Statistics Questions

Events A and B are independent if the occurrence of one does not affect the probability of the other occurring This statement is best represented in symbols by OP A B P A A B A OP A and B P A OA and B A

Events A and B are mutually exclusive if they can not occur simultaneously This statement is best represented in symbols by P A and B 0 A given B 0 A and B 0 P A B 0

Sara draws the 7 of hearts from a standard deck of 52 cards Without replacing the first card she then proceeds to draw a second card a Determine the probability that the second card is another 7 P 7 7 of hearts b Determine the probability that the second card is another heart P heart 7 of hearts c Determine the probability that the second card is a club P club 7 of hearts d Determine the probability that the second card is a 3 P 3 7 of hearts

Two dice are rolled What is the probability P sum is 3 OO 11 18 1 12 1 18 19 36

Let P U 0 15 and P V 0 44 Events U and V are mutually exclusive Find P U or V O 0 59 O 0 29 O 2 9333333333333 O 0 066 O 0 29 O 0 34090909090909

Table Frequencies by grades and shirt color Shirt Color A B Blue Shirt 20 10 Red Shirt Total C 4 7 3 11 27 13 15 Total 34 21 55 If one student is chosen at random from those who took the test Find the probability that the student was Red Shirt GIVEN they got an A

64 Suppose that a category of world class runners are known to run a marathon 26 miles in an average of 145 minutes with a standard deviation of 14 minutes Consider 49 of the races Let X be the average of the 49 races a X b Find the probability that the runner will average between 142 and 146 minutes in these 49 marathons c Find the 80th percentile for the average of these 49 marathons d Find the median of the average running times

65 The length of songs in a collector s online album collection is uniformly distributed from 2 to 3 5 minutes Suppose we randomly pick five albums from the collection There are a total of 43 songs on the five albums a In words X b X c In words X d X e Find the first quartile for the average song length f The IQR for the average song length is

Here is the probability model for the blood type of a randomly chosen person in the United States Blood type 0 A B AB Probability 0 49 0 17 0 03 0 31 What is the probability that a randomly chosen American does not have type O blood Round to the nearest 0 01

Events A and B are mutually exclusive if they can not occur simultaneously This statement is best represented in symbols by OA given B 0 OP A B 0 OP A and B 0 O A and B 0

There are 13 freshmen and 16 sophomore students in a classroom We randomly select two students Suppose the students were selected with replacement a If the first student was a freshman what is the probability that the 2nd student is also a freshman b If the first student was a sophomore what is the probability that the 2nd student was a freshman Suppose the students were selected without replacement c If the first student was a freshman what is the probability that the 2nd student is also a freshman d If the first student was a sophomore what is the probability that the 2nd student was a freshman

Let P U 0 3 and P V 0 34 Events U and V are mutually exclusive Find P U or V O 0 102 O 0 04 O 0 04 O 0 88235294117647 0 64 O 1 1333333333333

A certain virus infects one in every 600 people A test used to detect the virus in a person is positive 90 of the time if the person has the virus and 10 of the time if the person does not have the virus Let A be the event the person is infected and B be the event the person tests positive a Find the probability that a person has the virus given that they have tested positive b Find the probability that a person does not have the virus given that they have tested negative

The length of time it takes to find a parking space in a certain city at 9 A M follows a normal distribution with a mean of five minutes and a standard deviation of three minutes 4 Based upon the given information and numerically justified would you be surprised if it took less than one minute to find a parking space Many statisticians say that a probability below 0 05 is an unusual event Explain your answer including numbers that you calculated 5 Find the probability that it takes at least eight minutes to find a parking space 6 Seventy percent of the time it takes more than how many minutes to find a parking space 7 Please explain in your own words why the answers to 1 through 3 are different from the answers to 4 through 6

Which of the following scenarios represent random sampling without replacement 1 We visit a farmer s market and ask a vendor for four pears from a crate The vendor picks four pears and gives them to us II Each student in a class rolls a die Those students who get a 5 are selected to complete a special survey III Each day at a hospital ten patients are randomly selected from all patients who had an appointment that day This study is completed over 5 years to help ensure patients with a large number of ailments are considered including seasonal ailments like influenza Olonly Oll only OI and II O II and III OI II and III O None of the above

A poll showed that 58 4 of Americans say they believe that Marilyn Monroe had an affair with JFK What is the probability of randomly selecting someone who does not believe that Marilyn Monroe had an affair with JFK Round to 3 decimal places

62 Suppose that the distance of fly balls hit to the outfield in baseball is normally distributed with a mean of 250 feet and a standard deviation of 50 feet We randomly sample 49 fly balls a If X average distance in feet for 49 fly balls then X b What is the probability that the 49 balls traveled an average of less than 240 feet Sketch the graph Scale the horizontal axis for X Shade the region corresponding to the probability Find the probability c Find the 80th percentile of the distribution of the average of 49 fly balls

A company that makes cartons finds that the probability of producing a carton with a puncture is 0 05 the probability that a carton has a smashed corner is 0 08 and the probability that a carton has a puncture and has a smashed corner is 0 004 Answer parts a and b below a Are the events selecting a carton with a puncture and selecting a carton with a smashed corner mutually exclusive Explain OA Yes a carton can have a puncture and a smashed corner OB Yes a carton cannot have a puncture and a smashed corner OC No a carton can have a puncture and a smashed corner O D No a carton cannot have a puncture and a smashed corner I b If a quality inspector randomly selects a carton find the probability that the carton has a puncture or has a smashed corner The probability that a carton has a puncture or a smashed corner is Type an integer or a decimal Do not round EXCH

The length of time it takes to find a parking space in a certain city at 9 A M follows a normal distribution with a mean of five minutes and a standard deviation of two minutes 1 Based upon the given information and numerically justified would you be surprised if it took less than one minute to find a parking space Many statisticians say that a probability below 0 05 is an unusual event Explain your answer including numbers that you calculated 2 Find the probability that it takes at least eight minutes to find a parking space 3 Seventy percent of the time it takes more than how many minutes to find a parking space

A Manufacturing and the life times distribution with to set wants Only 2 of 2 of the bulbs what should hours period Period Company Produces light bulbs of these bulbs follow mean of 800 hours 9 warranty period bulbs fail a in a The Such that normal Comport 3 minimum pony fail during the warranty be the warranty

You may need to use the appropriate appendix table to answer this question According to Money magazine Maryland had the highest median annual household income of any state in 2018 at 75 847 t Assume that annual household income in Maryland follows a norma distribution with a median of 75 847 and standard deviation of 33 800 a What is the probability that a household in Maryland has an annual income of 90 000 or more Round your answer to four decimal places b What is the probability that a household in Maryland has an annual income of 50 000 or less Round your answer to four decimal places c What is the probability that a household in Maryland has an annual income between 40 000 and 70 000 Round your answer to four decimal places d What is the annual income in of a household in the ninety first percentile of annual household income in Maryland Round your answer to the nearest cent

We found P z 0 53 0 7019 and P z 1 21 0 8869 Subtract the smaller area from the larger area to find the requested probability P 0 53 sz 1 21 P 0 53 z 1 21 P z 1 21 P z 0 53 0 8869

Step 3 The probability statement was determined to be P z 2 Recall that the normal probability table gives the area under the curve to the left of a given z value An excerpt is given bela Z 2 1 0 02 0 00 0 01 0 03 0 0179 0 0174 0 0170 0 0166 0 0217 0 0212 0 0222 0 0268 0 0274 0 0281 2 0 0 0228 1 9 0 0287 0 04 0 0162 0 0207 0 0262 0 05 0 0158 0 0202 0 0256 0 06 0 0154 0 0197 0 0250 0 07 0 08 0 0146 0 0150 0 0192 0 0188 0 0244 0 0239 0 09 0 0143 0 0183 0 0233 Use the above table to find the probability of completing the exam in an hour or less P z 2 rounding the result to four decimal places P Z 2

The probability statement was determined to be P z 0 36 Recall that the normal probability table below gives the area under the curve to the left of a given z value and the entire area under this curve is 1 Here we want the area to the right of z 0 36 so we can subtract the area to the left of z 0 36 from 1 Z 0 00 0 2 0 5793 0 5832 0 4 0 01 0 3 0 6179 0 6217 0 6554 0 02 0 5871 0 03 0 5910 0 04 0 5948 0 6255 0 6293 0 6331 0 05 0 06 0 5987 0 6026 0 07 0 6064 0 6103 0 6368 0 6406 0 6443 0 6480 0 6517 0 6591 0 6628 0 6664 0 6700 0 6736 0 6772 0 08 0 6808 0 09 0 6844 0 6141 0 6879 Use the table excerpt above to find the probability that an individual large cap domestic stock fund has a three year return of at least 16 rounding to four decimal places P z 0 36 1

Converting an hour to minutes the probability of completing the exam in an hour or less is P x 60 Before finding this probability the random variable x needs to be converted to th normal random variable z so that a table of probabilities can be used Recall the formula to convert an x value to the normal random variable z where x is the value that needs to be converted u is the population mean and a is the population standard deviation Z The mean was given to be 100 minutes and the standard deviation was given to be 20 minutes Find the value of the z statistic corresponding to x 60 x G Z 60 X H 20 Thus the new probability statement using the normal random variable z is S

The average starting salary of this year s MBA students is 55 000 with a standard deviation of 5 000 Furthermore it is known that the starting salaries are normally distributed What are the minimum and the maximum starting salaries in dollars of the middle 95 of MBA graduates You may need to use the appropriate appendix table or technology to answer this question Round your answers to the nearest dollar minimum maximum

Use the table excerpt above to find the area under the standard normal curve to the left of z 0 46 P z 0 46 P z 0 46 0 6772 0 6772 Step 3 We found P z s 1 95 0 0256 and P z 0 46 0 6772 Subtract the smaller area from the larger area to find the requested probability P 1 95 sz s 0 46 P 1 95 sz 0 46 P Z 0 46 P Z 1 95 0 6772

b What is the probability that a Dutch male is taller than 194 cm Round your answer to four decimal places You may need to use the appropriate appendix table to answer this question Males in the Netherlands are the tallest on average in the world with an average height of 183 centimeters cm t Assume that the height of men in the Netherlands is normally distributed with an mean of 183 cm and standard deviation of 10 5 cm a What is the probability that a Dutch male is shorter than 176 cm Round your answer to four decimal places c What is the probability that a Dutch male is between 175 and 191 cm Round your answer to four decimal places ASK YOUR TEACHER d Out of a random sample of 1 000 Dutch men how many would we expect to be taller than 188 cm Round your answer to the nearest integer men PRACTICE ANOTH

You may need to use the appropriate appendix table to answer this question The average return for large cap domestic stock funds over three years was 14 4 Assume the three year returns were normally distributed across funds with a standard deviation of a What is the probability an individual large cap domestic stock fund had a three year return of at least 23 Round your answer to four decimal places b What is the probability an individual large cap domestic stock fund had a three year return of 10 or less Round your answer to four decimal places c How big does the return have to be to put a domestic stock fund in the top 25 for the three year period Round your answer to two decimal places

You may need to use the appropriate appendix table to answer this question Given that z is a standard normal random variable find z for each situation Round your answers to two decimal places a The area to the left of z is 0 9750 b The area between 0 and z is 0 4750 c The area to the left of z is 0 7454 d The area to the right of z is 0 1251 e The area to the left of z is 0 6293 f The area to the right of z is 0 3707

Stop We found P Z 1 95 0 0256 and P z 0 46 0 6772 Subtract the smaller area from the larger area to find the requested probability P 1 95 z 0 46 P 1 95 z 0 46 P z 0 46 P Z 1 95 0 6772

0 3 12 Points 0 66 0 60 0 60 0 66 You may need to use the appropriate appendix table or technology to answer this question Given that z is a standard normal random variable what is the value of z if the area to the left of z is 0 2546 x Need Help Submit Answer DETAILS Tutorial Exercise Given that in Read It 0 3 3 12 Points This question has several parts that must be completed sequentially If you skip a part of the question you will not rec skipped part PREVIOUS ANSWERS ASWSBE14 6 TB 2 039 DETAILS PREVIOUS ANSWERS ASWSBE14 6 E 013 MI SA tandard normal random variable compute the TA

You may need to use the appropriate appendix table or technology to answer this question Given that z is a standard normal random variable compute the following probabilities Round your answers to four decimal places a P 0 z 0 86 b P 1 52 z 0 c P Z 0 42 d P z 2 0 27 e P Z 1 20 f P z 0 76

Before finding this probability the random variable x needs to be converted to the normal random variable z so that a table of probabilities can be used Recall the formula to convert an x value to the normal random variable z where x is the value that needs to be converted u is the population mean and is the population standard deviation Z x x The mean was given to be 14 4 and the standard deviation was given to be 4 4 Find the value of the z statistic corresponding to x 16 rounding the result to two decimal places 4 4 6 Thus the new probability statement using the normal random variable z is P Z

DETAILS ASWSBE14 6 E 022 MY NOTES ASK YOUR TEACHER You may need to use the appropriate appendix table to answer this question Suppose that the mean daily viewing time of television is 8 35 hours Use a normal probability distribution with a standard deviation of 2 5 hours to answer the following questions about television viewing per household a What is the probability that a household views television between 5 and 11 hours a day Round your answer to four decimal places PRACTIC c What is the probability that a household views television more than 3 hours a day Round your answer to four decimal places b How many hours of television viewing must a household have in order to be in the top 2 of all television viewing households Round your answer to two decimal places hrs

will be for a range of values and represented by the area under the graph between the values The desired probability is for the region between z 1 95 and z 0 46 Since z 1 95 is to the left of z 0 46 we can find this probability by subtracting the area under the curve to the left of z 1 95 1 95 from the area under the curve to the left of z 0 46 0 46 Step 2 To find P 1 95 z 0 46 subtract the area to the left of z 1 95 from the area to the left of z 0 46 Tables can be used to find areas to the left of z values Along the leftmost column are values of z precise to one decimal place Trace along the necessary row until you get to the column for the needed hundredths place The value where the row and column intersect is the area under the curve to the left of that z value Z 2 0 0 08 0 09 0 0188 0 0183 0 0239 0 0233 0 00 0 01 0 02 0 03 0 04 0 05 0 06 0 07 0 0228 0 0222 0 0217 0 0212 0 0207 0 0202 0 0197 0 0192 2 9 0 0287 0 0281 0 0274 0 0268 0 0262 0 0256 0 0250 0 0244 1 8 0 0359 0 0351 0 0344 0 0336 0 0329 0 0322 0 0314 0 0307 0 0301 0 0294 Use the table excerpt above to find the area under the standard normal curve to the left of z 1 95 P z 1 95 P Z 1 95 Z 0 00 0 01 0 02 0 03 0 05 0 06 0 3 0 6179 0 6217 0 6255 0 6293 0 6368 0 6406 0 4 0 6554 0 6591 0 6628 0 6664 0 6736 0 6772 0 5 0 6915 0 6950 0 6985 0 7019 0 7054 0 7088 0 7123 0 7157 0 04 0 6331 0 6700 0 07 0 08 0 6443 0 6480 0 6808 0 6844 0 7190 0 09 0 6517 0 6879 0 7224 Use the table excerpt above to find the area under the standard normal curve to the left of z 0 46 P Z 0 46 P Z 0 46

You may need to use the appropriate appendix table or technology to answer this question Given that z is a standard normal random variable what is the value of z if the area to the left of z is 0 2546 O 0 66 0 60 0 60 0 66 Noor

You may need to use the appropriate appendix table to answer this question Given that z is a standard normal random variable compute the following probabilities Round your answers to four decimal places a P 1 98 z 0 49 b P 0 52 z 1 22 c P 1 75 sz 1 04

You may need to use the appropriate appendix table or technology to answer this question z is a standard normal random variable What is the value of z if the area to the right of z is 0 1271 1 14 O 0 55 O 0 55 1 14 X

According to a survey 54 of the residents of a city oppose a downtown casino Of these 54 about 8 out of 10 strongly oppose the casino Complete parts a through c a Find the probability that a randomly selected resident opposes the casino and strongly opposes the casino b Find the probability that a randomly selected resident who opposes the casino does not strongly oppose the casino c Would it be unusual for a randomly selected resident to oppose the casino and strongly oppose the casino Explain a The probability that a randomly selected resident opposes the casino and strongly opposes the casino is Round to three decimal places as needed b The probability that a randomly selected resident who opposes the casino does not strongly opposes the casino is Round to three decimal places as needed c Would it be unusual for a randomly selected resident to oppose the casino and strongly oppose the casino Explain Choose the correct answer below A No this is not unusual because the probability is less than or equal to 0 05 B Yes this is unusual because the probability is less than or equal to 0 05 OC No this is not unusual because the probability is not less than or equal to 0 05 D Yes this is unusual because the probability is not less than or equal to 0 05

The following data were collected by counting the number of operating rooms in use at Tampa General Hospital over a 20 day period On four of the days only one operating room was used on eight of the days two were used on five of the days three were used and on three days all four of the hospital s operating rooms were used a Use the relative frequency approach to construct an empirical discrete probability distribution for the number of operating rooms in use on any given day f x X 1 2 3 4 b Draw a graph of the probability distribution f x 0 5 0 4 0 3 0 2 0 1 O f x 0 5 0 4 0 3 1 2 3 4 X f x 0 5 0 4 0 3 0 2 0 1 1 2 3 4 X f x 0 5 0 4 0 3 0 2 0 1 1 2 3 4 S

The probability that a person in the United States has type B blood is 8 Three unrelated people in the United States are selected at random Complete parts a through a Find the probability that all three have type B blood The probability that all three have type B blood is Round to six decimal places as needed b Find the probability that none of the three have type B blood The probability that none of the three have type B blood is Round to three decimal places as needed c Find the probability that at least one of the three has type B blood The probability that at least one of the three has type B blood is Round to three decimal places as needed d Which of the events can be considered unusual Explain Select all that apply OA The event in part a is unusual because its probability is less than or equal to 0 05 OB The event in part b is unusual because its probability is less than or equal to 0 05 OC The event in part c is unusual because its probability is less than or equal to 0 05 OD None of these events are unusual

In this question we will formulate a measure to quantify the level of association between the two categorical variables Such a measure is often used in a statistical test called Chi square test for assessing whether there is an association between two categorical variables This question is also used to motivate the learning of independence and to connect the concept back to what we have learnt in the course Let s revisit the example we have looked at in the course How is diet type high cholesterol diet versus low cholesterol diet related to the risk of coronary heart disease Data of 23 individuals High cholesterol diet Low cholesterol diet Heart disease No heart disease iii 4 iv 6 10 i 11 ii 2 13 Total 15 8 23 From the table we find that the probability of having heart disease is 13 23 and the probability of having high cholesterol diet is 15 23 Similarly we can find the probability of not having heart disease and the probability of having low cholesterol diet Part a If there is no association between the two variables i e the two are independent the probability of having heart disease and high cholesterol diet is Round to four decimal places Part b If the two variables are independent we should expect the number of individuals with heart disease and high cholestoral diet to be the probability in Part a multiplied by 23 individuals which is Round to two decimal places

3 Points DETAILS Consider the experiment of a worker assembling a product a Define a random variable that represents the time in minutes required to assemble the product Let x the percentage the product assembly is complete Let x the total number of products assembled by the worker during four 6 hour shifts O Let x the time in hours to assemble six products Let x the time in minutes to assemble the product O Let x the average number of products assembled by the worker during six 4 hour shifts ASWSBE14 5 E 002 b What values may the random variable assume O It may assume values such as 1 2 99 100 It may assume three values 4 5 and 6 It may assume any value between 0 and 100 inclusive 0 x 100 O It may assume any positive value x 0 O It may assume any value between 0 and 60 inclusive 0 x 60 Need Help c Is the random variable discrete or continuous continuous discrete Submit Answer Read It

Select True or False from each pull down menu depending on whether the corresponding statement is true or false 1 If a sample has 18 observations and a 90 confidence estimate for u is needed the appropriate t score is 1 740 2 If a sample of size 30 is selected the value A for the probability P A t A 0 95 is 2 045 3 If a sample of size 20 is selected the value of A for the probability P t A 0 01 is 2 528 4 The lower limit of the 90 confidence interval for the population proportion p given that n 400 and p 0 10 is 0 1247 Note In order to get credit for this problem all answers must be correct

Odiscrete c Audit 40 tax returns random variable x number of returns containing errors Identify the values that the random variable can assume O 0 1 2 O 0 1 2 40 Ox 0 0 0 x 40 none of these State whether the random variable is discrete or continuous continuous discrete d Observe an employee s work random variable x number of nonproductive hours in a seven hour workday Identify the values that the random variable can assume O 0 1 2 O 0 1 2 7 Ox 0 00 x 7 O none of these State whether the random variable is discrete or continuous O continuous

c Compute o the standard deviation of x Recall that the standard deviation is the positive square root of the variance That is o 2 The variance was found to be o 8 Use this to find the standard deviation rounding the result to two decimal places 6 6

A random sample of 13 size AA batteries for toys yield a mean of 3 37 hours with standard deviation 1 16 hours a Find the critical value t for a 99 Cl t b Find the margin of error for a 99 CI

Katie thinks that people living in a rural environment have a healthier lifestyle than other people She believes the average lifespan in the USA is 77 years A random sample of 17 obituaries from newspapers from rural towns in Idaho give 80 63 and s 0 87 Does this sample provide evidence that people living in rural Idaho communities live longer than 77 years a State the null and alternative hypotheses Type mu for the symbol u e g mu 1 for the mean is greater than 1 mu 1 for the mean is less than 1 mu not 1 for the mean is not equal to 1 Ho Ha b Find the test statistic t c Answer the question Does this sample provide evidence that people living in rural Idaho communities live longer than 77 years Use a 10 level of significance Type Yes or No

Use the frequency distribution which shows the number of American voters in millions according to age to find the probability that a voter chosen at random is in the 18 to 20 years old age range The probability that a voter chosen at random is in the 18 to 20 years old age range is Round to three decimal places as needed Ages 18 to 20 21 to 24 25 to 34 35 to 44 45 to 64 65 and over Frequency 5 8 8 8 23 2 22 3 51 4 28 2