Probability Questions and Answers

Mr Janes can write 10 tests in 4 hours This weekend he needs to write 25 tests How long will it take him Select all of the proportions below that model this situation Note you do not need to solve for 10 tests 2 hours 25 tests 10 tests 4 hours 4 hours hours 25 tests 10 tests 25 tests 4 hours hours

n Newton s equation for universal gravitation is the force of gravity directly or inversely proportional to the square of the distance between the two masses

A coin is tossed 8 times What is the probability of getting at least one head Express your answer as a fraction or a decimal number rounded to four decimal places Ancwor E Tables F Koynar

Suppose that prices of recently sold homes in one neighborhood have a mean of 215 000 with a standard deviation of 8900 Using Chebyshev s Theorem what is the minimum percentage of recently sold homes with prices between 197 200 and 232 800 Round your answer to one decimal place

ancy distribution of the F scale intensities of recent tornadoes Use the frequency distribution to construct a frequency polygon Does the graph suggest that the distribution is skewed If so how Construct a frequency polygon using the data Choose the correct graph below Q IN OA 30 Frequency F Scale B 30 04 1 Frequency F Scale Does the graph suggest that the distribution is skewed If so how Q OA The distribution appears to be skewed to the right or positively skewed OB The distribution does not appear to be skewed OC The distribution appears to be skewed to both sides OD The distribution appears to be skewed to the left or negatively skewed D Frequency 30 I F Scale Q Tornado F Scale Frequency 01234 67231 1 D Frequency 30 0 F Scale Q

Hint s Check My Work Consider a sample with data values of 27 24 23 17 30 33 28 and 24 Compute the range interquartile range variance and standard deviation Round to 2 decimals if necessary Range Interquartile range Variance Standard deviation 16 5 00 X X

F The table below shows the frequency distribution of the rainfall on 52 consecutive Sundays in a certain city Use the frequency distribution to construct a histogram Do the data appear to have a distribution that is approximately normal Class 0 0 19 0 20 0 39 0 40 0 59 0 60 0 79 0 80 0 99 1 00 1 19 1 20 1 39 Frequency 26 6277ad 13 3 Choose the correct histogram below A B O C 30 20 10 0 0 0 0 2 0 4 0 6 0 8 1 0 1 2 1 4 30 20 10 10 0 0 02 04 06 08 10 12 14 30 20 10 0 0 0 02 04 06 08 10 12 14 C Q

data to uct the relative quency Natural births randomly selected from four hospitals in New York State occurred on the days of the week in the order of Monday through Sunday with the frequencies 54 66 72 55 53 46 54 Does it appear that such births occur on the days of the week with equal frequency Construct the relative frequency distribution Relative Frequency Day Monday Tuesday Wednesday Thursday Friday Saturday Sunday

Consider a sample with data values of 7 19 8 15 and 11 Compute the z score for each of the five observations z score for 7 z score for 19 Z score for 8 z score for 15 z score for 11

Consider a sample with data values of 20 26 28 15 30 34 24 and 26 Compute the range Compute the interquartile range Compute the sample variance Round your answer to two decimal places Compute the sample standard deviation Round your answer to two decimal places

f xxx x y for 2 x 6 02 y 24 Solve P x 2 2 X and

f x x x y for 22x26 02 y 24 Find Marginal distribution and probability X of

You are playing a game where you must roll two 4 sided dice each die can yield a value of 1 2 3 or 4 The goal is to obtain a score of at least 6 You initially roll both dice and are allowed to reroll one of them the other die value must be kept For example if you roll the dice and obtain the results of 1 and 3 you will want to reroll the die with a value of 1 to try to get a higher value You do not need to roll again if your initial roll obtains a combined score of at least 6 Find the following a The probability that you succeed without needing to reroll the dice the odds that the values sum to at least 6 when you first roll both dice b In what scenario would you be unable to succeed after your first roll think what values must be present on both dice in which rerolling one of them will never allow them to sum to at least 6 What is the likelihood of this c You discover that one of the die is rigged and will always roll a 2 Given this information what is the chance of each of the following i You succeed in any given turn your chance to obtain a total score of 6 or greater given the initial rules ii After 5 tries you succeed at least 3 times iii You succeed exactly 5 times after 7 tries

Briefly explain the error in the construction of the ogive below which illustrates the average quantitative GRE scores to the top 30 graduate schools of engineering Cumulative Frequency 2 25 20 15 10 S 0 744 756 770 777 784

Let s assume that the function f x y c 3x 2y represents a joint probability mass function for x 1 3 5 and y 2 4 6 Determine a The value of c b P X 3 5 Y 3 c P X 3 d E X e V Y f Marginal probability distribution of X g Are X and Y independent Why or why not

n average a certain intersection results in an average of 1 3 traffic accidents pe etermine the probability of the following situations for any given week a Exactly I accident occurs b No more than 2 accidents occur c At least 3 accidents occur

Forty plasma TVs were tested and the number of watts per hour they used recorded Given that the lowest value is 429 find the class limits to be used in the construction of a frequency distribution for the data with 6 classes and a class width of w 13 Begin the first class with the smallest data value given above and write the limits for the classes in the table provided in the answer box below

4 2 2 5 In example of lecture 5 Sec 2 2 An enterprising man proposes a game let the player throw a die Let the reward becomes Z u X X a Find the pmf of Z say h z b Find E Z c How much on average can the enterprising man expect to win on each play from players if he charges 10 dollars per play

Assume that the U S Mint manufactures dollar coins so that the standard deviation is 0 0390 g The accompanying list contains weights grams of dollar coins manufactured with a new process designed to decrease the standard deviation so that it is less than 0 0390 g This sample has these summary statistics n 16 x 8 066 g s 0 02 g A 0 05 significance level is used to test the claim that the sample is from a population with a standard deviation less than 0 0390 g The null and alternative hypotheses are Ho o 0 0390 g and H o 0 0390 g For this sample data we get a P value of 0 0099 when testing the claim that o 0 0390 g Complete parts a through c a What should we conclude about the null hypothesis 4 O A Accept the null hypothesis OB Fail to reject the null hypothesis OC Fail to accept the null hypothesis OD Reject the null hypothesis

Use the accompanying 200 Los Angeles commute times to test the claim that the mean Los Angeles commute time is less than 32 minutes Assume that a simple random sample has been selected Use a 0 10 significance level Compare the sample mean to the claimed mean of 32 minutes Is the difference between those two values statistically significant Click the icon to view the Los Angeles commute times Identify the null and alternative hypotheses Ho 32 H 32 Type integers or decimals Do not round Identify the test statistic Round to two decimal places as needed 4

In the following problem check that it is appropriate to use the normal approximation to the binomial Then use the normal distribution to estimate the requested probabilities Do you take the free samples offered in supermarkets About 58 of all customers will take free samples Furthermore of those who take the free samples about 39 will buy what they have sampled Suppose you set up a counter in a supermarket offering free samples of a new product The day you were offering free samples 309 customers passed by your counter Round your answers to four decimal places USE SALT a What is the probability that more than 180 will take your free sample b What is the probability that fewer than 200 will take your free sample c What is the probability that a customer will take a free sample and buy the product Hint Use the multiplication rule for dependent events Notice that we are given the conditional probability P buy sample 0 39 while P sample 0 58 0 2262 d What is the probability that between 60 and 80 customers will take the free sample and buy the product Hint Use the probability of success calculated in part c

A survey about social media reported that 77 of B2B marketers marketers that focus primarily on attracting businesses plan to increase their use of social media as compared to 56 of B2C marketers marketers that primarily target consumers The survey was based on 1 426 B2B marketers and 1 618 B2C marketers The accompanying table summarizes the results Complete parts a through d below Click the icon to view the contingency table about social media use and marketers a What is the probability that a randomly selected respondent plans to increase use of social media Round to three decimal places as needed U P O DE E SE ELEV

The height in feet of a football that is kicked can be modeled by the function 16a 64a where is the time in seconds after it is kicked for what time is the ball at or above a height of 48 feet above the ground

7 The height in feet of a football that is kicked can be modeled by the function f a 16z 6z where is the time in seconds after it is kicked For what time s is the ball at or above a height of 48 feet above the ground

weight all 26 5 ounces with a 95 of data range from 16 to 37 ounces Let x be a random variable that represents the weight in ounces of a healthy 10 week old kitten Assume that x has a distribution that is approximately normal USE SALT a The empirical rule indicates that for a symmetric and bell shaped distribution approximately 95 of the data lies within two standard deviations of the mean 20 Therefore a 95 range of data values extending from 20 to 4 20 is often used for commonly occurring data values Note that the interval from 20 to 20 is 40 in length This leads to a rule of thumb for estimating the standard deviation from a 95 range of data values 95 Estimating the standard deviation For a symmetric bell shaped distribution standard deviation H 20 where it is estimated that about 95 of the commonly occurring data values fall into this range X range high value low value Estimate the standard deviation of the x distribution in ounces Round your answer to two decimal places 5 25 oz b What is the probability that a healthy 10 week old kitten will weigh less than 16 ounces Round your answer to four decimal places x c What is the probability that a healthy 10 week old kitten will weigh more than 34 ounces Round your answer to four decimal places d What is the probability that a healthy 10 week old kitten will weigh between 16 and 34 ounces Round your answer to four decimal places e A kitten whose weight is in the bottom 10 of the probability distribution of weights is called undernourished What is the cutoff point for the weight of an undernourished kitten in ounces Round your answer to the nearest whole number

The profit in dollars from the sale of x lawn mowers is P x 30x 0 03x 650 A Find the average profit per mower if 50 mowers are produced B Find the marginal average profit at a production level of 50 mowers and interpret the results C Use the results from parts A and B to estimate the average profit per mower if 51 mowers are produced A If 50 mowers are produced the average profit per mower is Round to the nearest cent as needed B The marginal average profit at a production level of 50 units is per mower Round to the nearest cent as needed Choose the correct interpretation of P 50 0 23 per mower OA At a production level of 50 mowers the average profit is increasing at a rate of 0 23 per mower OB At a production level of 50 mowers the average profit is 0 23 per mower Oc At a production level of 50 mowers a unit increase in production will increase the total profit by 0 23 per mower 4 C Using parts A and B the estimate of the average profit per mower if 51 mowers are produced is Round to the nearest cent as needed

Isabella and Antonio are auto mechanics Isabella takes 8 hours to replace a clutch and 6 hours to replace a set of brakes Antonio takes 8 hours to replace a clutch and 4 hours to replace a set of brakes State whether anyone has an absolute advantage at either task and for each task identify who has a comparative advantage Instructions Enter your responses rounded to two decimal places The opportunity cost of replacing a set of brakes for Isabella is The opportunity cost of replacing a set of brakes for Antonio is has a comparative advantage in brake replacement has a comparative advantage in clutch replacement has an absolute advantage in brake replacement has an absolute advantage in clutch replacement Neither Neither Click to select Click to select

Use the Measures of Center flow chart interactive to answer the following question A data set consists of a list of eye colors from 250 randomly selected statistics students Which measure of center appears to be best Choose the correct answer below OA Median OB Mode OC Midrange KERUD

Use the Measures of Center flow chart interactive to answer the following question A data set consists of 100 amounts of annual income dollars and there are two outliers that are exceptionally high Which measure of center appears to be best Choose the correct answer below O A Mode OB Midrange O C Mean OD Median

Which of the ng is always true Choose the correct answer below CONCHHICHCHAD A In a symmetric and bell shaped distribution the mean median and mode are the same B Data skewed to the right have a longer left tail than right tail OC For skewed data the mode is farther out in the longer tail than the median D The mean and median should be used to identify the shape of the distribution

Use the magnitudes Richter scale of the earthquakes listed in the data set below Find the mean and median of this data set Is the magnitude of an earthqual measuring 7 0 on the Richter scale an outlier data value that is very far away from the others when considered in the context of the sample data given in this data set Explain Click the icon to view the earthquake Richter scale data CHERCHID Find the mean and median of the data set using a calculator or similar data analysis technology The mean of the data set is Round to three decimal places as needed

Find the mean of the data summarized in the given frequency distribution Compare the computed mean to the actual mear of 50 3 miles per hour 58 61 C Speed miles per hour 1 Frequency 42 45 27 The mean of the frequency distribution is miles per hour Round to the nearest tenth as needed 46 49 13 50 53 6 54 57 4

A defunct website listed the average annual income for Florida as 35 031 What is the role of the term average in statistics Should another term be used in place of average Choose the correct answer below O A The term average is often used in statistics to represent the mean OB The term average is not used in statistics The term median should be used for the result obtained by adding all of the sample values and dividing by the total number of sample values OC The term average is often used in statistics to represent the median O D The term average is not used in statistics The term mean should be used for the result obtained by adding all of the sample values and dividing by the total number of sample values

Find x in the following equalities a 12 Cx 12 C2x 3 b P x 2 90 c C 3p x 3

BFW Publishers All human blood can be typed as one of O A B or AB but the distribution of the types varies with race Here is the distribution of blood types of Black Americans Suppose we choose one Black American at random Blood Type 0 A B AB Probability 0 49 0 27 0 20 0 04 Rolanda has type B blood She can safely receive blood transfusions from people with blood types O and B a Find the probability that a randomly chosen Black American can donate blood to Rolanda Use two decimal places b Find the probability that a randomly chosen Black American cannot donate blood to Rolanda Use two decimal places

BFW Publishers How is the hatching of water python eggs influenced by the temperature of the snake s nest Researchers randomly assigned newly laid eggs to one of three water temperatures hot neutral or cold Hot duplicates the extra warmth provided by the mother python and cold duplicates the absence of the mother The two way table summarizes the data on whether or not the eggs hatched Of the eggs that hatched what proportion were randomly assigned to hot water Round to 3 decimal places Hatched Cold Yes 16 No 11 Total 27 Water Temperature Neutral Hot Total 38 75 129 18 29 58 56 104 187 Proportion R Shine et al The influence of nest temperatures and maternal brooding on hatchling phenotypes in water pythons Ecology 78 1997 pp 1713 1721

OBFW P The Pew Research Center asked a random sample of 2024 adult cell phone owners from the United States their age and which type of cell phone they own iPhone Android or other including non smartphones The two way table summarizes the data Type of cell phone Age 35 54 55 Total 127 467 18 34 iPhone 169 171 Android 214 189 100 503 134 277 643 Other 1054 Total 517 637 870 2024 Are the events iPhone and 18 34 independent Justify your answer De P iPhone and 18 34 0 083 and P iPhone and not 1834 0 147 Because these two probabilities are not equal the events iPhone and 18 34 are not independent P iPhone 0 231 and P iPhone 18 34 0 083 Because these two probabilities are not equal the events iPhone and 18 34 are not independent P iPhone 0 231 and PiPhone 18 34 0 327 Because these two probabilities are not equal the events iPhone and 18 34 are independent P iPhone 0 231 and PiPhone 18 34 0 327 Because these two probabilities are not equal the events iPhone and 18 34 are not independent

estion 3 of 13 BFW Publishers Almost everyone has one a drawer that holds miscellaneous batteries of all sizes Suppose that your drawer contains 8 AAA batteries but only 6 of them are good You have to choose 4 for your graphing calculator If you randomly select 4 batteries what is the probability that all 4 of them will work Use combinations to help answer this question Round your answer to 3 decimal places

of foreign tourists who travel to Canada and the United States The primary focus of Safe Trip is to act as an interface between local health care providers and the overseas insurance company that insured the tourist should the traveller find him or herself in need of medical attention Currently Safe Trip considers increasing the processing fee collected partly from medical providers and partly from the foreign insurers Although this would generate additional revenue it would also incur the risk of losing contracts with insurers and health care providers The Safe Trip estimates of the impact of the change in rates on the annual profit in millions depending on where from most of the tourists come According to the company s research the probability of most tourists coming not from Europe is about 0 39 and the probability of most tourists coming from Europe is approximately 0 61 Suppose that Safe Trip has decided to hire a professional analyst who will provide additional information about number of potential tourists from Europe Based on the analyst s past record the company has estimated 0 74 probability that the analyst would present a favorable report given most of the tourists come from Europe and 0 94 probability that the analyst would present an unfavorable report given not most of the tourists come from Europe a If the analyst provides a favorable report how then would the Safe Trip estimate the probability of most tourists coming from Europe Find the answer using the following revised probability table Please round the values of joint posterior probabilities and probability of favorable report to four decimal places State of Nature not Europe Europe Probability Revisions given a Favorable Report State of Nature Prior Conditional Probability Joint Posterior Probability Probability P Fav State of Nature Probability P State of Nature Fav P Most of the tourists come from Europe Favorable report not Europe Europe P Favorable report b If the analyst provides an unfavorable report how then would the Safe Trip estimate the probability of most tourists coming from Europe Find the answer using the following revised probability table Please round the values of joint posterior probabilities and probability of unfavorable report to four decimal places 1 0000 Probability Revisions given an Unfavorable Report Conditional Probability Joint Posterior Probability Prior Probability P Unfav State of Nature Probability P State of Nature Unfav P Unfavorable report 1 0000

Determine whether the study is an experiment or an observational study and then nuly a major problem with t study A medical researcher tested for a difference in systolic blood pressure levels between male and female students w are 12 years of age She randomly selected four males and four females for her study This is an experiment because the researcher observational study the individuals

described below A woman experienced a tax audit The tax department claimed that the woman was audited because she was randomly selected from all women in her age group Which type of sampling did the tax department use Stratified sampling O Systematic sampling Random sampling Cluster sampling Convenience sampling

K The diameter of a brand of tennis balls is approximately normally distributed with a mean of 2 72 inches and a standard deviation of 0 04 inch A random sample of 11 tennis balls is selected Ca through d below a What is the sampling distribution of the mean A Because the population diameter of tennis balls is approximately normally distributed the sampling distribution of samples of size 11 will also be approximately normal B Because the population diameter of tennis balls is approximately normally distributed the sampling distribution of samples of size 11 cannot be found C Because the population diameter of tennis balls is approximately normally distributed the sampling distribution of samples of size 11 will be the uniform distribution OD Because the population diameter of tennis balls is approximately normally distributed the sampling distribution of samples of size 11 will not be approximately normal b What is the probability that the sample mean is less than 2 71 inches P X 2 71 0 2035 Round to four decimal places as needed c What is the probability that the sample mean is between 2 70 and 2 74 inches P 2 70 x 2 74 0 9027 Round to four decimal places as needed d The probability is 68 that the sample mean will be between what two values symmetrically distributed around the population mean The lower bound is 2 71 inches The upper bound is Round to two decimal places as needed inches

In a study of a sample of babies born at hospitals in one state it was found that 49 of the babies were girls Identify whether this value is a statistic or a par Choose the correct answer below OA The value is a statistic because it describes some characteristic of a population OB The value is a parameter because it describes some characteristic of a population OC The value is a parameter because it describes some characteristic of a sample OD The value is a statistic because it describes some characteristic of a sample

When a car is randomly selected it is found to have an engine with 4 cylinders Choose the correct answer below OA A continuous data set because there are infinitely many possible values and those values can be counted OB The data set is neither continuous nor discrete OC A discrete data set because there are a finite number of possible values OD A continuous data set because there are infinitely many possible values and those values cannot be counted

users this question was posted online Have you ever been hit by a computer virus Among the 170 063 responses 63 answered yes What term is used to describe this t of survey in which the people surveyed consist of those who chose to respond What is wrong with this type of sampling method What term is used to describe this type of survey Select all that apply A The respondents are a voluntary response sample B The respondents are a self selected sample C The respondents are a population D The respondents are a census What is wrong with this type of sampling method Select all that apply A The survey question is loaded or intentionally worded to elicit a desired response B It is too time consuming C Many people may choose not to respond to the survey D Responses may not reflect the opinions of the general population E It is too expensive CIB

6 Consider a game where you have five distinct coins in a row and you are allowed to move them in pairs and insert them between other coins in the row without changing the order of the pair For example here is an allowable move As above you can slide the coins over horizontally to create space to insert the pair or to remove extra space as long as you do not change the order of the coins in the row a Suppose the coins are placed in order of increasing value is there a sequence of moves as described above that reverses the order so they are in order of decreasing value b Again suppose the coins are placed in order of increasing value is there a sequence of moves that switches the first coin with the last and where the middle three end up in their starting positions

The results of a common standardized test used in psychology research is designed so that the population mean is 115 and the standard deviation is 20 A subject earns a score of 123 What is the z score for this raw score Z score

The Wall Street Journal reported that approximately 25 of the people who are told a product is improved will believe that it is in fact improved The remaining 75 believe that this is just hype the same old thing with no real improvement Suppose a marketing study consists of a random sample of eleven people who are given a sales talk about a new improved product USE SALT a Make a histogram showing the probability that r 0 to 11 people believe the product is in fact improved 0 25 0 30 0 25 0 20 P r P r 0 20 0 0 15 0 10 0 05 0 00 0 6 0 5 0 4 0 3 0 2 0 1 0 0 20 MLA 0 1 2 3 4 5 6 7 8 9 10 11 r 0 1 2 3 4 5 6 7 8 9 10 11 r people people P r P r 0 15 0 10 0 05 0 00 0 25 0 20 0 15 0 10 0 05 0 00 0 1 2 3 4 5 6 7 8 9 10 11 T 0 1 2 3 4 5 6 7 8 9 10 11 b Compute the mean and standard deviation of this probability distribution Round your standard deviation to two decimal places H T c What is the smallest number of people needed in the marketing study to be at least 99 sure that at least one person believes the product to be improved Hint Note that P r 1 0 99 is equivalent to 1 P 0 0 99 or P 0 0 01 people

Using the following table what is the probability that if you randomly selected a student that student is between ages 14 17 and skips breakfast Students ages 10 13 Students ages 14 17 Totals Select one O a 48 b 27 O c 56 O d 88 Eat Breakfast 40 12 47 Skip Breakfast 14 24 43 Totals 54 36 90

from the set 3 4 5 82 Which event by definition covers the entire sample space of this experiment Choose Choose The number is neither prime nor composite Please answer all parts of the questi The square root of the number is less than 3 The number is divisible by 3 1 and loss than 99