puppy named Thunder from the shelter When Thunder was adopted he had been at the shelter for 57 days Find the z score of Thunder s length of stay at the shelter relative to the the lengths of stay for all the puppies at the shelter Round your answer to two decimal places X

Adult male chimpanzees in Gombe National Park have a population variance for weight of 132 3 kg2 In this national park chimpanzee populations are healthy and well protected Recently a new chimpanzee population has been found in marginal forests just outside the park Dr Jenna Goofball suspects that this new population is isolated and inbred and hypothesizes that it should have a lower variability for body weight compared to the one inside the park Nine 9 individuals are weighted from the new population obtaining a sample variance of 70 6 kg In the box below enter the left tail Chi square critical value from the Chi square distribution table keep in mind that this is a one tailed test

The batteries from a certain manufacturer have a mean lifetime of 860 hours with a standard deviation of 90 hours Assuming that the lifetimes are normally distributed complete the following statements a Approximately 95 of the batteries have lifetimes between hours and hours b Approximately 1130 hours of the batteries have lifetimes between 590 hours and

If b 3 X is 5 and Y is 10 what is the value of the constant O 5 5 0

rete random variable X is the number of students that show up for Professor Smith s office hours on Monday afternoons The table below shows the probability distribution for X What is the probability that at least 1 student comes to office hours on any given Monday X P X 0 1 40 30 Multiple Choice 40 50 60 2 20 3 10 Total 1 00

Ephemeral Services Corporation ESCO knows that nine other companies besides ESCO are bidding for a 900 000 government contract Each company has an equal chance of being awarded the contract If ESCO has already spent 100 000 in developing its bidding proposal what is its expected net profit Multiple Choice O O O O 90 000 0 100 000 10 000

The trade magazine QSR routinely checks the drive through service times of fast food restaurants A 90 confidence interval that results from examining 568 customers in Taco Bell s drive through has a lower bound of 153 8 seconds and an upper bound of 157 4 seconds Complete parts a through o a What is the mean service time from the 568 customers The mean service time from the 568 customers is Type an integer or a decimal Do not round seconds

Here are three histograms 60 50 40 30 20 10 0 Histogram A TH 0 2 4 6 8 10 12 14 16 18 20 22 24 60 50 40 30 20 Histogram B ind 10 0 0 2 4 6 8 10 12 14 16 18 20 22 24 a Which histogram is best described as unimodal OHistogram A OHistogram B OHistogram C b Which histogram is skewed to the right the most OHistogram A OHistogram B OHistogram C c Which histogram is the closest to being uniform OHistogram A OHistogram B OHistogram C d Which histogram is the closest to being symmetric Histogram C 60 50 40 30 20 10 0 0 2 4 6 8 10 12 14 16 18 20 22 24 X

The total blood cholesterol level was measured for each of 20 adults Here are the 20 measurements in 134 138 154 158 166 166 177 179 179 181 193 193 197 198 204 209 218 224 234 242 Send data to calculator Send data to Excel Frequency 8 6 4 N 125 2 150 00 175 200 Cholesterol level in mg dl Answer the questions below a Which measures of central tendency do not exist for this data set Choose all that apply b Suppose that the measurement 134 the smallest measurement in the data set were replaced by 50 Which measures of central tendency would be affected by the change Choose all that apply c Suppose that starting with the original data set the largest measurement were removed Which measures of central tendency would be changed from those of the original data set Choose all that apply 225 2 Mean Median Mode 250 All of these measures exist Mean Median Mode None of these measures Mean Median Mode None of these measures

d Could we have computed the confidence intervals in parts a c if the population had not been normally distributed O A No the population needs to be normally distributed B Yes the population does not need to be normally distributed OC No the population does not need to be normally distributed OD Yes the population needs to be normally distributed

In a survey conducted by a reputable marketing agency 268 of 1000 adults 19 years of age or older confessed to bringing and using their cell phone every trip to the bathroom confessions included texting and answering phone calls Complete parts a through f below Click here to view the standard normal distribution table page 1 Click here to view the standard normal distribution table page 2 a What is the sample in this study What is the population of interest Determine the sample in this study Select the correct choice below and if necessary fill in the answer box within your choice OA The sample is all adults with a cell phone B The sample is all adults OC The sample is the adults 19 years of age or older Type a whole number D The sample is all adults 19 years of age or older

Problems 3 and 4 are based on the following table which shows the stock market performance of 40 industries from five sectors of the U S economy as of noon on September 11 2015 Take S to be the set of all 40 industries represented in the table Hint Similar to Section 6 2 Exercises 47 and 51 in the book Decreased Y Financials F Manufacturing M Increased X OLUWBUD Unchanged Z Totals 4 1 8 Information Technology T Health Care H 4 Utilities U 3 Totals 24 3 10 6 1 1 3 6 1 1 3 40 8 0 6 5 14 7 3 Calculate the following probabilities for an industry selected at random from all those represented in the table a That its stock market performance increased given that it was a manufacturing industry b That it was a manufacturing industry given that its stock market performance increased industry and its stock market performance increased

A survey of 2296 adults in a certain large country aged 18 and older conducted by a reputable polling organization found that 427 have donated blood in the past two years Complete parts a through c below Click here to view the standard normal distribution table page 1 Click here to view the standard normal distribution table page 2 a Obtain a point estimate for the population proportion of adults in the country aged 18 and older who have donated blood in the past two years Round to three decimal places as needed

Construct a 99 confidence interval of the population proportion using the given information x 40 n 200 Click here to view the table of critical values The lower bound is The upper bound is Round to three decimal places as need Table of critical values Level of Confidence 1 a 100 90 95 Area in Each Tail 0 05 0 025

Determine the point estimate of the population proportion the margin of error for the following confidence interval and the number of individuals in the sample with the specified characteristic x for the sample size provided Lower bound 0 577 upper bound 0 813 n 1200 The point estimate of the population proportion is Round to the nearest thousandth as needed

A realtor is displaying the histogram below which summarizes the percentage of appreciation in value over the past five years for each of a sample of 25 houses in Newburg Park Relative frequency 0 4 0 3 0 2 0 1 0 0 0 04 10 0 16 0 24 20 30 Appreciation in percent 0 36 40 0 2 50 Based on the histogram find the proportion of appreciation percentages in the sample that are less than 10 percent Write your answer as a decimal and do not round your answer 9980 Aa

The following table gives the 52 week low and high prices the closing price and the change from the previous day for six stocks on June 23 2009 Let the universal set U consist of the six stocks listed in the table Let A contain all stocks with a high price greater than 50 and B all stocks with a closing price between 60 and 70 Describe the set ANB in words then list the elements of this set Click the icon to view the table of stocks ENCOD Describe the set AnB in words Choose the correct answer below A ANB is the set of all stocks on the list that have both a high price greater than 50 and a closing price between 60 and 70 B ANB is the set of all stocks on the list that do not have both a high price greater than 50 and a closing price between 60 and 70 OC ANB is the set of all stocks on the list that have a high price greater than 50 and do not have a closing price between 60 and 70 D ANB is the set of all stocks on the list that have either a high price greater than 50 or a closing price between 60 and 70 h

Insert nor U to make the statement true 5 7 8 5 7 8 5 7 8 Choose the correct answer below O Either n or U O Only U O Only n

Shade the Venn diagram to show the set DUF Which regions of the Venn diagram compose F Select all that apply A The region within U but outside F and D B The region in D but outside F C The region in both F and D D The region in F but outside D Use the graphing tool to shade in the set D U F Click to F LL e D

Use the union rule to answer the following question Suppose n B 5 n ANB 6 and n AUB 29 What is n A Determine which equation can be used to find n A O A n A 29 5 6 OB n A 29 5 OC n A 29 5 6 O D n A 29 5 6 n A Type a whole number

Let D 10 13 15 E 10 12 13 14 and F 9 11 12 13 15 List the elements in the set DUE NF DUE Use a comma to separate answers as needed List the elements in order with no duplicates DUE nF Use a comma to separate answers as needed List the elements in order with no duplicates

Let U universities in the United States A universities in the United States that have the name of a city in their name and B universities in the United States that have the word West in their name Describe the set in words AnB AnB is the set of universities in the United States that in their name A have the name of a city and have the word West B have the name of a city or have the word West or both OC do not have the name of a city and do not have the word West D do not have the name of a city