Question

paint color preference
liked new paint color: children 0.6, adults x, total 0.77
disliked new paint color: children 0.4, adults 0.15, total 0.23
total: children 1.0, adults 1.0, total 1.0
which value for x completes the conditional relative frequency table by column?
0.17
0.25
0.40
0.85

Explanation:

Step1: Recall column total rule

For the "Adults" column, total should be 1.0. So \( x + 0.15 = 1.0 \).

Step2: Solve for x

Subtract 0.15 from both sides: \( x = 1.0 - 0.15 = 0.85 \)? Wait, no, wait. Wait, the total for the "Liked + Disliked" for Adults should be 1.0. Wait, the total row for Adults is 1.0. So in the Adults column, Liked is \( x \), Disliked is 0.15, so \( x + 0.15 = 1.0 \)? Wait, no, wait the total for the "Liked New Paint Color" row total is 0.77, which is Children (0.6) + Adults (x) = 0.77. Oh! I misread. The table is by column? Wait, conditional relative frequency by column. So for each column (Children and Adults), the sum of the two rows (Liked and Disliked) should be 1.0? No, wait the total row for each column is 1.0. Wait, the "Total" row for Children is 1.0 (0.6 + 0.4 = 1.0), for Adults is 1.0 (x + 0.15 = 1.0)? But also, the "Total" column for Liked is 0.77 (0.6 + x = 0.77), and Disliked is 0.23 (0.4 + 0.15 = 0.55? Wait no, 0.4 + 0.15 is 0.55, but the total for Disliked is 0.23. Wait, no, I think the table is:

Rows: Liked, Disliked, Total

Columns: Children, Adults, Total

So for the "Liked New Paint Color" row: Children (0.6) + Adults (x) = Total (0.77)

For the "Disliked New Paint Color" row: Children (0.4) + Adults (0.15) = Total (0.23)

For the "Total" row: Children (0.6+0.4=1.0), Adults (x + 0.15), Total (0.77+0.23=1.0)

So to find x, use the "Liked" row: 0.6 + x = 0.77

So x = 0.77 - 0.6 = 0.17? Wait, no, 0.77 - 0.6 is 0.17? Wait 0.6 + 0.17 is 0.77. And then Adults column total: x + 0.15 = 0.17 + 0.15 = 0.32? But the total for Adults should be 1.0? Wait, no, I think I messed up the table structure.

Wait, conditional relative frequency by column means that each column (Children and Adults) is a separate group, so the relative frequencies are within each column. So for the Children column: Liked is 0.6, Disliked is 0.4, total 1.0. For the Adults column: Liked is x, Disliked is 0.15, total 1.0. Then the "Total" column is the sum of the two columns.

But also, the "Total" row for Liked is the sum of the two columns' Liked: 0.6 (Children) + x (Adults) = 0.77 (Total)

Ah, that's the key. So 0.6 + x = 0.77. Then x = 0.77 - 0.6 = 0.17? Wait, but then Adults column: x (0.17) + 0.15 (Disliked) = 0.32, which is not 1.0. Wait, that can't be. Wait, no, maybe the table is:

The conditional relative frequency by column means that for each column (Children and Adults), the frequencies are relative to the column total (which is 1.0 for each column). So for Children column: total is 1.0, so Liked is 0.6 (60% of Children), Disliked is 0.4 (40% of Children). For Adults column: total is 1.0, so Liked is x (x% of Adults), Disliked is 0.15 (15% of Adults), so x + 0.15 = 1.0 => x = 0.85. But then the total row for Liked would be 0.6 (Children) + 0.85 (Adults) = 1.45, which is more than 1.0. So that's wrong.

Wait, the problem says "conditional relative frequency table by column". So the total for each column is 1.0, and the total for each row is the sum across columns. So:

Children column: Liked = 0.6, Disliked = 0.4, Total = 1.0

Adults column: Liked = x, Disliked = 0.15, Total = 1.0

Total column: Liked = 0.6 + x, Disliked = 0.4 + 0.15 = 0.55, Total = 1.0

But the problem states the Total for Liked is 0.77 and Disliked is 0.23. So 0.6 + x = 0.77 => x = 0.17, and 0.4 + 0.15 = 0.55, but the problem says Disliked total is 0.23. Wait, that's a contradiction. Wait, maybe the table is:

Rows: Liked, Disliked

Columns: Children, Adults

With totals at the bottom and right.

So:

Children Adults Total

Liked 0.6 x 0.77

Disliked 0.4 0.15 0.23

Total 1.0 1.0 1.0

Ah! Now that makes sense. So the "Total" row for each column is 1.0 (Children: 0.6+0.4=1.0, Adults: x+0.15=1.0), and the "Total" column for each row is the sum of the two columns (Liked: 0.6 + x = 0.77, Disliked: 0.4 + 0.15 = 0.55? No, 0.4 + 0.15 is 0.55, but the total for Disliked is 0.23. Wait, no, the problem must have a typo, but according to the table given, the "Total" for Disliked is 0.23, which is 0.4 (Children) + 0.15 (Adults) = 0.55, which is not 0.23. Wait, I think I misread the table. Let me look again:

The table is:

Paint Color Preference

Children Adults Total

Liked New Paint Color 0.6 x 0.77

Disliked New Paint Color 0.4 0.15 0.23

Total 1.0 1.0 1.0

Ah! Now, the "Total" for Liked is 0.77, which is 0.6 (Children) + x (Adults) = 0.77. So solving for x: x = 0.77 - 0.6 = 0.17. Wait, but then Adults column: x (0.17) + 0.15 (Disliked) = 0.32, but the total for Adults is 1.0. That's a problem. Wait, no, the "Total" row for Adults is 1.0, so x + 0.15 should equal 1.0, but 0.17 + 0.15 = 0.32 ≠ 1.0. So there's a mistake in my understanding. Wait, no, the "Total" row is the sum of the two rows (Liked and Disliked) for each column. So for Children: 0.6 (Liked) + 0.4 (Disliked) = 1.0 (Total). For Adults: x (Liked) + 0.15 (Disliked) = 1.0 (Total). So x = 1.0 - 0.15 = 0.85. Then, the "Total" column for Liked is 0.6 (Children) + 0.85 (Adults) = 1.45, which is more than 1.0. But the "Total" column for Liked is given as 0.77. So there's a conflict. Wait, the problem says "conditional relative frequency table by column". So the conditional relative frequencies are calculated with respect to the column totals. So for each column (Children and Adults), the relative frequencies are (frequency in cell) / (column total). But in the table, the column totals are 1.0 (since the Total row for each column is 1.0). Wait, maybe the table is already a conditional relative frequency table by column, so for Children column: Liked is 0.6 (60% of Children), Disliked is 0.4 (40% of Children). For Adults column: Liked is x (x% of Adults), Disliked is 0.15 (15% of Adults). Then the total row for each column is 1.0 (0.6+0.4=1.0, x+0.15=1.0). The total column for Liked is the sum of the two columns' Liked: 0.6 (Children) + x (Adults) = 0.77. So now we have two equations:

1. \( x + 0.15 = 1.0 \) (from Adults column total)

2. \( 0.6 + x = 0.77 \) (from Liked row total)

Wait, these two equations give different x. That means I misinterpret the table. Wait, the problem says "conditional relative frequency table by column". So the conditional relative frequency is (frequency in cell) / (column total). So column total for Children is, say, C, and for Adults is A. Then:

For Children column:

Liked: \( \frac{\text{Children Liked}}{\text{C}} = 0.6 \)

Disliked: \( \frac{\text{Children Disliked}}{\text{C}} = 0.4 \)

So \( 0.6 + 0.4 = 1.0 \), so C is the total number of Children, and the relative frequencies sum to 1.0.

For Adults column:

Liked: \( \frac{\text{Adults Liked}}{\text{A}} = x \)

Disliked: \( \frac{\text{Adults Disliked}}{\text{A}} = 0.15 \)

So \( x + 0.15 = 1.0 \) => \( x = 0.85 \)

But then the total row for Liked is \( \frac{\text{Children Liked} + \text{Adults Liked}}{\text{Total}} = 0.77 \), and Disliked is \( \frac{\text{Children Disliked} + \text{Adults Disliked}}{\text{Total}} = 0.23 \)

Let’s denote:

Children Liked = 0.6C

Children Disliked = 0.4C

Adults Liked = 0.85A

Adults Disliked = 0.15A

Total Liked = 0.6C + 0.85A = 0.77(C + A) [since total is C + A]

Total Disliked = 0.4C + 0.15A = 0.23(C + A)

Let’s solve the first equation:

0.6C + 0.85A = 0.77C + 0.77A

0.85A - 0.77A = 0.77C - 0.6C

0.08A = 0.17C

\( \frac{A}{C} = \frac{0.17}{0.08} = 2.125 \)

Second equation:

0.4C + 0.15A = 0.23C + 0.23A

0.4C - 0.23C = 0.23A - 0.15A

0.17C = 0.08A

Which is the same as \( \frac{A}{C} = \frac{0.17}{0.08} = 2.125 \), so consistent.

But the problem is asking for x, which is the conditional relative frequency for Adults Liked, so x = 0.85? But the options include 0.85. Wait, the options are 0.17, 0.25, 0.40, 0.85. Oh! 0.85 is an option. Wait, earlier I thought x + 0.15 = 1.0, so x = 0.85. Let's check the Liked row total: 0.6 (Children) + 0.85 (Adults) = 1.45, but that's the sum of the two column's Liked relative frequencies. But the total row's Liked is 0.77, which is the relative frequency of Liked in the total population. So the table is:

- Columns: Children (total C), Adults (total A), Total (C+A)

- Rows: Liked, Disliked, Total

- Conditional relative frequency by column: so each cell in Children column is (frequency in Children) / C, Adults column is (frequency in Adults) / A, Total column is (frequency in Total) / (C+A)

So:

Children Liked: (Liked Children) / C = 0.6 => Liked Children = 0.6C

Children Disliked: (Disliked Children) / C = 0.4 => Disliked Children = 0.4C

Adults Liked: (Liked Adults) / A = x => Liked Adults = xA

Adults Disliked: (Disliked Adults) / A = 0.15 => Disliked Adults = 0.15A

Total Liked: (Liked Children + Liked Adults) / (C + A) = 0.77 => (0.6C + xA) / (C + A) = 0.77

Total Disliked: (Disliked Children + Disliked Adults) / (C + A) = 0.23 => (0.4C + 0.15A) / (C + A) = 0.23

Also, for Adults column, the sum of Liked and Disliked relative frequencies is 1.0: x + 0.15 = 1.0 => x = 0.85

Let’s verify with x = 0.85:

Liked total: (0.6C + 0.85A) / (C + A) = 0.77

Multiply both sides by (C + A): 0.6C + 0.85A = 0.77C + 0.77A

0.85A - 0.77A = 0.77C - 0.6C

0.08A = 0.17C => A = (0.17 / 0.08)C = 2.125C

Now, Disliked total: (0.4C + 0.15A) / (C + A) = (0.4C + 0.15*2.125C) / (C + 2.125C) = (0.4C + 0.31875C) / 3.125C = (0.71875C) / 3.125C = 0.23, which matches.

So x is 0.85.

Answer:

0.85