Probability Questions and Answers

1 point A newspaper conducted a statewide survey concerning the 1998 race for state senator The newspaper took a SRS of n 1100 registered voters and found that 570 would vote for the Republican candidate Let p represent the proportion of registered voters in the state who would vote for the Republican candidate We test a What is the z statistic for this test b What is the P value of the test this problem Ho p 50 Ha p 50

Physicians at a clinic gave what they thought were drugs to 860 patients Although the doctors later learned that the drugs were really placebos 56 of the patients reported an improved condition Assume that if the placebo is ineffective the probability of a patients condition improving is 0 52 Test the hypotheses that the proportion of patients improving is 0 52 Find the test statistics Z Find the p value p

Trials in an experiment with a polygraph include 99 results that include 22 cases of wrong results and 77 cases of correct results Use a 0 01 significance level to test the claim that such polygraph results are correct less than 80 of the time Identify the null hypothesis alternative hypothesis test statistic P value conclusion about the null hypothesis and final conclusion that addresses the original claim Use the P value method Use the normal distribution as an approximation of the binomial distribution Let p be the population proportion of correct polygraph results Identify the null and alternative hypotheses Choose the correct answer below OA Ho p 0 20 H p0 20 OC Ho p 0 80 H p 0 80 OE Ho p 0 20 H p 0 20 B Ho p 0 80 H p 0 80 O D Ho p 0 20 H p 0 20 OF Ho p 0 80 H p 0 80

Assume a significance level of 0 1 and use the giv parts a and b below Original claim The mean pulse rate in beats per minute of a certain group of adult males is 74 bpm The hypothesis test results in a P value of 0 0784 a State a conclusion about the null hypothesis Reject Ho or fail to reject Ho Choose the correct answer below A Reject Ho because the P value is less than or equal to x B Fail to reject Ho because the P value is greater than C Reject Ho because the P value is greater than D Fail to reject Ho because the P value is less than or equal to a b Without using technical terms state a final conclusion that addresses the original claim Which of the following is the correct conclusion A The mean pulse rate in beats per minute of the group of adult males is 74 bpm OB There is not sufficient evidence to warrant rejection of the claim that the mean pulse rate in beats per minute of the group of adult males is 74 bpm OC There is sufficient evidence to warrant rejection of the claim that the mean pulse rate in beats per minute of the group of adult males is 74 bpm OD The mean pulse rate in beats per minute of the group of adult males is not 74 bpm

Directions Identify the slope and y Intercept Then write an equation in y mx b form 9 Slope 0 3 2 1 0 1 2 y intercept 3 Equation N

12 y 3x 5 Which of the following scenarios could be described by the equation above A The number of cards Larry has if he had 3 collector cards and he adds 5 more cards to his collection every 3 days The distance of a tortoise from the starting line in a race if the tortoise started 5 miles ahead of the starting line and moved at a pace of 3 miles per hour B The number of points earned on a test if each question is worth 5 points and Mr Felder subtracted 3 points for every incorrect answer D The amount of money Miranda has if for every 5 Miranda earns she gives 3 to her mother

15 Write in slope intercept form an equation of the line that passes through the given points 4 1 8 2

21 Consider the graph below in 3 2 O 2 3 B The equation of the line is 5 6 2 What is the equation of the line in slope intercept form

18 Convert the equation standard form equation into Slope intercept form rewrite it as y mz b Standard form 4y 3x 16 Slope intercept form

Identify the type I error and the type II error that corresponds to the given hypothesis The proportion of people who write with their left hand is equal to 0 19 Which of the following is a type I error A Fail to reject the claim that the proportion of people who write with their left hand is 0 19 when the proportion is actually different from 0 19 B Reject the claim that the proportion of people who write with their left hand is 0 19 when the proportion is actually different from 0 19 OC Reject the claim that the proportion of people who write with their left hand is 0 19 when the proportion is actually 0 19 OD Fail to reject the claim that the proportion of people who write with their left hand is 0 19 when the proportion is actually 0 19

bsite receives an average of 10 visits every minute The site manager starts to monitor the number of visits at 10 00 am Part a 2 5 points What is the probability of receiving 30 visits in the period between 10 00 and 10 05 am Part b 2 5 points What is the probability of receiving 15 visits in the period between 10 03 and 10 05 am if 10 visits were received in the period between 10 00 and 10 02 Part c 5 points What is the expected number of arrivals in the period between 10 00 and 10 08 am if 10 visits were received by 10 03 am How about variance

Two health insurance compete for the business of 1000 customers Assume that each customer chooses between the insurance providers independently and with indifference It is also assumed that each company is capable of providing high quality service for at most N customers Parta 5 points Using a binomial model find an expression for N that will guarantee that the probability of turning away a customer because of the limited capability of the company is less than 1 Part b 5 points Use the normal approximation to get a numerical value for N

Nutrition labels The nutrition label on a bag of potato chips says that a serving of potato chips has 130 calories You believe this is incorrect and take a random sample of 53 bags that yielded a sample mean of 134 calories with a standard deviation of 17 calories Test the claim at a 5 level of significance that the number of calories listed on the nutrition label on a bag of potato chips is not correct a Write the hypotheses for the test Ho H 130 Ha 134 Ho 130 Ha H 134 O Ho o 17 Ha 0 17 Ho 130 Ha 130 b Compute the p value for the test p round to three sig figs c What is the final conclusion O There is not sufficient evidence to support the claim that the true average calorie content is different from 130 calories There is sufficient evidence to reject the claim that the true average calorie content is different from 130 calories There is not sufficient evidence to reject the claim that the true average calorie content is different from 130 calories O There is sufficient evidence to support the claim that the true average calorie content is different from 130 calories

Making handcrafted pottery generally takes two major steps wheel throwing and firing That is the time for a piece of pottery to be finished is the sum of the time associated with these two steps The time for wheel throwing and the time for firing are each normally distributed random variables with means of 50 min and 70 min and standard deviations of 5 min and 7 min respectively That is if we let the random variable X represent the time for wheel throwing then X N 50 25 Furthermore if we let the random variable Y represent the time for firing then Y N 70 49 Part a 5 points Assuming the time required for wheel throwing and firing are independent what is the probability that a piece of pottery will be finished within 110 minutes Part b 5 points Let X denote the wheel throwing time required for the i th piece of pottery produced Suppose X N 50 25 for each i and the X variables are independent of each other Let X denote the average wheel throwing time required by the first 5 pieces of pottery What is P 53

Suppose we have a random sample of size 2n from a population denoted by X and E X and V X 0 Let X X 2n 2n 2 i 1 n X X i 1 be two estimators of u Which is the better estimator of Explain your choice in terms of bias and variance

1 Travel Time to Work A study of commuting times reports the travel times to work of a random sample of 1000 employed adults in Seattle The mean is x 37 9 minutes and the standard deviation is s 27 2 minutes What is the standard error of the mean

20 49 Right versus 4step ments affects how easily people can use them Timothy Sturm investigated this effect in a course project asking 25 right handed students to turn a knob with their right hands that moved an indicator by screw action There were two identical instruments one with a right hand thread the knob turns clockwise and the other with a left hand thread the knob turns counterclockwise Table 20 4 gives the times in sec onds each subject took to move the indicator a fixed distance RTLFT TABLE 20 4 Performance times seconds using right hand and left hand threads Right Subject Thread 1 113 2 105 3 130 4 101 5 138 6 118 7 87 8 116 9 75 10 96 11 122 12 103 13 116 Left Thread 137 105 133 108 115 170 103 145 78 107 84 148 147 Subject 14 15 16 17 18 19 20 21 22 23 24 25 Right Left Thread 107 118 103 111 104 111 89 78 100 89 85 88 Thread 87 166 146 123 135 112 93 76 116 78 101 123

Cereal for Breakfast A cereal company announced 60 of Americans start the day with cereal for breakfast You believe the percentage of Americans that start the day with cereal is less To test your claim you randomly sample 100 Americans and ask them what they eat for breakfast 45 of them say cereal Test your claim that percent of Americans that start the day with cereal for breakfast is less than 60 at a 5 level of significance a Select the correct null and alternative hypotheses O Ho P 0 45 Ha p 0 45 O Ho P 0 6 H P 0 6 Ho P 0 45 Ha p 0 45 Ho p 0 6 Ha p 0 6 Ho p 0 45 Ha p 0 45 O Ho P 0 6 Ha p 0 6 b Find the p value round to four decimal places c Write the final conclusion using everyday language O There is sufficient evidence to reject your claim There is not sufficient evidence to support your claim There is not sufficient evidence to reject your claim There is sufficient evidence to support your claim

Test the claim that the proportion of people who own cats is significantly different than 30 at the 0 2 significance level The null and alternative hypothesis would be Ho Select an answer Select an answer H The test is left tailed right tailed two tailed O O Based on a sample of 600 people 24 owned cats The p value is Based on this we O Reject the null hypothesis O Fail to reject the null hypothesis at least 4 decimals

A researcher is interested in finding a 95 confidence interval for the mean number of times per day that college students text The study included 81 students who averaged 25 9 texts per day The standard deviation was 13 6 texts Round answers to 3 decimal places where possible a To compute the confidence interval which distribution should you use t distribution z distribution b With 95 confidence the population mean number of texts per day is between 22 941 and 28 589 texts c If many groups of 81 randomly selected members are studied then a different confidence interval would be produced from each group About percent of these confidence intervals will contain the true population number of texts per day and about percent will

Only about 18 of all people can wiggle their ears Is this percent higher for millionaires Of the 302 millionaires surveyed 64 could wiggle their ears Run a hypothesis test to see if the percent of millionaires who can wiggle their ears is more than 18 at the a 0 10 level of significance Use the classical approach a For this study we should use z test for a population proportion b The null and alternative hypotheses would be Ho p H P c The test statistic d The critical value 0 18 0 18 please enter a decimal Please enter a decimal please show your answer to 3 decimal places Please show your answer to 3 decimal places

Prior to a new drive through system the mean wait time at a certain fast food restaurant was 99 seconds Use the data in the table below to decide whether the evidence suggests the mean wait time is different than 99 seconds with their new system Assume wait times are normally distributed and use an a 0 05 significance level Use the p value approach 69 9 65 74 84 8 81 3 a For this study we should use t test for a population mean b The null and alternative hypotheses would be Ho H c The test statistic d The p value 99 99 93 6 86 84 8 107 6 please show your answer to 3 decimal places Please show your answer to 4 decimal places 97 5

rior to a new drive through system the mean wait time at a certain fast food restaurant was 99 seconds Ise the data in the table below to decide whether the evidence suggests the mean wait time is different han 99 seconds with their new system Assume wait times are normally distributed and use an a 0 05 ignificance level Use the p value approach 69 9 65 74 81 3 a For this study we should use Select an answer b The null and alternative hypotheses would be Ho Select an answer H Select an answer 84 8 93 6 86 84 8 107 6 97 5

In an Online Discussion one of our statistics classes sampled 81 student s resting heart rates and found a standard deviation of 11 71 beats per minute Construct a 99 confidence interval for o the true standard deviation of resting heart rates for Rio Hondo students Use 3 decimal places a Find the lower and upper x critical values at 99 confidence Lower Upper

1 Be sure to fully justify your responses when appropriate Don t forget units and use complete sentences when appropriate asked for Round probabilities to 4 decimal places 10 pts Assume the weights of NFL football players are normally distributed with a mean of 248 lbs and a standard deviation of 29 lbs A new private airplane which could carry all 53 players has a maximum weight capacity of 13250 lbs a Sketch the appropriate distribution below for the random variable X the weight in lbs of an individual NFL football player Be sure the mean and standard deviation are clearly labeled standard deviation 29 306 335 364 b Using this distribution what is the probability that a randomly selected NFL player will weigh over 250 lbs 55 87 248 mean 277

4 pts To estimate the mean heights which are normally distributed of female students on your campus you will measure a simple random sample of female students You know from government data that the standard deviation of the heights of young women is about 2 6 inches and the mean is 64 inches If we record the heights of 10 randomly selected female students what is the probability that their mean height is between 62 and 65 inches You must sketch the distribution including labels and shade in the corresponding region to your probability for full credit

2 n 1 p 3 41 Can we say that the sampling distribution of is approximately normal in this case If the answer is no state the first conditions in the list above that falled No since n 1 p 10 the second condition did not check out 1 np 10 7 which c If you take a sample of size 535 from the original population where p is 2 can we say that the sampling distribution of is approximately normal Why or why not Check conditions which 10 10 2 n 1 p 524 3 which Select an answer 10 Can we say that the sampling distribution of is approximately normal in this case If the answer is no state the first conditions in the list above that failed Select an answer Yes since all conditions check out No since np 10 the first condition did not check out No since n 1 p 10 the second condition did not check out dyw www vg VITVINY IN distribution to compute probabilities How are the problems in this section different was large enough to use the normal

Assume you have a population where 2 of the population has some particular disease List the given numeric information along with the correct symbols v 2 a If you take a sample of size 31 can we say that the sampling distribution of is approximately normal Why or why not Check conditions 1 np 2 n 1 P which Can we say that the sampling distribution of pis approximately normal in this case If the answer is no state the first conditions in the list above that failed Select an answer 1 np which v b Redo part a assuming p has increased to 89 Check conditions 2 n 1 P which which Can we say that the sampling distribution of p is approximately normal in this case If the answer is no state the first conditions in the list above that failed Select an answer

Bone review question Which of the following are bones in your leg mark all that apply O femur humerus tibia thoracic O radius

Ag 8 6 A large laboratory has four types of devices used to determine the pH of soil samples The laboratory wants to determine whether there are differences in the average readings given by these devices The lab uses 24 soil samples having known pH in the study and randomly assigns six of the samples to each device The soil samples are tested and the response recorded for each sample is the difference between the pH reading of the device and the known pH of the soil These values along with summary statistics are given in the following table Device A B C D 1 307 176 137 042 2 294 125 063 690 3 Sample 079 013 240 201 4 019 082 050 166 5 136 091 318 219 6 324 459 154 407 a Based on your intuition is there evidence to indicate any difference among the mean differences in pH readings for the four devices b Run an analysis of variance to confirm or reject your conclusion in part a Use 05 c Compute the p value of the F test in part b d What conditions must be satisfied for your analysis in parts b and c to be valid e Suppose the 24 soil samples have widely different pH values What problems may occur by simply randomly assigning the soil samples to the different devices

PECKSTAT3 5 2 012 A professor assigns five problems to be completed as homework At the next class meeting two of the five problems will be selected at random and collected for grading You have only completed th first three problems b Are the outcomes in the sample space equally likely O Yes No a What is the sample space for the chance experiment of selecting two problems at random Hint You can think of the problems as being labeled A B C D and E One possible selection of tw problems is A and B If these two problems are selected and you did problems A B and C you will be able to turn in both problems There are nine other possible selections to consider Ent your answers in the form AB Enter your answers as a comma separated list c What is the probability that you will be able to turn in both of the problems selected MY NOTES d Does the probability that you will be able to turn in both problems change if you had completed the last three problems instead of the first three problems Explain Yes the probability does not change The problems are being selected at random and each problem is equally likely Yes the probability does not change The problems are being selected at random and each problem is not equally likely O No the probability does not change The problems are being selected at random and each problem is equally likely No the probability does not change The problems are being selected at random and each problem is not equally likely e What is the probability that you will be able to turn in both problems selected if you had completed four of the problems rather than just three PRACTICE ANOTHER

A college job placement center has requests from five students for employment interviews Three of these students are math majors and the other two students are statistics majors Unfortunatel the interviewer has time to talk to only two of the students These two will be randomly selected from among the five a What is the sample space for the chance experiment of selecting two students at random Hint You can think of the students as being labeled A B C D and E One possible selection of tv students is A and B There are nine other possible selections to consider Enter your answers in the form AB Enter your answers as a comma separated list b Are the outcomes in the sample space equally likely o Yes No c What is the probability that both selected students are statistics majors d What is the probability that both students are math majors e What is the probability that at least one of the students selected is a statistics major f What is the probability that the selected students have different majors

A study of the impact of seeking a second opinion about a medical condition is described in a paper Based on a review of a certain number of patient initiated second opinions the paper state Second opinions often resulted in changes in diagnosis 13 8 treatment 36 4 or changes in both 9 6 Consider the following two events D second opinion results in a change in diagnosis T second opinion results in a change in treatment a What are the values of P D P T and P D n T P D 0 138 P T 0 364 P Dn T 0 096 b Use the given probability information to set up a hypothetical 1 000 table with columns corresponding to D and not D and rows corresponding to T and not T Change in Diagnosis D No Change in Diagnosis not D Change in Treatment 7 No Change in Treatment not T Total 96 42 138 268 594 862 364 d What is the probability that a second opinion results in a change in diagnosis or a change in treatment 0 452 X 636 c What is the probability that a second opinion results in neither a change in diagnosis nor a change in treatment 0 548 X Total MY M 1 000

Fiber content in grams per serving and sugar content in grams per serving for 18 high fiber cereals are shown below Fiber Content 7 10 10 Sugar Content 12 7 9 7 12 7 14 13 0 18 9 10 19 7 10 USE SALT 12 9 13 10 9 12 7 14 7 9 9 lower quartile upper quartile interquartile range a Find the median quartiles and interquartile range in grams serving for the fiber content data set median grams serving grams serving grams serving grams serving grams serving grams serving 17 10 10 0 b Find the median quartiles and interquartile range in grams serving for the sugar content data set median lower quartile upper quartile interquartile range grams serving grams serving 9 5 12 c Are there any outliers in the sugar content data set Yes there is one outlier on the high end OYes there are two outliers on the high end Yes there is one outlier on the low end O Yes there are two outliers on the low end O No there are no outliers Exam 1 MATH1123 sec d Explain why the minimum value for the fiber content data set and the lower quartile for the fiber content data set are equal the lower quartile are always the same value

2 Let X denote the proportion of allotted time that a ran domly selected student spends working on a certain aptitude test Suppose the pdf of X is 0 1 x 0 x 1 0 otherwise f x 0 where 1 0 A random sample of ten students yields data x 92 x 79 x 90 x 65 x 86 x 47 x 73 x 97 x 94 X 0 77 a Use the method of moments to obtain an estimator of 8 and then compute the estimate for this data b Obtain the maximum likelihood estimator of 8 and then compute the estimate for the given data

A study of the impact of seeking a second opinion about a medical condition is described in a paper Based on a review of a certain number of patient initiated second opinions the paper states Second opinions often resulted in changes in diagnosis 13 8 treatment 36 4 or changes in both 9 6 Consider the following two events D second opinion results in a change in diagnosis T second opinion results in a change in treatment a What are the values of P D P T and P Dn T P D P T P DnT b Use the given probability information to set up a hypothetical 1 000 table with columns corresponding to D and not D and rows corresponding to T and not T Change in Diagnosis D No Change in Diagnosis not D Change in Treatment 7 No Change in Treatment not 7 Total c What is the probability that a second opinion results in neither a change in diagnosis nor a change in treatment d What is the probability that a second opinion results in a change in diagnosis or a change in treatment Total MY N 1 000

accompanying table summarizes data from a survey of 3 400 parents with school aged children In this survey parents were asked if they were completely sausried with the education their child receives School Type Public Private Public School Private School Percentage of Parents Completely Total Satisfied Of the 3 400 parents surveyed 700 were parents whose oldest child attended a private school The parents participating in this survey were thought to be representative of U S parents of school aged children a Use the given information to determine the number of parents surveyed falling into each of the cells in the table below Round your answers to the nearest integer Completely Satisfied Not Completely Satisfied 25 65 Total 700 3 400 b Estimate the probability that a randomly selected parent of school aged children is completely satisfied with their oldest child s education Round your answer to four decimal places c Estimate the probability that a randomly selected parent of school aged children has an oldest child who attends a private school Round your answer to four decimal places d Estimate the probability that a randomly selected parent of school aged children is not completely satisfied with their oldest child s education given that the oldest child attends a private school e Estimate the probability that a randomly selected parent of school aged children is completely satisfied with their oldest child s education and the oldest child attends public school Round your answer to four decimal places

Internet Service A large cable company reports that 70 of its customers subscribe to its cable TV service 34 subscribe to its Internet service and 95 subscribe to at least one of these two services a Use the given probability information to set up a hypothetical 1 000 table Not Cable TV Not Internet Service Total b Use the table from part a to find the following probabilities Cable TV Need Help 1 the probability that a randomly selected customer subscribes to both cable TV and Internet service Read It Total ii the probability that a randomly selected customer subscribes to exactly one of these services Watch It 1 000 MY NOTES PRACTIC

ditional problem

Ninety three countries won medals at the Olympics in a particular year The results indicate the following 1 country won more than 100 medals 5 countries won between 51 and 100 medals 6 countries won between 31 and 50 medals 3 countries won between 21 and 30 medals 10 countries won between 11 and 20 medals 16 countries won between 6 and 10 medals 52 countries won between 1 and 5 medals Suppose one of the 93 countries winning medals at this Olympics is selected at random Round your answers to four decimal places a What is the probability that the selected country won more than 50 medals b What is the probability that the selected country did not win more than 100 medals c What is the probability that the selected country won 10 or fewer medals d What is the probability that the selected country won between 11 and 50 medals

12 4 Points A magazine reported the following information for Ph D degrees awarded by a certain country s colleges in the 2013 2014 academic year Round your answers to three decimal pl A total of 54 070 Ph D degrees were awarded 12 704 of these degrees were in the life sciences 9 659 of these degrees were in the physical sciences The remaining degrees were in majors other than life or physical sciences a What is the probability that a randomly selected Ph D student who received a degree in 2013 2014 received a degree in the life sciences DETAILS PECKSTAT3 5 2 501 XP MI b What is the probability that a randomly selected Ph D student who received a degree in 2013 2014 received a degree that was not in a life or a physical science c What is the probability that a randomly selected Ph D student who received a degree in 2013 2014 did not receive a degree in the physical sciences

The mean number of text messages sent per month by customers of a cell phone service provider is 1 800 and the standard deviation is 750 Find the z score associated with each of the numbers of text messages sent Round your answer for part b to three decimal places a 0 b 11 000 c 4 500 d 300 N

An appliance manufacturer offers extended warranties on its washers and dryers Based on past sales the manufacturer reports that of customers buying both a washer and a dryer 42 purci the extended warranty for the washer 46 purchase the extended warranty for the dryer and 58 purchase at least one of the two extended warranties a Use the given probability information to set up a hypothetical 1 000 table Let W be the event that the customer purchases an extended warranty for the washer Let D be the event the customer purchases an extended warranty for the dryer W Not W Total D Not D Total 1 000 b Use the table from part a to find the following probabilities i the probability that a randomly selected customer who buys a washer and a dryer purchases an extended warranty for both the washer and the dryer ii the probability that a randomly selected customer purchases an extended warranty for neither the washer nor the dryer

An article reported the following information for first time freshmen at four year colleges A total of 95 010 first time freshmen were surveyed 52 178 were enrolled in their first choice college 25 942 were enrolled at their second choice college The rest were enrolled at a college that was not their first or second choice college a What is the probability that a randomly selected person who participated in this survey was enrolled in their second choice college Round your answer to four decimal places b What is the probability that a randomly selected person who participated in this survey was not enrolled at their first choice college Round your answer to four decimal places c What is the probability that a randomly selected person who participated in this survey was not enrolled at either their first or second choice college Round your answer to four deci places Submit Answer DETAILS DECKSTOT 5 3 022 MY N

Treatment B c Now consider the summary data for the women who participated in the study Treatment A Treatment B Total 1 Find P S ii Find P SIA iii Find P SIB Survived 94 228 322 Died 6 32 38 iv Which treatment appears to be better O Treatment A O Treatment B Total 100 260 d You should have noticed from parts b and c that for both men and women Treatment A appears to be better But in part a when the data for Treatment B is better This is an example of what is called Simpson s paradox Write a brief explanation of why this apparent inconsistency occurs fo respond similarly to the two treatments The results are distorted in favor of Treatment A as women respond to both treatments better than men but Treatment A was given to far mo The results are distorted in favor of Treatment B as women respond to both treatments better than men but Treatment B was given to far mor The results are distorted in favor of Treatment B as women respond to these treatments better than men but Treatment A was given to far mo

A paper described the results of a medical study in which one treatment was shown to be better for men and better for women than a competing treatment However if the data for men and are combined it appears as though the competing treatment is better To see how this can happen consider the accompanying data tables constructed from information in the paper Subjects in the study were given either Treatment A or Treatment B and their s was noted Let S be the event that a patient selected at random survives A be the event that a patient selected at random received Treatment A and B be the event that a patient selected at received Treatment B Round your answers to three decimal places a The following table summarizes data for men and women combined Treatment A Treatment B Total i Find P S ii Find P SIA iii Find P SIB Survived Died Total 214 248 462 86 300 52 138 iv Which treatment appears to be better Treatment A O Treatment B 300 the study

Each person in a representative sample of 3 337 students at two year colleges was asked how they would describe their household s financial situation compared to before the COVID 19 pandemic Each person in the sample was also asked if they have children who depend on them for care living with them The responses are summarized in the table Children No Children Better than Before Same as Before 37 f 208 298 1 521 Since P C Worse than Before Suppose one of these 3 337 two year college students is to be selected at random a What is the probability that the selected student has children Round your answer to four decimal places 306 b What is the probability that the selected student responded that they are better off than before the pandemic 967 c What is the probability that the selected student has children and reported their financial situation was worse than before the pandemic d What is the probability that the selected student does not have children and reported that their financial situation was the same as before the pandemic e What is the probability that the selected student does not have children and reported their financial situation was worse than before the pandemic P C C W Are the events C selected student has children and W selected student responded they are worse off independent events Use probabilities calculated from the information in the given table to justify your answer P W we see that the events C and W Select independent

An article describes a survey of 2 096 American adults Survey participants were asked if they were football fans and also if they agreed or disagreed that the rules that the National Football Leag adopted in 2010 designed to limit head injuries have been effective Data from the survey are summarized in the table below Football Fan Not a Football Fan 1 P A Total ii P D Agree 683 199 882 Disagree Total 533 681 1 216 880 Suppose that a survey participant is to be selected at random Consider the following events A selected participant agreed that the rules have been effective D selected participant disagreed that the rules have been effective F selected participant was a football fan a Calculate the following probabilities Round your answers to four decimal places 1 214 2 096 iii P A F iv P A FC b re the events F and A independent events Justify your answer using relevant probabilities O Yes the events F and A are independent because P A F P A Yes the events F and A are independent because P AIF P A No the events F and A are not independent because P AIF P A DELAY

A website provides data on the on time performance of airlines Suppose that the tables below summarize data from this website for flights that occurred over a 1 year time period a The following table summarizes data for flights leaving the Phoenix and Seattle airports for two airlines Airline A and Airline B On Time Departure Airline A Airline B Total 33 001 46 787 79 788 Late Departure Total Airline A 4 550 37 551 6 526 53 313 Suppose that one of these flights is selected at random 1 What is the probability that the flight had an on time departure Round your answer to four decimal places 11 076 90 864 ii What is the probability that the flight had an on time departure given that it was an Airline A flight Round your answer to four decimal places iii What is the probability that the flight had an on time departure given that it was an Airline B flight Round your answer to four decimal places iv Which airline had the better on time performance Airline A Airline B 29 392 b Now consider the flight data for just the Phoenix airport On Time Departure Late Departure Total 3 792 33 184

1 The probability of getting a king when a card is selected at random from a standard deck of 52 playing cards is 13 a Give a relative frequency interpretation of this probability Enter your probability as a fraction In the long run of all cards selected at random from a standard deck will be kings b Express the probability as a decimal rounded to three decimal places Then complete the following statement Round your answer to the nearest integer If a card is selected at random I would expect to see a king about times in 1 000