# Go Math Grade 8 Answer Key Chapter 15 Two-Way Tables

## Go Math Grade 8 Chapter 15 Two-Way Tables Answer Key

Lesson 1: Two-Way Frequency Tables

Lesson 2: Two-Way Relative Frequency Tables

Model Quiz

Mixed Review

### Guided Practice – Two-Way Frequency Tables – Page No. 454

Question 1.
In a survey of 50 students, 60% said that they have a cat. Of the students who have a cat, 70% also have a dog. Of the students who do not have a cat, 75% have a dog. Complete the two-way table.

a. Enter the total number of students surveyed in the bottom right cell of the table.
Type below:
_______________

Explanation:
In a survey of 50 students, 60% said that they have a cat.
In mathematical terms:
Cat = 0.6×50 = 30
If 60% have a cat, then 40% don’t have a cat
No Cat = (1 – 0.6) × 50 = 20
Because there are 2 options, Adding them will give the total amount of students
Total = Cat + No Cat = 50
Of the students who have a cat, 70% also have a dog. Of the students who do not have a cat, 75% have a dog. So, in mathematical terms:
Dog = Cat × 0.7 + No Cat ×0.75 = 30 × 0.7 + 20 × 0.75 = 36
Following the same logic as before, if 70% of students who have a cat also have a dog, then 30% of them don’t have a dog. The same analysis for the students who do not have a cat.
No Dog = Cat × (1-0.7) + No Cat × (1-0.75) = 30 × (1-0.7) + 20 × (1-0.75) = 14
Again, the addition of the 2 options has to give the total amount of students
Total = 50

Question 1.
b. Fill in right column.
Type below:
_______________

Of the students who have a cat, 70% also have a dog. Of the students who do not have a cat, 75% have a dog. In mathematical terms:
Dog = Cat × 0.7 + No Cat ×0.75 = 30 × 0.7 + 20 × 0.75 = 36

Question 1.
c. Fill in top row.
Type below:
_______________

In a survey of 50 students, 60% said that they have a cat. In mathematical terms:
Cat = 0.6×50 = 30

Question 1.
d. Fill in second row.
Type below:
_______________

If 60% have a cat, then 40% don’t have a cat
No Cat = (1 – 0.6) × 50 = 20

Question 1.
e. Fill in last row.
Type below:
_______________

Because there are 2 options, the addition of them has to give the total amount of students
Total = Cat + No Cat = 50

Question 2.
The results of a survey at a school are shown. Is there an association between being a boy and being left-handed? Explain.

_______________

No, there isn’t any association between being a boy and being left-handed.
Boys are no more likely to be left-handed than right-handed.

ESSENTIAL QUESTION CHECK-IN

### 15.1 Independent Practice – Two-Way Frequency Tables – Page No. 455

Question 4.
Represent Real-World Problems One hundred forty students were asked about their language classes. Out of 111 who take French, only 31 do not take Spanish. Twelve take neither French nor Spanish. Use this information to make a two-way table.

Type below:
_______________

Question 5.
Represent Real-World Problems Seventh- and eighth-grade students were asked whether they preferred science or math.
a. Complete the two-way table.

Type below:
_______________

Question 5.
b. Is there an association between being in eighth grade and preferring math? Explain.
_______________

There is no association as such between being in eighth grade and preferring maths. But the total no. of eighth-grade students choosing maths is greater than the total no. of students in seventh grade-preferring science. So, the eighth-grade students preferred it.

Question 6.
Persevere in Problem-Solving The table gives partial information on the number of men and women who play in the four sections of the Metro Orchestra.
a. Complete the table.

Type below:
_______________

Question 6.
b. Is there an association between being a woman and playing strings? Explain.
_______________

There is no association between being a woman and playing strings since the number of men playing strings is less than women.

### Two-Way Frequency Tables – Page No. 456

FOCUS ON HIGHER ORDER THINKING

Question 7.
Multi-Step The two-way table below shows the results of a survey of Florida teenagers who were asked whether they preferred surfing or snorkeling.
a. To the right of the number in each cell, write the relative frequency of the number compared to the total for the row the number is in. Round to the nearest percent.

Type below:
_______________

Question 7.
b. Explain the meaning of the relative frequency you wrote beside 28.
Type below:
_______________

The relative frequency shows the percentage of people aged 16-18 that prefer snorkeling.

Question 7.
c. To the right of each number you wrote in part a, write the relative frequency of each number compared to the total for the column the number is in. Are the relative frequencies the same? Why or why not?
Type below:
_______________

Question 7.
d. Explain the meaning of the relative frequency you wrote beside 28.
Type below:
_______________

The relative frequency represents the percentage of people that prefer snorkeling that is aged 16-18.

### Guided Practice – Two-Way Relative Frequency Tables – Page No. 462

Question 1.
In a class survey, students were asked to choose their favorite vacation destination. The results are displayed by gender in the two-way frequency table.

a. Find the total for each gender by adding the frequencies in each row. Write the row totals in the Total column.
Type below:
_______________

Girl = 7 + 3 + 2 = 12
Boy = 5 + 2 + 6 = 13

Question 1.
b. Find the total for each preferred vacation spot by adding the frequencies in each column. Write the column totals in the Total row.
Type below:
_______________

Seashore = 7 + 5 = 12
Mountains = 3 + 2 = 5
Other = 2 + 6 = 8

Question 1.
c. Write the grand total (the sum of the row totals and the column totals) in the lower-right corner of the table.
Type below:
_______________

grand total = 25

Question 1.
d. Create a two-way relative frequency table by dividing each number in the above table by the grand total. Write the quotients as decimals.

Type below:
_______________

Explanation:
7/25 = 0.28, 3/25 = 0.12; 2/25 = 0.08; 12/25 = 0.48
5/25 = 0.2; 2/25 = 0.08; 6/25 = 0.24; 13/25 = 0.52
12/25 = 0.48; 5/25 = 0.2; 8/25= 0.32; 25/25 = 1

Question 1.
e. Use the table to find the joint relative frequency of students surveyed who are boys and who prefer vacationing in the mountains.
_________

Joint relative frequency of boys = 2/25 = 0.08
These boys who prefer vacationing in the mountains.

Question 1.
f. Use the table to find the marginal relative frequency of students surveyed who prefer vacationing at the seashore.
_________

The marginal relative frequency of students = 12/25 = 0.48
These are the number of students who prefer vacationing in the seashore.

Question 1.
g. Find the conditional relative frequency that a student surveyed prefers vacationing in the mountains, given that the student is a girl. Interpret this result.
_________

The condition the relative frequency of girls of row = 3/12 = 0.25
And that of the column is 3/5 = 0.6
These are the number of girls who preferred vacationing in the mountains.

ESSENTIAL QUESTION CHECK-IN

### 15.2 Independent Practice – Two-Way Relative Frequency Tables – Page No. 463

Stefan surveyed 75 of his classmates about their participation in school activities as well as whether they have a part-time job. The results are shown in the two-way frequency table. Use the table for Exercises 3–6.

Question 3.
a. Complete the table.
Type below:
_______________

Question 3.
b. Explain how you found the correct data to enter in the table.
Type below:
_______________

1) In the first row of yes the values of sports only, Neither nor total were provided. Also, in the 1st column of cubes, only the values of No and total were providers. So, these values were subtracted and the value of yes was known.
2) The values in the 1st row of yes were added and subtracted from the total column. Hence the value in both columns was known. So, similarly, by adding and subtracting the values in the rows and columns the vacant values were known.

Question 4.
Create a two-way relative frequency table using decimals. Round to the nearest hundredth.

Type below:
_______________

Question 5.
Give each relative frequency as a percent.
a. the joint relative frequency of students surveyed who participate in school clubs only and have part-time jobs
_________ %

13%

Explanation:
The joint relative frequency of students surveyed who participate in school clubs only and have part-time jobs 0.13 or 13%
(Job and clubs only: 10/75 = 0.13)

Question 5.
b. the marginal frequency of students surveyed who do not have a part-time job
_________ %

32%

Explanation:
The marginal frequency of students surveyed who do not have a part time job is 0.32 or 32%
(No job total: 24/75 = 0.32)

Question 5.
c. the conditional relative frequency that a student surveyed participates in both school clubs and sports, given that the student has a part-time job
_________ %

39%

Explanation:
The conditional relative frequency that a student surveyed participates in both school clubs and sports, given that the student has a part-time job is 0.39 or 39%
(20/51 = 0.39)

### Two-Way Relative Frequency Tables – Page No. 464

FOCUS ON HIGHER ORDER THINKING

Question 7.
The head of quality control for a chair manufacturer collected data on the quality of two types of wood that the company grows on its tree farm. The table shows the acceptance and rejection data.

a. Critique Reasoning To create a two-way relative frequency table for this data, the head of quality control divided each number in each row by the row total. Is this correct? Explain.
_______________

No, it is not correct for the head of quality control to divide each number in each row by the row total to create a two-way relative frequency table. Each data value should have been divided by 600, the grand total, not by the row total.

Question 7.
b. Draw Conclusions Are any of the data the head of quality control entered into the two-way relative frequency table correctly? If so, which is and which isn’t? Explain.
Type below:
_______________

Since the head of quality control is divided incorrectly, the top two rows are incorrect. However, the bottom row has correct data because each number in the bottom row was divided by the grand total.

### Ready to Go On? – Model Quiz – Page No. 465

15.1 Two-Way Frequency Tables

Martin collected data from students about whether they played a musical instrument. The table shows his results. Use the table for Exercises 1–4.

Question 1.
Of the students surveyed, how many played an instrument?
__________ students

90 students

Explanation:
Of the students surveyed, 90 students played an instrument

15.2 Two-Way Relative Frequency Tables

Students were asked how they traveled to school. The two-way relative frequency table shows the results. Use the table for Exercises 5–7. Write answers as decimals rounded to the nearest hundredth.

Question 5.
What is the joint relative frequency of high school students who ride the bus?
________

The joint relative frequency of high school students who ride the bus is 0.12

Question 6.
What is the marginal relative frequency of students surveyed who are in middle school?
________

The marginal frequency of students surveyed in middle school is 0.42

ESSENTIAL QUESTION

### Selected Response – Mixed Review – Page No. 466

The table gives data on the length of time that teachers at Tenth Avenue School have taught. Use the table for Exercises 1–5.

Question 1.
How many female teachers have taught for fewer than 10 years?
Options:
a. 4
b. 9
c. 21
d. 30

c. 21

Explanation:
(No. of male teachers who have taught for fewer than 10 years) + (No. of female teachers who have taught fewer than 10 years) = 30
9 + x = 30
x = 21
The number of female teachers who have taught for fewer than 10 years is 21.

Question 2.
What is the relative frequency of teachers who have taught for 10 or more years?
Options:
a. 10%
b. 25%
c. 30%
d. 60%

b. 25%

Explanation:
The relative frequency of teachers who have taught for more than 10 or more years.
Total no. of teachers = 40
No. of teachers who taught for more than 10 years = 10
Relative frequency = (10/40) . 100 = 25%

Question 3.
What is the relative frequency of having taught for fewer than 10 years among male teachers?
Options:
a. 0.09
b. 0.225
c. 0.6
d. 1.50

c. 0.6

Explanation:
The relative frequency of male teachers who have taught fewer than 10 or more years.
Total no. of teachers = 15
No. of male teachers who taught for fewer than 10 years = 9
Relative frequency = (9/15) = 0.6

Question 4.
What is the joint relative frequency of female teachers who have taught for more than 10 years?
Options:
a. 4%
b. 10%
c. 16%
d. 25%

b. 10%

Explanation:
The relative frequency of female teachers who taught for more than 10 years is 4/40 = 1/10 = 0.1 × 100 to calculate the data in percentage
10%

Question 5.
What is the marginal relative frequency of teachers who are female?
Options:
a. 0.16
b. 0.25
c. 0.4
d. 0.625

d. 0.625

Explanation:
The total no. of teachers who are female = 25
Total no. of teachers = 40
Marginal frequency = 25/40 = 0.625

Question 6.
A triangle has an exterior angle of x°. Which of the following represents the measure of the interior angle next to it?
Options:
a. (180 − x)°
b. (x − 180)°
c. (90 − x)°
d. (x − 90)°

a. (180 − x)°

Explanation:
The triangle has an exterior angle of x°. Let that angle be Angle ACD. So, the angle next to it is
Angle ACD + Angle ACB = 180º
Angle ACB = (180 − x)°

Question 7.
What is the volume of a cone that has a diameter of 12 cm and a height of 4 cm? Use 3.14 for π and round to the nearest tenth.
Options:
a. 25.12 cm3
b. 602.88 cm3
c. 150.72 cm3
d. 1,808.64 cm3

c. 150.72 cm3

Explanation:
Diameter = 12cm
Radius r = 6cm
height h = 4cm
So, the volume of the cone = 1/3 . π . r². h
= 1/3 . 6 . 6 . 4 . 3.14 = 150.72 cm³

Question 8.
The table gives data on books read by members of the Summer Reading Club.

a. Find the relative frequency of a club member reading fewer than 25 books.
________ %

25%

Explanation:
The relative frequency of a club member reading fewer than 25 books is
Total of 16 members read fewer than 25 books
16/64 = 0.25 or 25%

Question 8.
b. Find the relative frequency of reading fewer than 25 books among girl club members.
________ %

14%

Explanation:
The relative frequency of a girl club member reading fewer than 25 books is
9/64 = 0.14 or 14%

Question 8.
c. Is there an association between being a girl and reading fewer than 25 books? Explain.
____________