Question

paint color preference
children\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 rule

For the "Adults" column, the sum of "Liked" and "Disliked" should be 1.0. So, \( x + 0.15 = 1.0 \) or we can use the total row for "Liked New Paint Color": \( 0.6 + x = 0.77 \). Let's use the second approach.

Step2: Solve for x

From \( 0.6 + x = 0.77 \), subtract 0.6 from both sides: \( x = 0.77 - 0.6 = 0.17 \)? Wait, no, wait. Wait, the "Total" for "Liked" is 0.77, which is children's 0.6 plus adults' x. Wait, but also, adults' total is 1.0, so x + 0.15 = 1.0 → x = 0.85? Wait, no, I messed up. Wait, the table is conditional relative frequency by column. So for each column (Children, Adults), the sum of the two rows (Liked, Disliked) is 1.0. So for Adults column: Liked (x) + Disliked (0.15) = 1.0. So x = 1.0 - 0.15 = 0.85? Wait, but the total for "Liked" row is 0.77, which is 0.6 (children) + x (adults) = 0.77. So x = 0.77 - 0.6 = 0.17? Wait, there's a contradiction. Wait, no, the table is a conditional relative frequency table by column. So the "Total" row for each column is 1.0 (since it's column - wise). The "Total" column (the last one) is the grand total, which is 1.0. So let's check the "Liked" row: children's proportion is 0.6 (of children), adults' proportion is x (of adults), and the total proportion (of all) is 0.77. The "Disliked" row: children's proportion is 0.4 (of children), adults' proportion is 0.15 (of adults), and total proportion is 0.23. So for the adults column: x (liked by adults, as a proportion of adults) + 0.15 (disliked by adults, as a proportion of adults) = 1.0 (since it's column - wise, so proportions within adults sum to 1). So x = 1.0 - 0.15 = 0.85? Wait, but then 0.6 (children liked) + 0.85 (adults liked) would be 1.45, which is more than 0.77. So I must have misinterpreted the table. Wait, no, conditional relative frequency by column means that each cell is the proportion of the column total. So for example, the "Children" column: total is 1.0 (all children). So 0.6 is the proportion of children who liked, 0.4 is proportion of children who disliked. The "Adults" column: total is 1.0 (all adults). So x is proportion of adults who liked, 0.15 is proportion of adults who disliked. The "Total" column (the last one) is the proportion of all people. So the "Liked" row total is proportion of all people who liked: 0.6 (proportion of children who liked) * (proportion of children in total) + x (proportion of adults who liked) * (proportion of adults in total) = 0.77. But we don't know the proportion of children and adults. Wait, no, the table is a conditional relative frequency table by column, so the numbers in each column are proportions within the column. So the "Total" row for each column is 1.0, and the "Total" column is the sum of the column - wise proportions weighted by the column totals. But maybe the problem is that the "Total" row (the last row) is the sum of each column, so for the "Liked" row: 0.6 (children's liked proportion) + x (adults' liked proportion) = 0.77 (total liked proportion). So solving for x: x = 0.77 - 0.6 = 0.17. Wait, but then adults' column: 0.17 (liked) + 0.15 (disliked) = 0.32, which is not 1.0. So I see my mistake. The table is a joint relative frequency table? No, the problem says "conditional relative frequency table by column". So in a conditional relative frequency table by column, each entry in the column is the proportion of the column. So for the "Adults" column, the two entries (x and 0.15) must sum to 1.0, because they are the proportions of adults who liked and disliked. So x + 0.15 = 1.0 → x = 0.85. Then, the "Liked" row total is 0.6 (children's liked, which is proportion of children) + 0.85 (adults' liked, proportion of adults) = 1.45, which is more than 0.77. That can't be. Wait, no, the "Total" column is the grand total, which is the sum of the joint relative frequencies. So let's denote: let C be the proportion of children in the total population, A be the proportion of adults (C + A = 1). Then, the joint relative frequency for children who liked is 0.6 * C, adults who liked is x * A. The total for liked is 0.6C + xA = 0.77. For disliked: 0.4C + 0.15A = 0.23. And C + A = 1. Let's solve these equations. From C + A = 1, C = 1 - A. Substitute into disliked 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. Now, substitute into liked 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. So x is 0.85, because in the adults column, the conditional relative frequency (proportion of adults who liked) is x, and since A = 0.68 (proportion of adults in total), the joint frequency is x * 0.68, and with children's joint frequency 0.6 * 0.32 = 0.192, total liked is 0.192 + 0.68x = 0.77, solving gives x = 0.85. Alternatively, since in the adults column, the conditional relative frequency (column - wise) is x (liked) and 0.15 (disliked), so x + 0.15 = 1.0? No, that's only if it's a marginal relative frequency, but no, conditional relative frequency by column means that each cell is (frequency in cell) / (column total). So column total for adults is the number of adults, so (number of adults who liked) / (number of adults) = x, (number of adults who disliked) / (number of adults) = 0.15. So x + 0.15 = 1.0 → x = 0.85. That makes sense, because the column total (number of adults) is the denominator, so the two proportions (liked, disliked) within adults must sum to 1.0. So my initial mistake was thinking the "Total" row (the last row) was the sum of the column entries, but no, the last row is the column totals (1.0 for each column), and the last column is the grand total (1.0). So for the adults column: x (liked by adults, as a proportion of adults) + 0.15 (disliked by adults, as a proportion of adults) = 1.0. So x = 1.0 - 0.15 = 0.85.

Answer:

0.85