Probability Questions and Answers

eye gre 4ster Does eye grease work In one study 16 student subjects took a test of sensitivity to contrast after three hours fac ing into bright sun both with and without eye grease This is a matched pairs design If eye grease is effective subjects will be more sensitive to contrast when they use eye grease Here are the differences in sensitivity with eye grease minus without eye grease 2 EYEGRS 0 07 0 64 0 12 0 05 0 18 0 14 0 16 0 03 0 05 0 02 0 43 0 24 0 11 0 28 0 05 0 29 Kathy Willens AP Images W SUTO We want to know whether eye grease increases sensitiv ity on the average a What are the null and alternative hypotheses Say in words what mean u your hypotheses concern b Suppose that the subjects are an SRS of all young people with normal vision that contrast differences follow a Normal distribution in this population and that the standard deviation of differences is

7 31 Is this what P means A randomized comparative experiment examined the effect of the attractiveness of an instructor on the performance of students on a quiz given by the instructor The researchers found a statistically significant difference in quiz scores between students in a class with an instructor rated as attrac tive and students in a class with an instructor rated as unattractive P 0 005 When asked to explain the meaning of P 0 005 a student says This means there is only probability of 0 005 that the null hypoth esis is true Explain what P 0 005 really means in a way that makes it clear that the student s explanation is wrong

5 Too Early The examinations in a large multisection statistics class are scaled after grading so that the mean score is 75 The professor thinks that students in the 8 00 A M class have trouble paying attention because they are sleepy and suspects that these students have a lower mean score than the class as a whole The students in the 8 00 A M class this semester can be considered a sample from the population of all students in the course so the professor compares their mean score with 75 State the hypotheses Ho and Ha Agrote

80 Consider the system of components connected as in the accompanying picture Components 1 and 2 are connected in parallel so that subsystem works iff either 1 or 2 works since 3 and 4 are connected in series that subsystem works iff both 3 and 4 work If components work independently of one another and P component i works 9 for i 1 2 and 8 for i 3 4 calculate P system works 3 2

6 In October 1994 a flaw in a certain Pentium chip installed in computers was discovered that could result in a wrong answer when performing a division The manu facturer initially claimed that the chance of any particular division being incorrect was only 1 in 9 billion so that it would take thousands of years before a typical user encountered a mistake However statisticians are not typical users some modern statistical techniques are so computationally intensive that a billion divisions over a short time period is not outside the realm of possibility Assuming that the 1 in 9 billion figure is correct and that results of different divisions are independent of one another what is the probability that at least one error occurs in one billion divisions with this chip

4 pts The amount of time in minutes it will take students to answer a homework question follows the probability distribution given below Use the graph to answer the following questions P x 15 x time min to complete a homework question a What is the probability a student will take exactly 1 minute to complete the homework assignment P x 1 0 5 The probability that a student will take exactly 1 minute to complete the homework assignment is 50 b What s the probability a student will take between 1 and 2 minutes to complete the homework question P 14x2 5 5 0 25 12 5 The probability that a student will take between 1 and 2 minutes to complete the homework question is 25 c What s the probability a student will take less than 60 seconds to complete the question

8 pts The height of a certain plant after 1 week ranges from 1 cm to 5 cm with a uniform probability distribution The graph is given below Using this distribution answer the following questions 0 25 0 0 5 15 0 50 25 3 5 4 5 a What s the probability that after 1 week the plant will be between 2 and 4 cm tall P 2x34 0 25 0 25 4 2 The probability that the plant will be between 2cm and 4cm tall after 1 week is 50 0 25 2 1 the plant will be at least 3 5 cm tall 1 The probability that the plant will be at least 3 5cm tall is 75 b What s the probability P x23 5 1 P x 3 5 1 4 3 5 1 0 25 0 75 c What s the probability the plant will be under 1 5 cm or over 4 5 cm

8 pts The height of a certain plant after 1 week ranges from 1 cm to 5 cm with a uniform probability distribution The graph is given below Using this distribution answer the following questions 0 25 0 5 1 5 25 45

Be sure to fully justify your responses when appropriate Show where all your numbers came from if not explicitly written Don t forget Units and use complete sentences when appropriate asked for 1 4 pts The amount of time in minutes it will take students to answer a homework question follows the probability distribution given below Use the graph to answer the following questions P x 0 5 0 5 1 5 x time min to complete a homework question a What is the probability a student will take exactly 1 minute to complete the homework assignment

Using Fig1a png Fig1b png and Fig1c png answer the following question For large values of p the binomial distribution is skewed to the 1 left II right III symmetric IV none of the above

Using Question 11 calculate the probability of the Senate being made up of so few minority students purely by chance That is what is the value of P X S 2 for this binom problem 1 0 941 OII 0 0001941 III 0 41 IV 9 1941

Find mean standard deviation and z values 7 2 a From 2012 thru 2022 the weight of babies born in hospitals in Metropolis have a normal distribution of with a mean of 7 2 pounds and a standard deviation of 1 pound Create a graph of the distribution with an underlying standard normal distribution shown Illustrate the intervals for x 9 2 and z 1 5 Find the z value for a baby of 5 9 pounds b Assume that the following distribution is approximately normal A random sample of car values of 50 cars parked in the parking garage at UAH on the first Mondays of each month of Spring 2023 semester yielded these results car values 1 million and car values 50 000 million Find the mean standard deviation and the z value of a 30 000 for a car parked in the UAH garage on a first Monday during Spring 2023 semester c During a CoVid lockdown of a family of 8 the sum of their temperatures on the last day of lockdown was 300 degrees Celsius and the sum of their temperatures squared was 11 300 Find the mean standard deviation and the temperature of a family member on the last day of lockdown with a z score of 2

4 Find the number of samples needed for a given significance level historic p know and desired margin of error Find the number of samples needed for a given significance level historic p unknown and desired margin of error Give those formulas Find the sample size needed for a significance level of 2 given a margin of error is 02 and the historic p value is 6 Find the sample size needed for a confidence level of 99 with a desired margin of error of 02 and NO historic p value Make up values and complete 4 more problems

BFW Publishers A group of researchers is designing an experiment to test whether an herbal supplement will help people lose weight The researchers selected a sample of 500 adults who want to lose weight and over the next four months 250 of them will take the herbal supplement daily whereas the remaining 250 will take a placebo daily At the end of the four months the researchers will compare the weight loss for the two groups Why is it important that the researchers use randomization to assign each of the adults who want to lose weight to a treatment group Select all true statements Randomization prevents the adults who want to lose weight from selecting their own treatment group Randomization prevents the researchers from specifically assigning all adults who exercise regularly to take the herbal supplement daily Randomization ensures each treatment group will be similar in everything except the type of pill taken Randomization eliminates lurking variables from the experiment Randomization ensures that there will be an equal number of adults who want to lose weight in each treatment group

4 Assume each outcome is equally likely in the sample space S provided below Find P AC S 0 1 2 3 4 A 0 1 Answer Previous Answer 0 6 Return to overview

Sketch the region corresponding to the statement P Z c 0 25 Shade Left of a value 3 Click and drag the arrows to adjust the values 2 T 1 1 5 0 1 2 3 4

The mean cholesterol level of women age 45 59 in Ghana Nigeria and Seychelles is 5 1 mmol l and the standard deviation is 1 0 mmol l Assume that cholesterol levels are normally distributed Round the probabilities to four decimal places It is possible with rounding for a probability to be 0 0000 a State the random variable rv X a randomly selected woman age 45 59 in Ghana Nigeria or Seychelles b Find the probability that a randomly selected woman age 45 59 in Ghana Nigeria or Seychelles has a cholesterol level of 2 7 mmol l or more 0 9918 c Find the probability that a randomly selected woman age 45 59 in Ghana Nigeria or Seychelles has a cholesterol level of 7 6 mmol l or less 0 3821 d Find the probability that a randomly selected woman age 45 59 in Ghana Nigeria or Seychelles has a cholesterol level between 2 7 and 7 6 mmol l 0 3714 X 0 9998 e Find the probability that randomly selected woman age 45 59 in Ghana Nigeria or Seychelles has a cholesterol level that is at least 8 6 mmol l X Why X f is a cholesterol level of 8 6 mmol l unusually high for a randomly selected woman age 45 59 in Ghana Nigeria or Seychelles

Identify the null hypothesis Ho and the alternative hypothesis H Use u for a claim about a mean p for a claim about a proportion and o for a claim about variation An entomologist writes an article in a scientific journal which claims that fewer than 8 in ten thousand male fireflies are unable to produce light due to a genetic mutation Use the parameter p the true proportion of fireflies unable to produce light O Ho p 0 0008 H p 0 0008 O Ho p 0 0008 H p 0 0008 O Ho p 0 0008 H p 0 0008 O Ho p 0 0008

4 A game is played where a 4 sided die is rolled and then a spinner with 10 equal sized sections labelled 1 thru 10 is spun The player wins 14 if the sum of the two is a 2 3 or 4 The player wins 28 if the sum of the two is 12 or greater a What is the expected winning for each play b What would be a good price to charge for the game Why

1 The circus club has 22 members There are 9 jugglers and 13 tumblers The club will select 5 members to perform in the upcoming variety show a What is the probability that 4 jugglers are chosen b What is the expected number of tumblers in a group this size c The group wants to be fair and randomly select members but agrees that the president who is a tumbler should go and at least 2 jugglers What is the probability that this happens Hint You ware more likely to get this question correct if you use direct reasoning 2 The Birket Lottery hopes to sell two million tickets prizes of 50000 and 20 third prizes of 1000 grand prize is four million dollars there will be 5 second a Find the expected winnings for each ticket b How much do you think the Birket Lottery should set the ticket price at Why kot prico

The time it takes me to wash the dishes on a randomly selected night is uniformly distributed between 7 minutes and 19 minutes a State the random variable in the context of this problem Orv X a uniform distribution Orv X washing dishes rv X the time it takes me to wash dishes on a randomly selected night Orv X a randomly selected night b Compute the height of the uniform distribution Leave your answer as a fraction 1 13 c Which of the following is the correct description of the picture of the probability distribution of X X O a bell shaped curve that starts at 7 and ends at 19 a rectangle with edges at 7 and 19 d What is the probability that washing dishes tonight will take me between 8 and 16 minutes Give your answer as a fraction 7 13 X Give your answer accurate to three decimal places

3 It is estimate that 40 of the general population wears glasses If a random group of 15 people are selected what is the probability that a 11 People will wear glasses b Over 11 people will wear glasses C What is the expected number of people wearing glasses in a group this size 4 A game is played where a 4 sided die is rolled and then a spinner with 10 equal sized sections labelled 1 thru 10 is spun The player wins 14 if the sum of the two is a 2 3 or 4 The player wins 28 if the sum of the two is 12 or greater a What is the expected winning for each play h What would be a good price to charge for the game Why

Four brands of flashlight batteries are to be compared by testing each brand in five flashlights Twenty flashlights are randomly selected and divided randomly into four groups of five flashlights each Then each group of flashlights uses a different brand of battery The lifetimes of the batteries to the nearest hour are as follows Brand Brand Brand Brand A B C D 42 28 24 20 30 36 36 32 39 31 28 38 28 32 28 28 29 27 33 25 Preliminary data analyses indicate that the independent samples come from normal populations with equal standard deviations At the 5 significance level does there appear to be a difference in mean lifetime among the four brands of batteries Type in SSB SSW and F test statistic Type your conclusion in the space below

Engineers must consider the breadths of male heads when designing helmets The company researchers have determined that the population of potential clientele have head breadths that are Normally distributed with a mean of 5 8 inches and a standard deviation of 1 2 inches Using the Empirical Rule what is the approximate percentage of head breadths that are between 2 2 and 3 4 inches Do not enter the percent symbol ans

13 11 c Which of the following is the correct picture of the probability distribution of X O a bell shaped curve that starts at 0 and ends at 11 a rectangle with edges at 0 and 11 d What is the probability that a randomly selected commuter on the Red Line during peak rush hours wait between 2 and 5 minutes Give your answer as a fraction 11 X Give your answer accurate to three decimal places 0 181 X e What is the probability that a randomly selected commuter on the Red Line during peak rush hours wa exactly 2 minutes

The time it takes me to wash the dishes on a randomly selected night is uniformly distributed between 7 minutes and 19 minutes a State the random variable in the context of this problem Orv X a uniform distribution Orv X washing dishes Orv X the time it takes me to wash dishes on a randomly selected night Orv X a randomly selected night b Compute the height of the uniform distribution Leave your answer as a fraction c Which of the following is the correct description of the picture of the probability distribution of X O a bell shaped curve that starts at 7 and ends at 19 O a rectangle with edges at 7 and 19 d What is the probability that washing dishes tonight will take me between 8 and 16 minutes Give your answer as a fraction Give your answer accurate to three decimal places

Use the following information to answer the next eight exercises Carbon 14 is a radioactive element with a half life of about 5 730 years Carbon 14 is said to decay exponentially The decay rate is 0 000121 We start with one gram of carbon 14 We are interested in the time years it takes to decay carbon 14 64 What is being measured here 65 Are the data discrete or continuous 66 In words define the random variable X 67 What is the decay rate m 68 The distribution for X is

Sometimes professional basketball players aren t consistent in their performance game to game Suppose that the standard deviation of the number of points scored by shooting guards in the NBA is 8 3 A random sample of 25 games played by Derrick Rose results in a sample standard deviation of 6 7 points Assume that a normal probability plot indicates that the points scored are approcimately normally distributed Is Derrick Rose more consistent than other shooting guards in the NBA at the 0 10 level of significance The null and alternative hypothesis would be Ho o 8 3 Ho 0 25 Ho o 6 7 Ho 8 3 Ho 8 3 H 0 8 3 H 0 25 H 0 6 7 H 8 3 H 8 3 The test is right tailed left tailed two tailed O Based on a sample of 25 games The test statistic is The p value is to 2 decimals to 2 decimals

20 of all college students volunteer their time Is the percentage of college students who are volunteers different for students receiving financial aid Of the 318 randomly selected students who receive financial aid 76 of them volunteered their time What can be concluded at the 0 05 level of significance Use the p value approach a For this study we should use z test for a population proportion b The null and alternative hypotheses would be Ho pv H pv c The test statistic z d The p value 0 20 0 20 please enter a decimal Please enter a decimal Please round your answer to 2 decimal places Please round your answer to 2 decimal places

The average amount of time it takes for couples to further communicate with each other after their first date has ended is 3 15 days Is this average longer for blind dates A researcher interviewed 57 couples who had recently been on blind dates and found that they averaged 3 5 days to communicate with each other after the date was over Their standard deviation was 1 245 days What can be concluded at the the x 0 05 level of significance Use the p value approach a For this study we should use t test for a population mean b The null and alternative hypotheses would be Ho P H P

Decoration Pumpkin Ghost Spiders Candy 5 2 iv not offer raisins 9 Chips 1 7 6 ii be decorated with ghosts or offer chips iii offer candy and be decorated with pumpkins Raisins 8 3 a The table above represents a record of last year s lawn decorations and Halloween treats offered at 45 houses in my neighbourhood From this information calculate the probability that a house chosen at random will i offer raisins if it is decorated with spiders 4 Time left 2 13 19 b If I roll 3 dice and select 1 card from a regular deck what is the probability that I get two 5 s or a heart

Simplify the following 2g 2h No Negative Exponer

In a class of 300 students Jean s standard score of 2 corresponds to a grade of 85 and Anne s standard score of 1 corresponds to a grade of 67 a What are the mean grade and the variance of the grades for the whole class

Simplify the expression com 6a76 2a b A 12a 610 B 8a b 12a f D 8a4f 0

Solve the following equation for using Fraction Busters 2 89 CT 5 355 5

Pumpkin5 Decoration Ghost 2 Spiders 9 a The table above represents a record of last year s lawn decorations and Halloween treats offered at 45 houses in my neighbourhood From this offer chips 1 7 6 information calculate the probability that a house chosen at random will 8 3 4 i offer raisins if it is decorated with spiders ii be decorated with ghosts or with pumpkins iii offer candy and be decorated iv not offer raisins b If I roll 3 dice and select 1 card from a

A group of researchers conducted some research in which they identified a significant correlation r 0 42 between the number of children a person has and her his life satisfaction Which of the following is inappropriate to conclude from this research All of the above are appropriate b That the directional influences of having children on life satisfaction are unclear a c 42 of variation in life satisfaction has been association with variation in the number of children d That there is a positive relationship between the number of children a person has and his her life satisfaction

Flag question If a distribution has no mode what does this mean O a The data has been entered incorrectly O b It will not be a normal distribution C It will be a normal distribution O d None of the above

Use the following information to answer the next eight exercises A distribution is given as X Exp 0 75 56 What is m 57 What is the probability density function 58 What is the cumulative distribution function 59 Draw the distribution 60 Find P x 4 61 Find the 30th percentile 62 Find the median 63 Which is larger the mean or the median

87 Suppose that the useful life of a particular car battery measured in months decays with parameter 0 025 We are interested in the life of the battery a Define the random variable X b Is X continuous or discrete C X d On average how long would you expect one car battery tollast e On average how long would you expect nine car batteries to last if they are used one after another f Find the probability that a car battery lasts more than 36 months 9 Seventy percent of the batteries last at least how long

96 At a 911 call center calls come in at an average rate of one call every two minutes Assume that the time that elapses from one call to the next has the exponential distribution a On average how much time occurs between five consecutive calls b Find the probability that after a call is received it takes more than three minutes for the next call to occur c Ninety percent of all calls occur within how many minutes of the previous call d Suppose that two minutes have elapsed since the last call Find the probability that the next call will occur within the next minute e Find the probability that fewer than 20 calls occur within an hour

Use the following information to answer the next three exercises The Sky Train from the terminal to the rental car and long term parking center is supposed to arrive every eight minutes The waiting times for the train are known to follow a uniform distribution 79 What is the average waiting time in minutes a zero b two c three d four 80 Find the 30th percentile for the waiting times in minutes a two b 2 4 c 2 75 d three

84 A fireworks show is designed so that the time between fireworks is between one and five seconds and follows a uniform distribution a Find the average time between fireworks b Find the probability that the time between fireworks is greater than four seconds

92 What is the probability that a phone will fail within two years of the date of purchase a 0 8647 b 0 4866 c 0 2212 d 0 9997

92 A student takes a 10 question true false quiz but did not study and randomly guesses each answer Find the probability that the student passes the quiz with a grade of at least 70 percent of the questions correct

2 Which expression is equivalent to 21 A B D 1 1 I 27

Simplify the radical Give the exact answer in simplified radical form 27

Graph x 6 O UND

3 2 x 17 A 34 B 19 19 D 34

Solve the equation for x You can use fraction busters to help you remember the LCM HIN 56 43