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# Two-Way Frequency Tables (Grade 10)

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## Two-Way Frequency Tables

1.
Fill in the following two-way frequency table for the given example.

A soft drink company conducts a survey about people's preferences for one of the company's existing drinks and a new drink they are coming out with. 50 people were surveyed. There were two groups of people, one with people 35 years of age and older (group A), and the other with people younger than 35 (group B). Each person was asked which drink they liked better. For the existing drink, 10 of group A preferred it while 17 of group B preferred it. For the new drink, 20 people from group A preferred it.

  $"Existing Drink"$ $"New Drink"$ $\mathbf{ Total}$ $"Group A"$    $"Group B"$    $\mathbf{Total}$   

2.
A survey was done in the town of Milton to see which diner people prefer most out of the three diners in town. The survey also asked whether the person usually ate out between 6 a.m. and 2 p.m., or between 2 p.m. and 10 p.m.

  $" 6 a.m. - 2 p.m."$ $"2 p.m. - 10 p.m."$ $\mathbf{ Total}$ $"Mary's Diner"$ $23$ $40$ $63$ $"Flo's Diner"$ $25$ $8$ $33$ $"Spoons Diner"$ $35$ $19$ $54$ $\mathbf{Total}$ $83$ $67$ $150$
A.
What is the probability that someone chosen randomly prefers Flo's Diner?
1. 0.76
2. 0.30
3. 0.22
4. 0.12
B.
What is the probability that someone chosen at random eats out at a diner between 2 p.m. and 10 p.m.?
1. 0.60
2. 0.45
3. 0.35
4. 0.24
C.
What is the conditional probability that someone who eats out between 6 a.m. and 2 p.m. prefers Mary's Diner?
1. 0.28
2. 0.37
3. 0.42
4. 0.55
D.
What is the conditional probability that someone who prefers Spoon's Diner eats out between 6 a.m. and 2 p.m.?
1. 0.28
2. 0.36
3. 0.55
4. 0.65
E.
What is the conditional probability that someone prefers Flo's, given that they eat out between 2 p.m. and 10 p.m.?
1. 0.05
2. 0.12
3. 0.22
4. 0.24
F.
Is going to Spoons Diner independent of eating out between 2 p.m. and 10 p.m.? Why or why not? There may be more than one correct answer.
1. The probability that someone eats out between 2 p.m. and 10 p.m. is 0.45. The probability that someone likes Spoons, given that they eat out between 2 p.m. and 10 p.m. is 0.28. Because these are not equal, the two events are not independent.
2. The probability that someone like's Spoons is 0.36, while the conditional probability that someone prefers Spoons given that they eat between 2 p.m. and 10 p.m. is 0.28. Since these are not equal, the events are not independent.
3. The probability that someone eats out between 2 p.m. and 10 p.m. is 0.45. The conditional probability that someone eats out between 2 p.m. and 10 p.m. given that they prefer Spoons is 0.35. Since these are no equal, the events are not independent.
4. Because the probability that someone prefers Spoons is 0.36, and this is almost equal to the conditional probability that someone eats out between 2 p.m. and 10 p.m. given that they like Spoons (which is 0.35), the events are independent.
3.
Residents of two different neighborhoods within a city were asked what their favorite sport is, basketball or soccer, with the results in the two-way table below. Is living in Leaside independent of someone's favorite sport, and why or why not?

  $"Soccer"$ $"Basketball"$ $\mathbf{ Total}$ $"Leaside"$ $51$ $26$ $77$ $"Danforth"$ $15$ $8$ $23$ $\mathbf{Total}$ $66$ $34$ $100$
1. No, they are not independent, because a different number of people chose each sport.
2. No, they are not independent, since P(Basketball) $!=$ P(Leaside | Basketball).
3. Yes, they are independent, because P(Soccer | Leaside) $~~$ P(Soccer) and P(Leaside | Soccer) $~~$ P(Leaside).
4. Yes, they are independent, since P(Soccer and Leaside) $~~$ P(Soccer) $xx$ P(Leaside) and P(Basketball and Leaside) $~~$ P(Basketball) $xx$P(Leaside).
4.
The following two-way table shows how many people own a truck versus a car, and of these people, which type of bread they normally eat. There are 4 numbers missing, starting with the value for those who own a car and eat whole wheat bread, those who own a truck and eat whole wheat bread, the total number of people who own a car, and the total number of people who own a truck. Which of the following answers, with numbers listed in the same order as just mentioned, would make owning a car or truck independent of eating white or whole wheat bread?

  $"Car"$ $"Truck"$ $\mathbf{ Total}$ $"White Bread"$ $13$ $17$ $30$ $"Whole Wheat Bread"$   $45$ $\mathbf{Total}$   $75$
1. 10, 35, 23, 52
2. 14, 31, 27, 48
3. 19, 26, 32, 43
4. 24, 21, 37, 38
5.
95 people were asked what type of soft serve ice cream they were most likely to buy. Their choices were vanilla, chocolate, and swirl (a combination of vanilla and chocolate). The chart shows the results. Which of the following statements are correct? There may be more than one correct answer.

  $"Vanilla"$ $"Chocolate"$ $"Swirl"$ $\mathbf{ Total}$ $"Male"$ $6$ $22$ $9$ $37$ $"Female"$ $10$ $27$ $21$ $58$ $\mathbf{Total}$ $16$ $49$ $30$ $95$
1. There is almost no difference in probability in choosing someone who likes vanilla, whether choosing from all people, only men, or only women.
2. It is nearly equally likely that you will choose someone at random who likes swirl, whether choosing from all people, only men, or only women.
3. It is more likely that you will choose a man from people who like swirl, than if you choose from people who like vanilla.
4. Randomly choosing a woman from people who like vanilla is more likely than choosing a woman from people who like chocolate.
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