Probability Questions and Answers

Suppose 16 coins are tossed Find the probability of getting the following result using the binomial probability formula and the normal curve approxim Exactly 4 heads Use the table of areas under the standard normal curve given below Click here to view page 1 Click here to view page 2 Click here to view page 3 Click here to view page 4 Click here to view page 5 Click here to view page 6 Binomial probability Round to 4 decimal places

A certain type of light bulb has an average life of 1900 hours with a standard deviation of 50 hours The length of life of the bulb can be closely approximated by a normal curve An amusem buys and installs 30 000 such bulbs Find the total number that can be expected to last less than 1960 hours Click here to view page 1 of the Areas under the Normal Curve Table Click here to view page 2 of the Areas under the Normal Curve Table The number of light bulbs that can be expected to last less than 1960 hours is BEEB

Find the percent of the total area under the standard normal curve between the following z scores This problem can be solved using technology or the table for the standard normal cur text Click here to view page 1 of the table Click here to view page 2 of the table z 1 1 and z 0 8 The percent of the total area between z 1 1 and z 0 8 is Round to the nearest whole percent as needed C

7 10 15 17 26 The mean is Round to the nearest tenth as needed

for normal distributions where 0 oro 1 hoose the correct answer below OA Az score is found with the formula z A OB Az score is found with the formula z OC Az score is found with the formula z A X H OD A z score is found with the formula z x b PIO X H X

Find the range and standard deviation of the set of numbers 34 28 38 36 34 29 32 The range is

certain distribution of numbers the mean is 70 with a standard deviation of 4 Chebyshev s rem states that for any set of numbers the fraction that will lie within k standard deviations of 1 mnean for k 1 is at least 1 Use Chebyshev s theorem to tell the probability that a k ber lies between 58 and 82 The probability a number lies between 58 ar Round to the nearest thousandth as needec

nd the median for the following list of numbers 103 118 131 139 148 164 he median is Type an integer or a decimal

4 Suppose that the number of cars visiting New Paltz has Poisson distribution with a parameter A 0 There are three types of cars diesel sedan and truck The probability of randomly chosen car is a diesel sedan or truck is P P2 and p3 with p P2 P3 1 and the event of each car being a certain type is independent of each other Let X Y Z be numbers of diesel cars sedans or trucks during one day a Suppose that there are n distinct items We want to divide n items into three groups group A B and C with a b and c elements with a b c n Show that the number of different grouping is c n a b c alxblxc b Show that a b c a b c P1P2P3 X a b c a b c 1

The shelf life of a battery produced by one major company is known to be Normally distributed with a mean life of 3 years and a standard deviation of 0 9 years What value of shelf life do 16 of the battery shelf lives fall below Round your answer to one decimal place Answer year

Complete parts a c y X y y 1 2 2 1 3 0 a Find a cubic function that models the data in the table Report the model with three decimal pla Linear 4 4 Cubic 5 8 b Find a linear function that models the data Report the model with three decimal places 6 16 c Visually determine which model is the better fit for the data

D Cost Model Find a linear regression model for the weekly cost data using as the independent variable C x mx k Round m to 1 decimal place and round to the nearest integer Use this cost model to answer questions 9 15 below Question 9 Use the cost model above to answer this question What is m Round to 1 decimal place 2 pts

Revenue Model Use the price model above to construct a revenue model Find a model for the weekly revenue using a as the independent variable NOTE Do not calculate another regression Use the price equation to find a model for revenue R x p z R z p z a bx cz z az bz cz Use this revenue model to answer questions 3 8 below Question 3 Use the revenue model above to answer this question What is a Round to the nearest integer 3 pts

Show that T PP defined by T p x p x p x is an isomorphism Since the dimensions of the domain and codomain are the same it suffices to show that T is one to one Let p x a a x ax Then p x p x a a a 2a x x 1 x But then the coefficient of x 1 is 0 an 1 so that an 1 If p x p x 0 then the coefficient of x is Select so that a a for all i Thus p x so that T is one to one and is thus an isomorphism We continue this process determinim

In a particular survey of internet users 3633 respondents say that they use social networking sites and 1402 respondents say that they do not use social networking sites What is the probability that a randomly selected person uses a social networking site Does that result suggest that it is likely with a probability of 0 5 or greater for someone to use social networking sites The probability that a randomly selected person uses a social networking site is Round to three decimal places as needed

Households are randomly selected and partitioned into groups of four For those groups the random variable x is the number of households with a printer Determine whether a probability distribution is given If a probability distribution is given find its mean and standard deviation If a probability distribution is not given identify the requirements that are not satisfied Does the table show a probability distribution Select all that apply 0 A Yes the table shows a probability distribution B No not every probability is between 0 and 1 inclusive C No the numerical values of the random variable x are not associated with probabilities D No the random variable x is categorical instead of numerical E No the sum of all the probabilities is not equal to 1 0 1 2 3 4 0 145 0 335 0 357 0 138

Groups of adults are randomly selected and arranged in groups of three The random variable x is the number in the group who say that they would feel comfortable in a self driving vehicle Determine whether a probability distribution is given If a probability distribution is given find its mean and standard deviation If a probability distribution is not given identify the requirements that are not satisfied X 0 1 2 3 P x 0 363 0 427 0 183 0 027 Does the table show a probability distribution Select all that apply A Yes the table shows a probability distribution B No not every probability is between 0 and 1 inclusive C No the random variable x s number values are not associated with probabilities D No the sum of all the probabilities is not equal to 1 E No the random variable x is categorical instead of numerical

Mr Sanderson marks his class on a normal curve Those with z scores above 2 2 will receive an A those between 2 2 and 0 8 will receive a B those etween 0 8 and 0 9 will receive a C those between 0 9 and 2 2 will receive a D and those under 2 2 will receive an F Determine the percent o rades that will be A B C D and F Click here to view page 1 of the standard normal distribution table Click here to view page 2 of the standard normal distribution table of the grades are A s Round to two decimal places as needed

Use the set of data below to do the following a Draw a scatter diagram b Find the value of r c Determine whether a correlation exists at 0 05 d Determine whether a correlation exists at x 0 01 Click the icon to view the correlation coefficient r for various values of n and a X 23 34 32 42 50 y 25 36 27 32 38 The formula for the correlation coefficient is n Exy Ex Ey r n Ex Ex n Ey Ey C a Choose the scatter plot that best represents the given data OA Ay 50 40 30 20 10 0 O C Ay 50 40 30 20 10 PIT 0 O 0 8 16 24 32 40 O O 00 O X O 60 A 0 10 20 30 40 50 X 2 Q OB Ay 40 30 20 10 ott D Ay 0 10 2030 40 50 40 O b O 30 20 10 0 O O d o d O Po X 0 10 20 30 40 50 b The value of ris Round to the nearest thousandth as needed N X

A 95 percent confidence interval for the true mean time spent preparing and recording a lecture is reported to be 75 to 95 minutes The point estimate for the true mean is Multiple Choice 90 minutes 80 minutes 85 minutes impossible to determine

The research department of a board game company surveyed some shoppers at a local mall The team asked shoppers how often they read the news among other questions Then the team played a memory card game with the shoppers They recorded how many cards each shopper remembered 0 times a week 1 2 times a week 1 3 cards 4 6 cards 7 5 2 3 What is the probability that a randomly selected shopper remembered 1 3 cards given that the shopper reads the news 0 times a week Simplify any fractions

4 18 Matching Pick the letter of the graph that goes with each numerical value listed below for the correlation Correlations 0 903 0 374 0 777 A Acreage 6 5 4 3 2 1 1 2 3 4 567 Full Bath

4 7 Sisters and Brothers The scatterplot shows the numbers of brothers and sisters for a large number of students Do you think the trend is somewhat positive or somewhat negative What does the direction positive or negative of the trend mean Does the direc tion make sense in this context Number of Brothers 9876543210

4 4 Shoe Size and GPA The figure shows a scatterplot of shoe size and GPA for some college students Does it show an increasing trend a decreasing trend or no trend Is there a strong relationship GPA 4 0 3 5 3 0 2 5 2 0 5 6 7 8 9 Shoe Size 10 11 12

Proceed as in Example 4 in Section 11 3 to find a particular solution xo t of equation 11 in Section 11 3 kx f t 11 m xp t dt k when m k 4 and the driving force f t is as given Assume that when f t is extended to the negative t axis in a periodic manner the resulting function is even f t 2mt 1 0 t 2m f t 2 f t 2 1 n 1

The Abco Company manufactures electrical assemblies The current process uses 10 workers and produces 200 units per hour You are considering changing the process with new assembly methods that increase output to 300 units per hour but will require 14 workers Workers are paid at a rate of 10 per hour Finished switches sell for 20 unit Using the information provided in Table below what is the multifactor productivity of the current process and the new process Based on your productivity calculations what decision should you make Should you go with the new process and why Show all your solution work and answers CURRENT PROCESS OUTPUT UNITS HOUR 200 NUMBER OF WORKERS 10 MATERIAL COST HOUR 120 NEW PROCESS 300 14 150

Consider the function C f x 1 0 x 1 1 20 0 x 1 0 Find the half range cosine expansion of the given function 1 200 f x n 1 Find the half range sine expansion of the given function f x 1 2 n 1

3 58 of students at a certain college are female A random sample of 50 students is to be selected Let the random variable X be the number of female students in a sample Round to 4 decimal places a What is the probability that exactly 49 of the students are female 4 pts b What is the probability that 30 or fewer of the students are female 4 pts

4 Assume that the finishing times in a New York City 10 kilometer road race are normally distributed with a mean of 63 minutes and a standard deviation of 8 minutes If one entrant in the race is randomly selected what is the probability that the entrant s finishing time is 66 minutes or more 6 pts Round to 4 decimal places

5 To qualify for security officers training recruits are tested for stress tolerance The scores are normally distributed with a mean of 72 and a standard deviation of 9 If only the top 10 of recruits are selected find the cut off score for qualifying 6 pts Round to the nearest whole number

In 2012 the U S Food and Drug Administration approved the first over the counter HIV testing kit the OraQuick In Home HIV Test The sensitivity and specificity of a diagnostic test are determined experimentally The OraQuick test was approved based on a study that examined the test results of a sample of 4410 adults from a high risk population who did not yet know their HIV status Each person s actual HIV status was later confirmed via several other medical examinations The table contains the findings 105 true positives HIV and positive test 8 false negatives HIV and negative test 1 false positive no HIV and positive test 4296 true negatives no HIV and negative test Macmillan Learning What is the specificity P negative test no HIV of OraQuick This represents the fraction of all individuals with no HIV who received a negative test result Enter you answer rounded to four decimal places P negative test no HIV

Hemophilia refers to a group of rare hereditary disorders of blood coagulation Because the disorder is caused by defective genes on the X chromosome hemophilia affects primarily men According to the Centers for Disease Control and Prevention CDC the prevalence of hemophilia the number in the population with hemophilia at any given time among American males is 13 in 100 000 What does this mean The prevalence of 13 in 100 000 means that if you repeatedly sampled 100 000 males on average you would have 13 individuals with diagnosed with hemophilia per sample For one random sample of 100 000 males you hemophilia

A large governmental survey of dental care among children 2 to 17 years old found the distribution of times since the last visit to the dentist Time since last visit 6 months or less More than 6 months but no more than 1 year More than 1 year but no more than 2 years More than 2 years but no more than 5 years More than 5 years P more than 5 years Probability 0 57 P not within last 6 months 0 18 0 08 0 03 a What is the probability that a child has not seen a dentist in more than five years Enter your answer rounded to two decimal places b What is the probability that a randomly chosen child has not seen a dentist within the last six months Enter your answer rounded to two decimal places c What is the probability that the child has seen a dentist within the last year Enter your answer rounded to two decimal places

An article reported that for a sample of 41 kitchens with gas cooking appliances monitored during a one week period the sample mean CO level ppm was 654 16 and the sample standard deviation was 166 25 a Calculate and interpret a 95 two sided confidence interval for true average CO level in the population of all homes from which the sample was selected Round your answers to two decimal places 601 69 X I 706 63 X ppm Interpret the resulting interval We are 95 confident that this interval contains the true population mean We are 95 confident that this interval does not contain the true population mean We are 95 confident that the true population mean lies below this interval We are 95 confident that the true population mean lies above this interval b Suppose the investigators had made a rough guess of 179 for the value of s before collecting data What sample size would be necessary to obtain an interval width of 40 ppm for a confidence level of 95 Round your answer up to the nearest whole number kitchens

A sample of ultimate tensile strength observations ksi was taken Use the accompanying descriptive statistics output from Minitab to calculate a 99 lower confidence bound for true average ultimat tensile strength and interpret the result Round your answer to two decimal places N 153 Mean 131 36 Median 131 37 TrMean 131 38 St Dev 4 59 SE Mean 0 37 Minimum 123 10 Maximum 148 70 We are 99 confident that the true average ultimate tensile strength is greater than 131 28 You may need to use the appropriate table in the Appendix of Tables to answer this question Q1 132 75 X ksi Q3 139 45

A graduate student wanted to estimate the average time spent studying among graduate students at her school She randomly sampled graduate students from her school and obtained a 99 confidence interval of 17 3 22 5 hours week In the context of the problem which of the following interpretations is correct Choose the correct answer below OA Graduate students at this student s school study between 17 3 and 22 5 hours per week 99 of all weeks during the year OB There is a 99 chance that the average amount of time spent studying for all graduate students at this student s school is between 17 3 and 22 5 hours per week OC Approximately 99 of all random samples of graduate students from this school will give a mean number of hours studied per week between 17 3 and 22 5 OD We are 99 sure that the average amount of time spent studying among graduate students at this student s school is between 17 3 and 22 5 hours per week OE At this student s school 99 of all graduate students study between 17 3 and 22 5 hours per week on average te ter tem tem temp

A 98 confidence interval for the proportion of people who performed volunteer work in the past year is Part 2 3 Part 3 of 3 X p c A sociologist states that 45 of Americans perform volunteer work in a given year Does the confidence interval contradict this statement Explain The confidence interval Choose one contradict the claim because 0 45 Choose one contained in the confidence interval Espa

Testing testing An instructor believes that the mean time that students will spend on the take home test is 8 hours A test is made of H 8 versus H 8 The null hypothesis is rejected State an appropriate conclusion There Choose one enough evidence to conclude that the mean time spent is Choose one 8 hours 13 X

Part 0 5 7 Part 1 of 5 8 9 X 10 11 SAT scores Assume that in a given year the mean mathematics SAT score was 467 and the standard deviation was 105 A sample of 59 scores is chosen Use Excel a What is the probability that the sample mean score is less than 455 Round the answer to at least four decimal places The probability that the sample mean score is less than 455 is 12 13

c Find the probability that the proportion of individuals in the sample of 83 who have high blood pressure is between 0 20 and 0 32 Round the answer to at least four decimal places The probability that the proportion of individuals in the sample of 83 who have high blood pressure is between 0 20 and 0 32 is 0 6311 Part 3 5 Part 4 of 5 d Find the probability that less than 21 of the people in the sample of 83 have high blood pressure Round the answer to at least four decimal places The probability that less than 21 of the people in the sample of 83 have high blood pressure is 8 M ve

Send data to Excel Part 1 of 2 a Construct a 95 confidence interval for the mean weight Round the answers to at least three decimal places A 95 confidence interval for the mean weight is 14 0555 u 14 1283 Part 1 2 Part 2 of 2 b The quality control manager is concerned that the mean weight is actually less than 14 ounces Based on the confidence interval is there a reason to be concerned Explain The quality control manager should Choose one concerned that the mean weight is actually less than 14 ounces X S

Internet tax The Gallup Poll asked 1033 U S adults whether they believed that people should pay sales tax on items purchased over the internet Of these 434 said they supported such a tax Does the survey provide convincing evidence that less than 46 of U S adults favor an internet sales tax Use the a 0 10 level of significance and the P value method with the TI 84 Plus calculator Part 0 5 Part 1 of 5 State the appropriate null and alternate hypotheses Ho H 0 0 0 0 0 P 8 20

TV sets According to the Nielsen Company the mean number of TV sets in a U S household in 2013 was 2 24 Assume the standard deviation is 1 4 A sample of 90 households is drawn Part 1 of 5 Question Attemp For a What is the probability that the sample mean number of TV sets is greater than 2 Round your answer to at least four decimal places The probability that the sample mean number of TV sets is greater than 2 is 0 9481 Part 1 5 Part 2 of 55 b What is the probability that the sample mean number of TV sets is between 2 5 and 3 Round your answer to at least four decimal places The probability that the sample mean number of TV sets is between 2 5 and 3 is m

1 3 4 5 6 7 8 9 10 11 There Choose one enough evidence to conclude that the mean number of days is Choose one 50 12 Business trips A sales manager believes that the mean number of days per year her company s sales representatives spend traveling is different than 50 A test is made of Ho H 50 versus H 50 The null hypothesis is rejected State an appropriate conclusion X 13 5 Espa ol 14 8E8

Part 0 4 Part 1 of 4 State the appropriate null and alternate hypotheses Ho H This hypothesis test is a Choose one test Height and age Are older men shorter than younger men According to a national report the mean height for U S men is 69 4 inches In a sample of 119 men between the ages of 60 and 69 the mean height was x 69 3 inches Public health officials want to determine whether the mean height u for older men is less than the mean height of all adult men Assume the population standard deviation to be a 2 58 Use the a 0 05 level of significance and the P value method with the TI 84 calculator O 0 0 0 0 13 3 14 15 16 17 Espa o 18 8 Co E 9 d

5 Part 0 5 6 Part 1 of 5 7 8 9 a State the null and alternate hypotheses 10 Ho H This hypothesis test is a Choose one test 11 0 0 0 0 0 12 Tweet tweet An article reported that 73 of companies have Twitter accounts An economist thinks the percentage is higher at technology companies She samples 70 technology companies and finds that 57 of them have Twitter accounts Can she conclude that more than 73 of technology companies have Twitter accounts Use the a 0 05 level of significance with the P value method and Excel P 13 14 15 16 17 Espa ol 18 8 E 1 dla E

Part 0 5 Part 1 of 5 e Blood pressure High blood pressure has been identified as a risk factor for heart attacks and strokes The proportion of U S adults with high blood pressure is 0 30 A sample of 31 U S adults is chosen Use Excel as needed 12 Choose one 10 13 a Is it appropriate to use the normal approximation to find the probability that more than 47 of the people in the sample have high blood pressure If so find the probability If not explain why not It Choose one appropriate to use the normal curve since np 14 8 h 244

The FDA regulates that a fish that is consumed is allowed to contain at most 1 mg kg of mercury In Florida bass fish were collected in 51 different lakes to measure the amount of mercury in the fish from each of the 51 lakes Do the data provide enough evidence to show that the fish in all Florida lakes have a mercury level higher than the allowable amount State the random variable population parameter and hypotheses a The symbol for thiandom variable involved in this problem is The wording for the random variable in context is as follows b The symbol for the parameter involved in this problem is X The wording for the parameter in context is as follows c Fill in the correct null and alternative hypotheses Ho 1 X the mercury level in fish of a randomly selected Florida lake the mean mercury level in fish of all Florida lakes

sample of 13 joint specimens of a particular type gave a sample mean proportional limit stress of 8 41 MPa and a sample standard deviation of 0 73 MPa a Calculate and interpret a 95 lower confidence bound for the true average proportional limit stress of all such joints Round your answer to two decimal places MPa Interpret this bound With 95 confidence we can say that the value of the true mean proportional limit stress of all such joints is centered around this value With 95 confidence we can say that the value of the true mean proportional limit stress of all such joints is greater than this value With 95 confidence we can say that the value of the true mean proportional limit stress of all such joints is less than this value What if any assumptions did you make about the distribution of proportional limit stress O We must assume that the sample observations were taken from a chi square distributed population We must assume that the sample observations were taken from a uniformly distributed population We do not need to make any assumptions We must assume that the sample observations were taken from a normally distributed population b Calculate and interpret a 95 lower prediction bound for proportional limit stress of a single joint of this type Round your answer to two decimal places MPa Interpret this bound If this bound is calculated for sample after sample in the long run 95 of these bounds will provide a lower bound for the corresponding future values of the proportional limit stress of a single joint of this type If this bound is calculated for sample after sample in the long run 95 of these bounds will provide a higher bound for the corresponding future values of the proportional limit stress of a single joint of this type If this bound is calculated for sample after sample in the long run 95 of these bounds will be centered around this value for the corresponding future values of the proportional limit stress of a single joint of this type

Part 3 of 5 c Using the result from part b compute the P value Round the final answer to at least four decimal places P value 0 0036 Part 3 5 Part 4 of 5 d Determine whether to reject Ho Choose one the null hypothesis Reject Do not reject X