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12-8 Two-Way Frequency Tables, Study notes of Reasoning

Community College of BaltimoreReasoning

Make a two-way frequency table to organize the given data. 1. MEALS Students in elementary school and high school were surveyed about eating habits. Thirty-.

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Makeatwo-wayfrequencytabletoorganizethe
givendata.
1.MEALSStudentsinelementaryschoolandhigh
schoolweresurveyedabouteatinghabits.Thirty-
eightstudentsinelementaryschoolatebreakfast,and
12skippedbreakfast.Twenty-twostudentsinhigh
schoolatebreakfast,and28skippedbreakfast.
SOLUTION:
ANSWER:
2.CLASSTRIPSArandomsampleofhighschool
seniorswasgivenasurveytodeterminethelocation
oftheirclasstrip.Thetwochoiceswerean
amusementparkoranaquarium.Theamusement
parkwasthechoiceof28girlsand23boys.The
aquariumwasthechoiceof9girlsand8boys.
SOLUTION:
ANSWER:
Mayumisurveyedarandomsampleofpeople
whocommutetoworkonabusoronatrain,
andsheaskedthemwhetherornottheyowna
car.Thedataareshowninthetwo-way
frequencytable.Answereachquestionandtell
whetheryouareusingamarginalfrequencyor
jointfrequency.
3.Howmanycommutersrideatrainandownacar?
SOLUTION:
Lookatthetwo-wayfrequencytable,specificallythe
intersectionoftherowlabeled"Train"andthecolumn
labeled"Car."Thereare16commutersinthis
intersection.Thisissolvedusingajointfrequency.
ANSWER:
16;jointfrequency
4.Howmanypeoplewhocommuteonatrainwere
surveyed?
SOLUTION:
Lookatthetwo-wayfrequencytable,specificallythe
rowlabeled"Train."Thereare50commutersinthis
row.Thisissolvedusingamarginalfrequency.
ANSWER:
50;marginalfrequency
5.Howmanycarownersweresurveyed?
SOLUTION:
Lookatthetwo-wayfrequencytable,specificallythe
columnlabeled"Car."Thereare24commutersinthis
column.Thisissolvedusingamarginalfrequency.
ANSWER:
24;marginalfrequency
eSolutionsManual-PoweredbyCogneroPage1
12-8Two-WayFrequencyTables
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Make a two-way frequency table to organize the given data.

  1. MEALS Students in elementary school and high school were surveyed about eating habits. Thirty- eight students in elementary school ate breakfast, and 12 skipped breakfast. Twenty-two students in high school ate breakfast, and 28 skipped breakfast. SOLUTION: ANSWER:
  2. CLASS TRIPS A random sample of high school seniors was given a survey to determine the location of their class trip. The two choices were an amusement park or an aquarium. The amusement park was the choice of 28 girls and 23 boys. The aquarium was the choice of 9 girls and 8 boys. SOLUTION: ANSWER: Mayumi surveyed a random sample of people who commute to work on a bus or on a train, and she asked them whether or not they own a car. The data are shown in the two-way frequency table. Answer each question and tell whether you are using a marginal frequency or joint frequency.
  3. How many commuters ride a train and own a car? SOLUTION: Look at the two-way frequency table, specifically the intersection of the row labeled "Train" and the column labeled "Car." There are 16 commuters in this intersection. This is solved using a joint frequency. ANSWER: 16; joint frequency
  4. How many people who commute on a train were surveyed? SOLUTION: Look at the two-way frequency table, specifically the row labeled "Train." There are 50 commuters in this row. This is solved using a marginal frequency. ANSWER: 50; marginal frequency
  5. How many car owners were surveyed? SOLUTION: Look at the two-way frequency table, specifically the column labeled "Car." There are 24 commuters in this column. This is solved using a marginal frequency. ANSWER: 24; marginal frequency
  1. How many car owners ride a bus? SOLUTION: Look at the two-way frequency table, specifically the intersection of the row labeled "Bus" and the column labeled "Car." There are 8 commuters in this intersection. This is solved using a joint frequency. ANSWER: 8; joint frequency
  2. Convert the two-way frequency table of data about commuters to a table of relative frequencies. SOLUTION: ANSWER:
  3. Determine whether commuting to work on a bus or train is independent of owning a car. Explain. SOLUTION: No; the expected and actual joint relative frequencies are not the same. ANSWER: No; the expected and actual joint relative frequencies are not the same. Use the two-way frequency table of data about commuters to find each conditional probability.
  4. the probability that a commuter rides a bus given that he or she owns a car SOLUTION: P (bus|car) = or 33.3% ANSWER: 33.3%
  5. the probability that a commuter does not own a car given that he or she rides a train SOLUTION: P (not car|train) = or 68% ANSWER: 68%
  6. the probability that a car owner rides a train SOLUTION: P(train|car) = or 66.7% ANSWER: 66.7%
  7. the probability that a train rider owns a car SOLUTION: P (car|train) = or 32% ANSWER: 32%
  1. How many members of the softball team were surveyed? SOLUTION: Look at the two-way frequency table, specifically the row labeled "Softball." There are 90 athletes in this row. This is solved using a marginal frequency. ANSWER: 90; marginal frequency
  2. How many team members who prefer water water are soccer players? SOLUTION: Look at the two-way frequency table, specifically the column labeled "Water." There are 36 soccer players in this column. This is solved using a joint frequency. ANSWER: 36; joint frequency
  3. How many softball players do not prefer a sports drink? SOLUTION: Look at the two-way frequency table, specifically the row labeled "Softball." There are 54 players that don't prefer a sports drink in this row. This is solved using a joint frequency. ANSWER: 54; joint frequency
  4. Convert the two-way frequency table of data about sports teams and beverages to a table of relative frequencies. SOLUTION: ANSWER:
    1. Determine whether a team member's beverage preference is independent of his or her team. Explain. SOLUTION: Yes; the expected and actual joint relative frequencies are the same. ANSWER: Yes; the expected and actual joint relative frequencies are the same. Use the two-way frequency table of data about sports teams and beverages to find each conditional probability.
    2. the probability that a team member prefers water, given that he or she is on the soccer team SOLUTION: P (water|soccer) = or 60% ANSWER: 60%
    3. the probability that a team member who prefers a sports drink is on the softball team SOLUTION: P (softball|sports drink) = or 60% ANSWER: 60%
    4. the probability that a member of the soccer team prefers a sports drink SOLUTION: P (sports drink|soccer) = or 40% ANSWER: 40%
  1. the probability that a team member who prefers water is not on the soccer team SOLUTION: P (not soccer|water) = or 60% ANSWER: 60%
  2. FLOWERS A florist surveyed a random sample of customers, by asking 20 men and 20 women whether they would rather receive one dozen roses or one dozen tulips. Of the women surveyed, 15 chose one dozen tulips. a. What is the probability that a customer chose tulips given that she is a woman? b. What is the joint relative frequency of women who chose roses? c. Can you determine the conditional probability of a customer choosing one dozen roses, given that he is man? If so, explain how to find the probability. If not, explain why not and describe any additional information you would need. SOLUTION: a. P (tulips|woman) = or 75% b. or 12.5% c. No, you cannot determine the conditional probability of a customer choosing one dozen roses, given that he is a man because there is not enough information. You need to know the number of men in the sample who chose roses or tulips. ANSWER: a. 75% b. 12.5% c. No; there is not enough information; need to know the number of men in the sample who chose roses or tulips. REASONING The two-way frequency tables show data from a school survey about preferred school colors. Copy and complete each table.

SOLUTION: ANSWER:

SOLUTION: ANSWER:

62.5%, which is greater than the marginal relative frequency for AP Classes.

  1. ANIMAL SHELTERS An employee at an animal shelter collected data on a random sample of cats and dogs as they were brought to the shelter. Copy and complete the table assuming that having fleas is independent of being a cat or a dog. SOLUTION: ANSWER:
    1. SENSE-MAKING Darius surveyed a random sample of 250 employees at the JQP Corporation to find out what the employees do for lunch. Of the 122 men that he surveyed, 43 bring their lunch, 56 eat at the company cafeteria, and the rest go out to get lunch. Of the women that he surveyed, 46 bring their lunch and 30 go out to get lunch. a. Make a two-way frequency table to organize the data. b. What is the probability that a randomly chosen employee is a man, given that the employee eats lunch at the company cafeteria? SOLUTION: a. b. P(man|cafeteria) = or 51.9% ANSWER: a. b. 51.9%
  1. OPEN-ENDED Make a two-way frequency table that shows the results of a survey in which 420 people were surveyed and 30% of the people surveyed were male. SOLUTION: ANSWER:
  2. ERROR ANALYSIS The two-way frequency table shows data about a random sample of apples at a supermarket. Kaci was asked to find the probability that a randomly chosen apple is red, given that it is organic. Her work is shown below. Explain her error and find the correct answer. SOLUTION: Kaci found the probability that a randomly chosen apple is organic, given that it is red. The correct answer is = 60%. ANSWER: Kaci found the probability that a randomly chosen apple is organic, given that it is red. The correct answer is = 60%.
    1. REASONING A standard two-way frequency table, in which each variable has two categories, contains 9 cells with numerical values. What is the minimum number of these values that you need to know in order to fill in the rest of the table? SOLUTION: 4 ANSWER: 4
    2. CHALLENGE In a two-way relative frequency table in which each variable has two categories, all of the joint relative frequencies are equal. What is the value of the joint relative frequencies? SOLUTION: 25% ANSWER: 25%
    3. WRITING IN MATH Describe the steps for using a two-way frequency table to determine whether two variables are independent of each other. SOLUTION: Compare the joint relative frequencies with the products of the corresponding marginal relative frequencies. If these are equal, the variables are independent. ANSWER: Compare the joint relative frequencies with the products of the corresponding marginal relative frequencies. If these are equal, the variables are independent.
  1. The manager of a theater asked members of a ballet audience and members of a jazz concert audience whether they would like to learn about upcoming events by a text message or by an email. The variables in the table represent relative frequencies for these data. Which expression can the manager use to find the probability that someone is a member of the ballet audience, given that he or she prefers a text message? A B C D SOLUTION: P (ballet|text) = There are a total of g people who prefer a text message and there are a people in that column that are in the ballet audience. ANSWER: C
    1. MULTI-STEP Malik surveyed vegetarians and nonvegetarians about their peanut butter preferences. He recorded some of the data in the two-way frequency table shown. a. Copy and complete the table. b. Convert the table to a table of relative frequencies. c. Determine whether someone's peanut butter preference is independent of whether or not he or she is a vegetarian. Explain. SOLUTION: a. b. c. Yes; the expected and actual joint relative frequencies are the same. ANSWER: a. b. c. Yes; the expected and actual joint relative frequencies are the same.