Question

video lesson 5.3 example 3 – finding conditional relative frequencies

1. use the debate competition information to the right (from extra example 2) to make a two - way table that shows the conditional relative frequencies based on the column totals.

(there are two tables in the image: one is a blank two - way table with rows junior, senior and columns qualified, did not qualify; the other is a filled two - way table with rows junior, senior, total and columns qualified, did not qualify and values: junior - qualified: 0.175, junior - did not qualify: 0.225, senior - qualified: 0.475, senior - did not qualify: 0.125, total - qualified: 0.65, total - did not qualify: 0.35)

Explanation:

Step1: Understand Conditional Relative Frequency

Conditional relative frequency for a cell in a two - way table (based on column totals) is calculated by dividing the frequency of the cell by the total of its column.

Step2: Calculate for Junior - Qualified

The column total for "Qualified" is \(0.65\), and the frequency for Junior - Qualified is \(0.175\). To get the conditional relative frequency, we divide \(0.175\) by \(0.65\): \(\frac{0.175}{0.65}=\frac{175}{650}=\frac{7}{26}\approx0.269\) (wait, no, actually, the given table on the right already has the conditional relative frequencies based on column totals? Wait, no, the left table is to be filled with conditional relative frequencies based on column totals. Wait, the right table has the joint relative frequencies? Wait, no, let's re - examine.

Wait, the problem says "make a two - way table that shows the conditional relative frequencies based on the column totals". So for the "Qualified" column:

- Junior in Qualified column: conditional relative frequency \(=\frac{\text{Junior and Qualified}}{\text{Total Qualified}}=\frac{0.175}{0.65}\approx0.269\)? Wait, no, maybe the right table is the joint relative frequencies, and we need to find conditional relative frequencies by dividing each cell by its column total.

Wait, column 1: Qualified, total \(0.65\)

- Junior - Qualified: \(0.175\div0.65=\frac{0.175}{0.65}=\frac{175}{650}=\frac{7}{26}\approx0.269\)

- Senior - Qualified: \(0.475\div0.65=\frac{0.475}{0.65}=\frac{475}{650}=\frac{19}{26}\approx0.731\)

Column 2: Did Not Qualify, total \(0.35\)

- Junior - Did Not Qualify: \(0.225\div0.35=\frac{0.225}{0.35}=\frac{225}{350}=\frac{9}{14}\approx0.643\)

- Senior - Did Not Qualify: \(0.125\div0.35=\frac{0.125}{0.35}=\frac{125}{350}=\frac{5}{14}\approx0.357\)

Wait, but maybe I misread. Wait, the right table has "Qualified" column total \(0.65\) (which is \(0.175 + 0.475\)), and "Did Not Qualify" column total \(0.35\) (which is \(0.225+0.125\)). So to find conditional relative frequencies based on column totals, we divide each cell by its column total.

So for Junior - Qualified: \(\frac{0.175}{0.65}=\frac{7}{26}\approx0.269\)

Junior - Did Not Qualify: \(\frac{0.225}{0.35}=\frac{9}{14}\approx0.643\)

Senior - Qualified: \(\frac{0.475}{0.65}=\frac{19}{26}\approx0.731\)

Senior - Did Not Qualify: \(\frac{0.125}{0.35}=\frac{5}{14}\approx0.357\)

Now, let's fill the left table:

| | Qualified | Did Not Qualify |
|----------|-----------|-----------------|
| Junior | \(\frac{0.175}{0.65}\approx0.269\) | \(\frac{0.225}{0.35}\approx0.643\) |
| Senior | \(\frac{0.475}{0.65}\approx0.731\) | \(\frac{0.125}{0.35}\approx0.357\) |

Wait, but maybe the numbers in the right table are the joint frequencies (relative to the grand total), and we need to find conditional relative frequencies (relative to column totals). Let's confirm:

Grand total is \(0.65 + 0.35=1\). So the joint frequencies are:

- Junior - Qualified: \(0.175\) (of grand total)

- Junior - Did Not Qualify: \(0.225\) (of grand total)

- Senior - Qualified: \(0.475\) (of grand total)

- Senior - Did Not Qualify: \(0.125\) (of grand total)

Column 1 (Qualified) total: \(0.175 + 0.475 = 0.65\) (of grand total, which is the column total for conditional relative frequency calculation)

Column 2 (Did Not Qualify) total: \(0.225+0.125 = 0.35\) (of grand total)

So conditional relative frequency for a cell in column 1 is (cell value) / 0.65, and in column 2 is (cell value)/0.35.

So:

- Junior - Qualified: \(0.175\div0.65=\frac{0.175}{0.65}=\frac{7}{26}\approx0.269\) (or \(0.175/0.65 = 0.2692\))

- Senior - Qualified: \(0.475\div0.65=\frac{0.475}{0.65}=0.7308\)

- Junior - Did Not Qualify: \(0.225\div0.35=\frac{0.225}{0.35}=0.6429\)

- Senior - Did Not Qualify: \(0.125\div0.35=\frac{0.125}{0.35}=0.3571\)

So the two - way table with conditional relative frequencies based on column totals is:

| | Qualified | Did Not Qualify |
|----------|-----------|-----------------|
| Junior | \(\approx0.269\) | \(\approx0.643\) |
| Senior | \(\approx0.731\) | \(\approx0.357\) |

Answer:

The two - way table with conditional relative frequencies based on column totals is:

	Qualified	Did Not Qualify
Senior	\(\frac{19}{26}\approx0.731\)	\(\frac{5}{14}\approx0.357\)

(Or in decimal form, rounded to three decimal places:

	Qualified	Did Not Qualify
Senior	\(0.731\)	\(0.357\)	)