Back to Exercise: Summarize categorical data in two-way frequency tables

Exercises: Summarize Categorical Data in Two-Way Frequency Tables

Work through each section in order. Many problems use a two-way frequency
table; read it carefully and note the row totals, column totals, and grand
total. Before computing any relative frequency, ask "out of whom?" — the
denominator is the grand total for joint and marginal frequencies, but a row
or column total for a conditional frequency. Express relative frequencies as
percents rounded to the nearest tenth of a percent unless told otherwise.

Grade 9·20 problems·~35 min·Common Core Math - HS Statistics and Probability·group·hss-id-b-5

Printable layout

Warm-Up: Reading a Two-Way Table

These problems review reading counts and totals from a two-way frequency table.

A survey of 80 students recorded grade level (9th or 10th) and learning preference (online or in-person):

	Online	In-person	Total
9th	18	22	40
10th	28	12	40
Total	46	34	80

How many students are 10th graders who prefer online learning?

Using the same table:

	Online	In-person	Total
9th	18	22	40
10th	28	12	40
Total	46	34	80

What does the number 46 represent?

The marginal count of all students who prefer online learning.

The joint count of 9th graders who prefer online learning.

The grand total of all students surveyed.

The number of 10th graders who prefer in-person learning.

A relative frequency is a count divided by a total. For the table above, a student wants "the fraction of ALL surveyed students who prefer online." Which denominator should they divide by?

The grand total, 80.

The 9th-grade row total, 40.

The Online column total, 46.

Any total will work, since they all describe the same students.

Fluency Practice

Use the survey table below for all problems in this section. Round percents to the nearest tenth.

A different survey of 50 adults recorded coffee preference and age group. The body cells are filled in; you must complete the totals.

	Decaf	Regular	Total
Under 40	6	19	̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲
40+	11	14	̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲
Total	̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲	̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲	̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲

What is the grand total (bottom-right corner) of this table?

Use the student survey table:

	Online	In-person	Total
9th	18	22	40
10th	28	12	40
Total	46	34	80

Compute the joint relative frequency of "9th grader who prefers online," as a percent rounded to the nearest tenth.

Use the student survey table:

	Online	In-person	Total
9th	18	22	40
10th	28	12	40
Total	46	34	80

Among 9th graders, what fraction prefer online learning?

$18/40 = 45\%$

$18/80 = 22.5\%$

$18/46 \approx 39.1\%$

$40/80 = 50\%$

Use the student survey table:

	Online	In-person	Total
9th	18	22	40
10th	28	12	40
Total	46	34	80

A report states: " $57.5\%$ of the students prefer online learning." Which type of relative frequency is this, and what is its denominator?

Marginal relative frequency; denominator is the grand total, 80 ( $46/80$ ).

Conditional relative frequency; denominator is the 9th-grade total, 40.

Joint relative frequency; denominator is the cell count, 18.

Conditional relative frequency; denominator is the Online total, 46.

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