Question

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
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 property

For the "Adults" column, the sum of "Liked" and "Disliked" relative frequencies 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, also, looking at the total row for "Liked New Paint Color", the total is 0.77, which is the sum of children's liked (0.6) and adults' liked (x). So \( 0.6 + x = 0.77 \).

Step3: Correct equation and solve

So \( x = 0.77 - 0.6 = 0.17 \)? Wait, no, confusion. Wait, conditional relative frequency by column: for each column (Children, Adults), the sum of the two rows (liked, disliked) should be 1.0. Wait, no, the "Total" column is the overall total. Wait, the problem says "conditional relative frequency table by column", so each column (Children, Adults) is a conditional distribution, so sum of each column is 1.0. Wait, but also, the "Total" row is the marginal distribution. Wait, let's check the numbers. For Children column: 0.6 (liked) + 0.4 (disliked) = 1.0, which is good. For Adults column: x (liked) + 0.15 (disliked) = 1.0, so x = 1 - 0.15 = 0.85? But that contradicts the "Total" row for "Liked New Paint Color": 0.6 (children) + x (adults) = 0.77. So 0.6 + x = 0.77 => x = 0.77 - 0.6 = 0.17. Wait, now there's a conflict. Wait, no, the table is a conditional relative frequency table by column, meaning that each column (Children, Adults) is the proportion within that group (Children or Adults). So the "Total" row is the marginal relative frequency. So for the "Liked New Paint Color" row, the total is the proportion of all people who liked, which is 0.77. So the sum of children who liked (0.6 of children) and adults who liked (x of adults) should equal 0.77. But also, for the Adults column, the sum of liked (x) and disliked (0.15) should be 1.0 (since it's a conditional distribution for Adults). Wait, that can't be, unless the table is misconstructed. Wait, no, maybe the "Total" column is the marginal relative frequency, not the sum of the columns. Wait, let's recast: Let's denote:

Let \( C \) be the proportion of Children, \( A \) be the proportion of Adults, so \( C + A = 1 \).

Let \( L_C = 0.6 \) (proportion of Children who liked), \( D_C = 0.4 \) (proportion of Children who disliked), so \( L_C + D_C = 1 \) (good).

Let \( L_A = x \) (proportion of Adults who liked), \( D_A = 0.15 \) (proportion of Adults who disliked), so \( L_A + D_A = 1 \) => \( x + 0.15 = 1 \) => \( x = 0.85 \). But then the marginal proportion of Liked is \( L_C \times C + L_A \times A = 0.77 \).

And marginal proportion of Disliked is \( D_C \times C + D_A \times A = 0.23 \).

We know \( C + A = 1 \).

From Disliked: \( 0.4C + 0.15A = 0.23 \).

Substitute \( A = 1 - C \):

\( 0.4C + 0.15(1 - C) = 0.23 \)

\( 0.4C + 0.15 - 0.15C = 0.23 \)

\( 0.25C + 0.15 = 0.23 \)

\( 0.25C = 0.08 \)

\( C = 0.08 / 0.25 = 0.32 \)

Then \( A = 1 - 0.32 = 0.68 \)

Then \( L_C \times C + L_A \times A = 0.6 \times 0.32 + x \times 0.68 = 0.77 \)

\( 0.192 + 0.68x = 0.77 \)

\( 0.68x = 0.77 - 0.192 = 0.578 \)

\( x = 0.578 / 0.68 = 0.85 \)

Wait, but that's conflicting with the earlier. Wait, no, I think I messed up the definition. Conditional relative frequency by column means that each cell is the proportion of the column total. Wait, no, conditional relative frequency by column: for each column (group: Children, Adults), the entries are the proportion within that group. So for Children, 0.6 is the proportion of Children who liked, 0.4 is proportion of Children who disliked. For Adults, x is proportion of Adults who liked, 0.15 is proportion of Adults who disliked. Then the "Total" row is the proportion of all people who liked (0.6*C + x*A) and disliked (0.4*C + 0.15*A), where C is proportion of Children, A is proportion of Adults, C + A = 1.

Given that the "Total" for disliked is 0.23, so 0.4C + 0.15A = 0.23, and C + A = 1.

Solving:

0.4C + 0.15(1 - C) = 0.23

0.4C + 0.15 - 0.15C = 0.23

0.25C = 0.08

C = 0.32, A = 0.68

Then total liked is 0.6*0.32 + x*0.68 = 0.192 + 0.68x = 0.77

0.68x = 0.77 - 0.192 = 0.578

x = 0.578 / 0.68 = 0.85

But that's one of the options? Wait, no, the options are 0.17, 0.25, 0.40, 0.85. Wait, 0.85 is an option. But wait, earlier when I thought of the "Total" row for liked as 0.6 + x = 0.77, that was wrong, because 0.6 is the proportion of Children who liked, not the proportion of all people who liked. The proportion of all people who liked is 0.6*C + x*A, where C and A are the proportions of Children and Adults in the total population.

But the problem says "conditional relative frequency table by column", so each column is a conditional distribution (within Children, within Adults), so the sum of each column is 1 (0.6 + 0.4 = 1, x + 0.15 = 1 => x = 0.85). That makes sense for the column totals. Then the marginal totals (Total row) are the sum of the column proportions weighted by the column sizes (C and A). But maybe the problem is simpler: maybe the table is a joint relative frequency table, and we need to find x such that the column for Adults has sum 1, or the row for Liked has sum 0.77. Wait, let's check the numbers again. The "Liked New Paint Color" row has total 0.77, which is 0.6 (Children) + x (Adults). So 0.6 + x = 0.77 => x = 0.17. But then the Adults column would be 0.17 (liked) + 0.15 (disliked) = 0.32, which is not 1.0. That can't be. So there's a mistake in my initial understanding.

Wait, the table is a conditional relative frequency table by column, meaning that each cell is the relative frequency within the column. So for the Children column: 0.6 (liked) and 0.4 (disliked) are the proportions of Children who liked and disliked, so sum to 1. For the Adults column: x (liked) and 0.15 (disliked) are proportions of Adults who liked and disliked, so sum to 1. Therefore, x + 0.15 = 1 => x = 0.85. Then the "Total" row is the marginal relative frequency, which is the sum of (proportion of Children * proportion of Children who liked) and (proportion of Adults * proportion of Adults who liked). Let’s denote the total number of people as N, number of Children as C, number of Adults as A. Then:

Proportion of Children who liked: 0.6 = (number of Children who liked)/C => number of Children who liked = 0.6C

Proportion of Children who disliked: 0.4 = (number of Children who disliked)/C => number of Children who disliked = 0.4C

Proportion of Adults who liked: x = (number of Adults who liked)/A => number of Adults who liked = xA

Proportion of Adults who disliked: 0.15 = (number of Adults who disliked)/A => number of Adults who disliked = 0.15A

Total number of people who liked: 0.6C + xA = 0.77N

Total number of people who disliked: 0.4C + 0.15A = 0.23N

Total number of people: C + A = N

Divide the first two equations by N:

0.6(C/N) + x(A/N) = 0.77

0.4(C/N) + 0.15(A/N) = 0.23

Let p = C/N (proportion of Children), q = A/N (proportion of Adults), so p + q = 1.

Then:

0.6p + xq = 0.77

0.4p + 0.15q = 0.23

From the second equation: 0.4p = 0.23 - 0.15q => p = (0.23 - 0.15q)/0.4

Substitute into p + q = 1:

(0.23 - 0.15q)/0.4 + q = 1

0.23 - 0.15q + 0.4q = 0.4

0.23 + 0.25q = 0.4

0.25q = 0.17

q = 0.68

Then p = 1 - 0.68 = 0.32

Now substitute p and q into the first equation:

0.6*0.32 + x*0.68 = 0.77

0.192 + 0.68x = 0.77

0.68x = 0.77 - 0.192 = 0.578

x = 0.578 / 0.68 = 0.85

Ah, so that's correct. The confusion was thinking that the "Total" row is the sum of the column entries, but actually, the column entries are conditional relative frequencies (within the column), so the "Total" row is the marginal relative frequency, which is the sum of each column's conditional frequency multiplied by the column's proportion in the total population. But in the problem, maybe they made a mistake, or maybe I misread. Wait, the options include 0.85, which is what we got from the column total (x + 0.15 = 1). So the correct answer is 0.85? But wait, let's check the "Disliked" row total: 0.4 (Children) + 0.15 (Adults) = 0.55, but the table says 0.23. No, that's not. Wait, no, the "Total" column is the sum of the joint relative frequencies. Wait, I think the table is a joint relative frequency table, where each cell is the proportion of the total population. So:

- Children who liked: 0.6 (of total)
- Children who disliked: 0.4 (of total)
- Adults who liked: x (of total)
- Adults who disliked: 0.15 (of total)
- Total who liked: 0.77 (of total)
- Total who disliked: 0.23 (of total)
- Total Children: 0.6 + 0.4 = 1.0 (of total? No, that can't be, because Children and Adults are two groups, so their totals should add to 1.0. Wait, the "Total" column for Children is 1.0, Adults is 1.0, and overall total is 1.0. So:

Children: 0.6 (liked) + 0.4 (disliked) = 1.0 (so Children are 1.0 of themselves, i.e., the column is conditional on being a Child)

Adults: x (liked) + 0.15 (disliked) = 1.0 (conditional on being an Adult)

Total who liked: 0.6 (Children's liked) + x (Adults' liked) = 0.77 (of total population)

Total who disliked: 0.4 (Children's disliked) + 0.15 (Adults' disliked) = 0.23 (of total population)

Let C be the proportion of Children in the total population, A be the proportion of Adults (C + A = 1).

Then:

- Children's liked: 0.6 * C (of total)
- Children's disliked: 0.4 * C (of total)
- Adults' liked: x * A (of total)
- Adults' disliked: 0.15 * A (of total)

Now, total who liked: 0.6C + xA = 0.77

Total who disliked: 0.4C + 0.15A = 0.23

And C + A = 1.

Solving:

From disliked: 0.4C + 0.15(1 - C) = 0.23

0.4C + 0.15 - 0.15C = 0.23

0.25C = 0.08

C = 0.32, A = 0.68

Then liked: 0.6*0.32 + x*0.68 = 0.77

0.192 + 0.68x = 0.77

0.68x = 0.578

x = 0.85

Yes, that makes sense. So the value of x is 0.85.

Answer:

0.85