Question

5.4.3 quiz: two-way frequency tables
a survey asked 60 students if they play an instrument and if they are in band.

1. 35 students play an instrument.
2. 30 students are in band.
3. 30 students are not in band.

which table shows these data correctly entered in a two-way frequency table?

option a:

	band	not in band	total
dont play instrument	5	25	30
total	35	25	60

option b:

	band and play instrument	not in band and play instrument	total
band and dont play instrument	5	25	30
total	35	25	60

Explanation:

Answer:

First, let's analyze the given data:

• Total students: 60
• Students who play an instrument: 35
• Students in band: 30
• Students not in band: 30

Now let's check each option:

Option A:

• Play instrument row:
• Band column: 30
• Not in band column: 0
• Total for play instrument: \(30 + 0=30\). But we know 35 students play an instrument, so this is incorrect.
• Don't play instrument row:
• Band column: 5
• Not in band column: 25
• Total for don't play instrument: \(5 + 25 = 30\)
• Total column:
• Band total: \(30+5 = 35\) (incorrect, should be 30)
• Not in band total: \(0 + 25=25\) (incorrect, should be 30)
• Grand total: \(35 + 25=60\) (correct, but other totals are wrong)

Option B:

Wait, there seems to be a mis - labeling in option B's table structure. Let's re - evaluate. Wait, maybe there was a typo in the original problem's option B. Let's go back to the correct two - way table structure. The correct two - way table should have rows as "Play instrument" and "Don't play instrument" and columns as "Band" and "Not in band" with proper totals.

Wait, let's recalculate the correct cell values:

Let \(a\) be the number of students who play an instrument and are in band.

Let \(b\) be the number of students who play an instrument and are not in band.

Let \(c\) be the number of students who don't play an instrument and are in band.

Let \(d\) be the number of students who don't play an instrument and are not in band.

We know that:

1. \(a + b=35\) (play an instrument)
2. \(a + c = 30\) (in band)
3. \(b + d=30\) (not in band)
4. \(a + b + c + d=60\) (total students)

From \(a + b + c + d = 60\) and \(a + b=35\), we get \(c + d=25\). But we also know that \(b + d = 30\) and \(a + c=30\).

From \(a + b=35\) and \(a + c = 30\), we get \(b - c=5\).

From \(b + d=30\) and \(c + d=25\), we get \(b - c = 5\) (which is consistent).

Let's assume the correct table has:

• Play instrument row:
• Band: Let's say \(a\), Not in band: \(b\), Total: 35
• Don't play instrument row:
• Band: \(c\), Not in band: \(d\), Total: \(60 - 35=25\)
• Band column total: \(a + c=30\)
• Not in band column total: \(b + d = 30\)

Now, looking at option A (after correcting our initial mis - reading), wait no, option A's "Play instrument" total is 30 which is wrong. Wait, maybe there was a mistake in the original problem's option presentation. Wait, the user might have made a typo. But according to the given options, let's re - check.

Wait, the first option (A) has:

• Play instrument: Band = 30, Not in band = 0, Total = 30 (wrong, should be 35)
• Don't play instrument: Band = 5, Not in band = 25, Total = 30 (correct, since \(60 - 35 = 25\)?) Wait, no, \(60-35 = 25\), so "Don't play instrument" total should be 25, but in option A it's 30. So option A is wrong.

Wait, maybe the correct option is not among the two? No, the user provided two options. Wait, maybe I misread option B. Let's re - read option B:

Option B's rows are "Not in band and don't play instrument" and "Band and don't play instrument", which is an incorrect way to structure a two - way table for this problem. The correct rows should be "Play instrument" and "Don't play instrument", and columns "Band" and "Not in band".

Since option A has the wrong total for "Play instrument" (30 instead of 35) and option B has an incorrect table structure, but maybe there was a typo. Wait, maybe the correct table (the one that should be chosen) is the one where:

• Play instrument row: Band = 30, Not in band = 5 (since \(30 + 5=35\))
• Don't play instrument row: Band = 0, Not in band = 25 (since \(0+25 = 25\))
• Band column total: \(30 + 0=30\)
• Not in band column total: \(5+25 = 30\)
• Grand total: \(35+25 = 60\)

But none of the given options match this. Wait, maybe the user made a mistake in presenting the options. But according to the given options, if we have to choose between A and B, and considering the totals:

In option A, the "Don't play instrument" total is 30 (which is wrong, should be 25) and "Play instrument" total is 30 (wrong, should be 35). In option B, the table structure is wrong. But maybe the intended correct option is A (even with the error) or maybe there was a typo.

Wait, the problem says "35 students play an instrument", "30 in band", "30 not in band".

Let's calculate the correct cells:

Total students: 60.

Number of students not in band: 30, so number in band: 30.

Number of students who play an instrument: 35, so number who don't: \(60 - 35=25\).

Let \(x\) be the number of students who play an instrument and are in band. Then the number of students who play an instrument and are not in band is \(35 - x\).

The number of students who don't play an instrument and are in band is \(30 - x\).

The number of students who don't play an instrument and are not in band is \(30-(35 - x)=x - 5\).

Since the number of students who don't play an instrument and are not in band can't be negative, \(x-5\geq0\Rightarrow x\geq5\). Also, \(30 - x\geq0\Rightarrow x\leq30\) and \(35 - x\geq0\Rightarrow x\leq35\).

If we assume \(x = 30\) (as in option A's "Play instrument - Band" cell), then:

• Play instrument - Band: 30
• Play instrument - Not in band: \(35 - 30 = 5\) (but option A has 0)
• Don't play instrument - Band: \(30 - 30=0\) (but option A has 5)
• Don't play instrument - Not in band: \(30 - 5 = 25\) (option A has 25)

Ah! So option A has a mistake in the "Play instrument - Not in band" cell (it should be 5, not 0) and "Don't play instrument - Band" cell (it should be 0, not 5) and "Play instrument - Total" (should be 35, not 30) and "Band - Total" (should be 30, not 35).

But since option B has an incorrect table structure (rows are not "Play instrument" and "Don't play instrument"), the intended answer is likely option A (even with the typos, maybe it's a formatting error in the problem).

So the answer is A.