Question

weekend curfews
determine the values of the letters to complete the conditional relative frequency table by column.
16 years old\t17 years old\ttotal
a =
b =
before 10 p.m.\t0.9\tb\t0.88
after 10 p.m.\ta\t0.15\t0.12
total\t1.0\t1.0\t1.0

Explanation:

Step1: Find the value of \( a \)

For the column of "16 Years Old", the sum of the relative frequencies in the "Before 10 p.m." and "After 10 p.m." rows should be equal to the total (which is 1.0). So we have the equation: \( 0.9 + a = 1.0 \). Solving for \( a \), we subtract 0.9 from both sides: \( a = 1.0 - 0.9 = 0.1 \).

Step2: Find the value of \( b \)

For the column of "17 Years Old", the sum of the relative frequencies in the "Before 10 p.m." and "After 10 p.m." rows should be equal to the total (which is 1.0). So we have the equation: \( b + 0.15 = 1.0 \). Solving for \( b \), we subtract 0.15 from both sides: \( b = 1.0 - 0.15 = 0.85 \). We can also verify using the "Total" row for "Before 10 p.m.": \( 0.9 + b = 0.88 \)? Wait, no, actually, the "Total" row for each row (Before 10 p.m., After 10 p.m.) is the sum of the two columns. Wait, let's re - check. The table is a conditional relative frequency table by column. So for the "Before 10 p.m." row, the total is the sum of the relative frequencies for 16 - year - olds and 17 - year - olds in that row. So \( 0.9 + b=0.88 \)? Wait, that can't be, because 0.9 is already greater than 0.88. I must have misinterpreted the table. Wait, no, the table is a conditional relative frequency table by column. So each column (16 Years Old, 17 Years Old) has a total of 1.0, and each row (Before 10 p.m., After 10 p.m.) has a total which is the sum of the two column entries. So for the "After 10 p.m." row: \( a + 0.15=0.12 \)? No, that also doesn't make sense. Wait, I think I made a mistake in the first approach. Let's start over.

In a conditional relative frequency table by column, for each column (e.g., 16 Years Old), the sum of the relative frequencies in the rows (Before 10 p.m., After 10 p.m.) is 1.0. So for the 16 Years Old column: \( 0.9 + a=1.0\), so \( a = 1.0 - 0.9=0.1\). For the 17 Years Old column: \( b + 0.15 = 1.0\), so \( b=1.0 - 0.15 = 0.85\). Now, let's check the row totals. For the Before 10 p.m. row: \( 0.9 + b=0.9 + 0.85 = 1.75\), which is not 0.88. Wait, that's a problem. Wait, no, the "Total" row for each row is the sum of the two column entries. So the "Before 10 p.m." row total is \( 0.9 + b=0.88\)? But 0.9 is in the 16 Years Old column for Before 10 p.m., and \( b \) is in the 17 Years Old column for Before 10 p.m. So \( 0.9 + b = 0.88\)? That would mean \( b=0.88 - 0.9=- 0.02\), which is impossible. Wait, I think the table is structured such that each column is a conditional distribution given the age (16 or 17), and each row is a conditional distribution given the curfew time (Before or After). Wait, maybe the "Total" row for each row is the marginal relative frequency. Let's use the fact that for the "After 10 p.m." row, the marginal relative frequency is 0.12, which is \( a+0.15 = 0.12\)? No, that would give \( a=0.12 - 0.15=- 0.03\), which is impossible. I must have misread the table. Wait, looking at the table again:

| | 16 Years Old | 17 Years Old | Total |
|----------------|--------------|--------------|-------|
| Before 10 p.m. | 0.9 | \( b \) | 0.88 |
| After 10 p.m. | \( a \) | 0.15 | 0.12 |
| Total | 1.0 | 1.0 | 1.0 |

Ah! Now I see. The "Total" column is the marginal relative frequency, and each column (16 and 17) is a conditional distribution (so their totals are 1.0). The "Total" row for each row is the marginal relative frequency (sum of the two column entries for that row). So for the "After 10 p.m." row: \( a + 0.15=0.12\)? No, that's not possible. Wait, no, the numbers in the table must satisfy two things:

1. For each column (16 Years Old, 17 Years Old): the sum of the row entries is 1.0. So for 16 Years Old: \( 0.9 + a = 1.0\) (so \( a = 0.1\)). For 17 Years Old: \( b+0.15 = 1.0\) (so \( b = 0.85\)).

2. For each row (Before 10 p.m., After 10 p.m.): the sum of the column entries is the total in the "Total" column. So for Before 10 p.m.: \( 0.9 + b=0.88\). But if \( b = 0.85\), then \( 0.9+0.85 = 1.75 eq0.88\). There is a contradiction here, which means I must have misinterpreted the table. Wait, maybe the table is a joint relative frequency table? No, the problem says "conditional relative frequency table by column". In a conditional relative frequency table by column, the entries in each column are the relative frequencies of the row categories given the column category. So for column 1 (16 Years Old), the relative frequency of "Before 10 p.m." given "16 Years Old" is 0.9, and the relative frequency of "After 10 p.m." given "16 Years Old" is \( a\), so \( 0.9 + a=1.0\) (since it's a conditional distribution), so \( a = 0.1\). For column 2 (17 Years Old), the relative frequency of "Before 10 p.m." given "17 Years Old" is \( b\), and the relative frequency of "After 10 p.m." given "17 Years Old" is 0.15, so \( b + 0.15=1.0\), so \( b = 0.85\). The "Total" column for each row is the marginal relative frequency, which is the sum of the conditional relative frequencies multiplied by the relative frequency of the column. Wait, maybe the problem has a typo, but according to the column - wise conditional relative frequency, we calculate \( a \) and \( b \) as follows:

For column "16 Years Old":
The sum of the relative frequencies in the two rows (Before 10 p.m. and After 10 p.m.) must be 1.0 (since it's a conditional distribution given the column category). So:
\( 0.9+a = 1.0\)
Subtract 0.9 from both sides:
\( a=1.0 - 0.9 = 0.1\)

For column "17 Years Old":
The sum of the relative frequencies in the two rows (Before 10 p.m. and After 10 p.m.) must be 1.0. So:
\( b + 0.15=1.0\)
Subtract 0.15 from both sides:
\( b=1.0 - 0.15 = 0.85\)

Even though the row totals don't match, the problem says "conditional relative frequency table by column", so we go by the column - wise conditional distributions.

Answer:

\( a = 0.1\), \( b = 0.85\)