Question

paint color preference
\t\t\tchildren\tadults\ttotal
liked new paint color\t0.6\tx\t0.77
disliked new paint color\t0.4\t0.15\t0.23
total\t1.0\t1.0\t1.0
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 relative frequencies should be 1.0. So, \( x + 0.15 = 1.0 \).

Step2: Solve for x

Subtract 0.15 from both sides: \( x = 1.0 - 0.15 \).
\( x = 0.85 \)? Wait, no, wait. Wait, the total for the "Liked" row is 0.77. Wait, maybe I misread. Wait, the table is conditional relative frequency by column? Wait, no, the "Adults" column: Liked is \( x \), Disliked is 0.15, so column total is \( x + 0.15 = 1.0 \). So \( x = 1 - 0.15 = 0.85 \)? But wait, the total for "Liked" row is 0.6 (Children) + \( x \) (Adults) = 0.77. Let's check that. \( 0.6 + x = 0.77 \), so \( x = 0.77 - 0.6 = 0.17 \)? Wait, now I'm confused. Wait, the problem says "conditional relative frequency table by column". So in a column - conditional table, each column sums to 1. So for Adults column: Liked (\( x \)) + Disliked (0.15) = 1.0. So \( x = 1 - 0.15 = 0.85 \)? But also, the total for the "Liked" row is 0.6 (Children) + \( x \) (Adults) = 0.77. So \( x = 0.77 - 0.6 = 0.17 \). Wait, there's a conflict. Wait, no, maybe I misinterpret the table. Let's look again. The table has rows: Liked, Disliked, Total. Columns: Children, Adults, Total. The "Total" row for each column is 1.0, so Children column: 0.6 (Liked) + 0.4 (Disliked) = 1.0, which is correct. Adults column: \( x \) (Liked) + 0.15 (Disliked) = 1.0, so \( x = 1 - 0.15 = 0.85 \). But the "Total" row for "Liked" is 0.6 + \( x \) = 0.77, so \( x = 0.77 - 0.6 = 0.17 \). Wait, this is a problem. Wait, no, maybe the table is a joint relative frequency table? No, the problem says "conditional relative frequency table by column". In a column - conditional (conditional on column) table, each column is a conditional distribution, so sums to 1. So Adults column: P(Liked | Adult) + P(Disliked | Adult) = 1. So \( x + 0.15 = 1 \), so \( x = 0.85 \). But then the total for "Liked" row would be P(Liked | Child)*P(Child) + P(Liked | Adult)*P(Adult), but no, in a conditional by column table, the columns are conditional, so the total row is the marginal. Wait, maybe the problem is that the "Total" row for "Liked" is the marginal relative frequency. So marginal for Liked is 0.77, which is P(Liked) = P(Liked | Child)*P(Child) + P(Liked | Adult)*P(Adult). But if it's a conditional by column table, P(Child) and P(Adult) are the proportions of children and adults in the sample. Wait, maybe the table is a two - way table with conditional relative frequencies by column. So for each column (Children and Adults), the relative frequencies are conditional on being in that column. So for Children column: 0.6 is P(Liked | Child), 0.4 is P(Disliked | Child). For Adults column: \( x \) is P(Liked | Adult), 0.15 is P(Disliked | Adult). Then the marginal total for Liked is P(Liked) = P(Liked | Child)*P(Child) + P(Liked | Adult)*P(Adult) = 0.77. And P(Child) + P(Adult) = 1. Let's let P(Child) = c, P(Adult) = a, so c + a = 1. Then 0.6c + xa = 0.77, and 0.4c + 0.15a = 0.23 (from Disliked row). We can solve these two equations. From the first equation: 0.6c + xa = 0.77. From the second: 0.4c + 0.15a = 0.23. Multiply the second equation by 1.5: 0.6c + 0.225a = 0.345. Subtract from the first equation: (0.6c + xa) - (0.6c + 0.225a) = 0.77 - 0.345. So (x - 0.225)a = 0.425. But also, c = 1 - a. Substitute c = 1 - a into the second equation: 0.4(1 - a)+0.15a = 0.23. 0.4 - 0.4a + 0.15a = 0.23. 0.4 - 0.25a = 0.23. - 0.25a = 0.23 - 0.4 = - 0.17. a = (- 0.17)/(- 0.25)=0.68. Then c = 1 - 0.68 = 0.32. Then from 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. Wait, that's 0.85. But the answer options include 0.85. Wait, but earlier when I thought x + 0.15 = 1, that's because in a column - conditional table, the column sums to 1. So for Adults column, conditional on being an adult, the probability of liking is x, disliking is 0.15, so x + 0.15 = 1, so x = 0.85. That makes sense. The confusion was about the marginal total. So the correct way is: in a conditional relative frequency table by column, each column (Children and Adults) is a conditional distribution, so sums to 1. Therefore, for Adults column: x (Liked | Adult) + 0.15 (Disliked | Adult) = 1. So x = 1 - 0.15 = 0.85.

Answer:

0.85