Probability Questions and Answers

1 a b Ryann observed 500 vehicles in three different colors Her results are summarized in the two way relative frequency table Gray Blue White Total Type of Vehicle Sedan Truck SUV Total 0 10 0 08 0 12 0 30 0 03 0 01 0 06 0 10 0 06 0 18 0 36 0 60 0 19 0 27 0 54 1 00 Part A What is the probability that an observed vehicle is not an SUV Part B What is the probability that an observed sedan is blue c Part C How many observed vehicles are white trucks

Consider the following dataset and determine the median 95 86 78 90 62 73 89 92 84 76

A company has both male and female employees The company has shirts and jackets with the company logo to give away to employees For each of the company s 193 employees a manager asked which piece of clothing the employee prefers The preferences based on gender are summarized in the table below Shirt Jacket 45 66 Female 7 75 Male Suppose an employee of the company is chosen at random Answer each part Do not round intermediate computations and round your answers to the nearest hundredth a What is the probability that the employee is female b What is the probability that the employee is female or prefers a jacket X

Pre school Attendance and Pre algebra Achievement In these times of educational reform attention has been focused on pre school for all children Since many districts are facing budget cuts funding pre school programs may impact other offerings Before making their recommendations administrators in a large urban district take a random sample of 50 seventh graders and compare the pre algebra achievement levels of those who attended pre school and those who did not If achievement is independent of attending pre school then the proportions at each level should be equal Use the counts in the frequency table to determine if there is an association between attending pre school and pre algebra achievement Pre school No Pre school DF 1 2 3 4 5 6 7 8 9 10 P Below grade level 8 6 0 995 0 975 0004 0 01 0 0717 0 207 0 412 0 676 0 989 1 344 1 735 2 156 0 2 00016 1 642 0 0506 3 219 0 216 4 642 0 484 0 831 1 237 1 69 9 803 2 18 11 03 2 7 3 247 At grade Level 6 15 5 989 7 779 7 289 8 558 0 1 2 706 3 841 5 024 4 605 5 991 7 378 6 251 7 815 9 348 9 488 11 149 9 236 11 07 12 833 10 645 12 592 14 449 12 017 14 067 16 013 13 362 15 507 17 535 16 919 19 023 18 307 20 483 0 05 12 242 14 684 13 442 15 987 Advanced 6 19 0 025

Suppose you are planning to collect data for a catering company Identify a potential quantitative variable that you could collect

B 80 200 240 Weight Proble 4 User Settings Grades Problems 300 Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 Problem 6 B 900 1 point The histogram below displays the weights for a certain group o a What is the width of the bins in this histogram b The bar centered over 160 includes people whose weights are between and c Is this distribution symmetric skewed to the left or skewed to the right E if symmetric 2 if skewed to the left and 3 if skewed to the right Note You can earn partial credit on this problem

BONUS Consider the following situation You want to survey a sample of registered voters about their opinions on raising taxes Please explain how you could use a stratified sampling approach to accomplish this

The width of the box is Part 1 of 2 feet An open rectangular box is 3 feet long and has a surface area of 16 square feet Find the dimensions of the box for which the volume is as large as possible O Points 0 of 1 4 of 23 points

Criminal justice researchers want to know if prison inmates in general have a lower average reading level compared to the general population What is the response variable in this situation O Classification of Reader inmate or general population Reading level Number of books Prison education

To determine customers satisfaction with a new website management attempts to survey every 5th customer as they checkout This is an example of what type of sampling O Stratified sampling Cluster sampling Systematic sampling Random sampling Convenience sampling

In May 2020 Pew Research Center surveyed 10 957 Americans The results indicated that 65 of Americans think the U S government is doing too little to reduce the effects of climate change This statement represents a Statistic Parameter

Compute d dx f g x where f x MIX 3 and g x 8 x

Fifteen randomly selected passengers on a flight are asked about the quality of the flight attendants service Identify the population All the passengers that were on that flight The flight attendants that were on duty for that flight The fifteen randomly selected passengers

Find the point s on the graph of y x 7x 5 X where the slope is 7 ACCED Type an ordered pair using integers or decimals Round the final answer to two decimal places as needed Round all intermediate values to two decimal places as needed comma to separate answers as needed

Find the dimensions of a closed rectangular box with a square base and volume 3375 in that can be constructed with the least amount of material What are the dimensions of the box The length of one side of the base is in

Sketch the graph of the following function for x 0 4 X y x 7 4 O A 14 30 X 30 OB Ay 30 X 30 5 O C Ay 20 15 X oo Q G OD 15 20 X 30

Sketch the following curve indicating all relative extreme points and inflection points y x 3x 7 The relative extreme points are Type an ordered pair Use a comma to separate answers as needed

The graph of the function has one relative extreme point Plot this point and check the concavity there Using only this information sketch the graph f x 2x 8 The relative extreme point on the graph is Type an ordered pair

In 2022 the CDC published data on the average weight of American female children by year Using ages 1 7 of this data researchers made a linear regression with age in years as the variable and weight in pounds as the y variable The equation of the line of best fit is 5 78x 17 56 According to the line of best fit what would you predict the weight would be for a 14 year old female Is it reasonable to use this line of best fit to make this prediction Select the correct answer below The predicted weight is 124 75 and the estimate is reliable and reasonable The predicted weight is 124 75 and the estimate is less reliable and unreasonable The predicted weight is 124 75 and the estimate is less reliable and reasonable The predicted weight is 98 48 and the estimate is reliable and reasonable O The predicted weight is 98 48 and the estimate is less reliable and reasonable O The predicted weight is 98 48 and the estimate is less reliable and unreasonable

The number of farms in the United States t years after 1925 is f t million where f t is graphed in Figure A and f t and f t are graphed in Figure B Click the icon to view Figure A Click the icon to view Figure B a Approximately how many farms were there in 1975 million farms Type an integer or decimal rounded to the nearest tenth as needed Figure A 6 5 4 3 2 1 Y 0 1925 10 20 y f t 30 40 50 a 60 70 t

Won a free ride to four years of college During these four years Jenn could also do any number of things including 1 Travel the world II Work to gather savings and career experience III Become a professional dancer Which of the following is the opportunity cost of Jenn attending college a Zero b Il only c The sum of I II III and any other opportunities she passed up d Whichever of I II III or any other opportunities she passed up is most valuable to her

11 Suppose the society produces one good corn using corn seed and labor Suppose they produce 500 bushels of corn output per year using 50 corn seed and 400 bushels of corn food to feed population and reproduce the labor force How much is the economic surplus in that society After that if 10 bushels of the corn surplus were added to the existing seed bin all other things remaining the same then 60 bushel of corn seed would produce 600 bushel of corn output with technology and population being same as before How much is the economic surplus in that society after using surplus to increase the size of the inputs

4 Interpret the production function Y f N L K 5 What causes economic growth

8 What are the three economic systems in history and what is the main institution that governs an economy in each system

7 Why does distribution matter

Opening Activity 13 A researcher surveyed 80 adults and found that 22 regularly run for exercise and 18 regularly swim The probability that they both run and swim is What is the probability that they run or swim 1

Using the linear relationship graphed above estimate the percent of repaired machines if 35 of the machines have defects Provide your answer below

2 A card is drawn at random from a standard deck of cards What is the probability that the card is a face card or a diamond card given that it is red ashamed overjoyed bewildered shocked A 8 13 B 1 2 C 17 26 D 6 13 E 4 13 nervous Gina Wilson All Things Algebra LLC 2018

A 1 4 B 1 3 C 5 24 D 3 8 E 7 24 7 14 15 8 21 222 A 9 10 16 17 23 24 28 29 30 31 11 12 18 19 25 26 Sara chose a date from the calendar shown above What is the probability that the date she chose is a prime number given that the date is after the 7th of the month 13 20 27 Mr Wright Ms Gilmore Mrs Silvertooth Mrs Van Duyn Mr Bowen Gina Wilson All Things Algebra LLC 2018

Upon landing airplane passengers were asked if they checked their bags and if this was their final destination Results are shown in the table below If a passenger is chosen at random find P checked bags final destination 10 5 Yes Bags No Checked A 72 4 B 68 1 C 65 3 D 64 7 E 63 2 Final Destination Yes 64 30 No 90 16 trolley bounce house haunted house space shuttle planetarium Gina Wilson All Things Algebra LLCJ 2018

Members at a gym were asked if they track their fitness and if they attend classes offered at the gym The results are given in the table below as relative joint and marginal frequencies If a person from the gym is chosen at random find the probability that they do not attend classes given they track their fitness Track Fitness Attend Classes A 31 7 B 32 5 C 34 8 D 36 1 07 Yes No Total Yes 0 384 0 24 0 624 No 0 256 0 12 0 376 Total 0 64 0 36 1 doing crunches playing with playdoh arm wrestling writing rap songs na hanios

A school survey reported that 80 of the students have cell phones 45 have part time jobs and 30 have both a cell phone and a part time job If a student is chosen at random what is the probability that the student has a part time job given that they have a cell phone Graduation Day Open House Night A 25 B 32 5 C 35 D 37 5 E 40 Halloween Valentine s Day April Fools Day Gina Wilson All Things Algebra LLC 2018

A restaurant sells chocolate and vanilla milkshakes in three sizes Milkshake orders for a certain day are given in the table below as relative joint and marginal frequencies Based on this data if a milkshake order is placed find P vanilla medium or large Cup Size Flavor Chocolate Vanilla Total A 39 5 B 41 2 C 43 7 D 44 9 E 46 4 Small 0 10 0 05 0 15 Medium 0 30 0 225 0 525 Large 0 20 0 125 0 325 Total 0 60 0 40 1 were live on Instagram lost a bet won a radio contest were on vacation wanted to get a laugh

4 Of the 100 employees at a company 75 are married and 57 have at least one child Thirty three of the employees married do not have any children If an employee is chosen at random find P has at least one child married A 14 25 B 7 15 C 2 5 D 11 25 E 19 30 Mr Van Duyn Mr Kissick Mrs Van Duyn Mr Kronstedt Ms McClanahan Gina Wilson All Things Algebra LLCJ 2018

3 Of the 60 students in orchestra 24 are in chorus and 15 are in photography Four of the students are in chorus and photography If one student is chosen at random what is the probability that this student is in chorus given that they are not in photography A 9 20 B 2 5 C 4 9 D 5 12 E 1 3 playing basketball mini golfing doing leg extensions snowboarding driving bumper cars Gina Wilson All Things Algebra LLC 2018

The Venn diagram shows the frequency of events A and B 625 A Part A What is P A and B Part B What is P A or B 20 255 B 135

The number of questions marked incorrect on a final y is dependent on the number of hours the student studies during the week I and can be modeled by the function g z 40 2 5 Draw the graph of the function by plotting its y intercept and another point Provide your answer below Number of Incorrect Problems 60 50 40 30 20 10 0 16 2640 F

A city conducted a census to determine the types of schools attended by the city s students Hannah finds some of the data from the census on different websites and constructs the two way relative frequency table shown Use the table to complete the problem Grade Level Elementary Middle school High school Total Public non charter 26 6 23 2 16 2 66 0 Type of School Public charter 5 0 3 5 5 5 14 0 Private 6 4 5 3 8 3 20 0 Total 38 0 32 0 30 0 100 What is the probability expressed as a percentage that a student in the city attends a public non charter high school Enter the percent in the box Round to the nearest percent as needed

Live in the city took public transit 125 Live in the city took no public transit 65 Live outside the city took public transit 40 Live outside the city took no public transit 270 The photo shows the breakdown of responses Live in City Yes Live in City No Total Yes No Took Public Transit Yes No Total 125 Total 40 165 The two way frequency table of the data is given above Yes 0 25 65 0 08 270 0 33 335 190 Took Public Transit 310 500 No 0 13 0 54 0 67 Total 0 38 0 62 1 00

Live in City Yes No Total 165 Live in City 125 Yes 40 Show Hints No Total Make a two way relative frequency table of the data Enter the frequencies in the boxes Give your answers as decimals without rounding 65 270 Yes 190 310 335 500 Took Public Transit 0 33 No Total 0 38 0 62 1 00

Probability O 1 120 O 1 220 O 1 320 1 420 1 520 1 620 Permutation Formula n nP n r Number of Favorable Outcomes Total Number of Outcomes Combination Formula nC n n r r A math question has 6 steps and you must place them in order If you guess the order of the steps what is the probability you guess the correct order

O 1 10 1 5 1 20 1 40 Probability Permutation Formula n n r A You have 5 pair of flip flops A green pair a red pair a blue pair a whi pair and a black pain If you randomly select 3 pair to take on vacation what is the probability you choose the black white and red flip flops 1 45 O 1 60 of Favorable Outcomes Total Number of Outcomes Combination Formula n n r r nC

O 1 10 O 1 5 O 1 20 O 1 40 1 45 O 1 60 Permutation Formula n nPr n r A bucket contains the names of 10 people including you and your friend If 2 names are randomly selected find the probability the first name selected is yours and the second name selected is your friend s name 1 65 Combination Formula n Cr n r r

1 480 O 1 680 1 880 Permutation Formula n n r 1 1080 O 1 1280 Pr 1 1480 Probability If a lock code uses the digits 1 8 what is the probability the 4 digit code is 5555 Number of Favorable Outcomes Total Number of Outcomes Combination Formula n nC n r r

1 110 O 1 150 O 1 180 O 1 210 O 1 240 Permutation Formula n nPr n r 1 270 Total Number of Outcomes At a talent show the order in which people perform is chosen randomly If there are 7 people performing what is the probability the first three people are you then your sister then your best friend Combination Formula nC n n r r

Permutation Formula nPr n n r A daycare has 8 girls and 6 boys Five children are randomly selected to form a leader group What is the probability all 5 are boys O 1 1000 O 1 1001 O 1 1002 O 1 1003 O 3 1000 O 3 1001 011000 Combination Formula n nC n r r

O 1 3 O 1 4 O 1 5 O 1 6 O 1 7 O 1 8 Probability 1 9 Permutation Formula nPr n n r Number of Favorable Outcomes Total Number of Outcomes A basketball team has 9 players Seven players are 13 years old anc 2 players are 12 years old If the starting 5 players is randomly selected find the probability all 5 are 13 years old Combination Formula n nCr n r r

O 1 10 O 1 5 O 1 20 1 75 O 1 100 O 1 120 Probability 1 130 Permutation Formula nPr n n r Number of Favorable Outcomes Total Number of Outcomes A bag a candy contains 3 red 2 blue 4 yellow and I purple If you randomly select 3 candies from this bag what is the probability that all 3 are red Combination Formula n nC n r r

Give a practical interpretation of the 99 confidence interval For example you might Say I am 99 confident that the true mean height of all of the people in my company is between 64 inches and 68 inches White a complete Sentence

Confidence interval The range of values consistent with the data that is believed to encompass the actual or true population value Sauke TA Sex M 2000 How to Repost Statistics in Medicine Chd ed Philadelphia American College of Pesicians To draw conclusions about a study population researchers use samples that they assume truly represent the population The confidence interval CI is among the most reliable indicators of the soundness of their assumption A CI is the range of values within which the population value being studied is believed to fall Cls are reported in the results section of published research and are often calculated either for mean or proportion data calculation details are beyond the scope of this article A 95 CI which is the most common level used others are 90 and 99 means that if researchers were to sample numerous times from the same population and calculate a range of estimates for these samples 95 of the intervals within the lower and upper limits of this range will include the population value To illustrate the 95 CI of a mean value say that a sample of patients with hypertension has a mean blood pressure of 120 mmHg and that the 95 CI for this mean was calculated to range from 110 to 130 mmHg This might be reported as mean 120 mmHg 95 CI 110 130 mmHg It indicates that if other samples from the same population of patients were generated and intervals for the mean blood pressure of these samples were estimated 95 of the intervals between the lower limit of 110 mmHg and the upper limit of 130 mmHg would include the true mean blood pressure of the population Notice that the width of the Cl range is a very important indicator of how reliably the sample value represents the population in question If the Cl is narrow as it is in our example of 110 130 mmHg then the upper and lower limits the sample This sample mean value is probably a more reliable estimate of the true mean value of the population than a sample mean value with a wider CI of for example 110 210 mmHg With such a wide CI the population mean could be as high as 210 mmHg which is far from the sample mean of 120 mmHg In fact a very wide CI in a study should be a red flag it indicates that more data should have been collected before any serious conclusions were drawn about the population Remember the narrower the Cl the more likely it is that the sample value represents the population value NurseONE resources ON THIS TOPIC EBSCO MEDLINE FULL TEXT ARTICLES Hildebrandt M Vervolgyl E Bender R 2009 Calculation of NNTs in RCTs with time to event outcomes A literature review BMC Medical Research Methodology 9 21 Hildebrandt M Bender R Gehrmann U Blettner M 2006 Calculating confidence intervals for impact numbers BMC Medical Research Methodology 6 32 Altman D G 1998 Confidence intervals for the number needed to treat BM Clinical Research Ed 317 7168 1309 1312 MYILIBRARY Campbell M J Machin D Walters S L 2010 Medical statistics A textbook for the health sciences 4th ed Mateo M A Kirchhoff K T Eds 2009 Research for advanced practice nurses From evidence to practice