Question

lisa is examining a two - way table representing the sales of juice packaged in bottles with an old or new label. in the table, p, q, r, and s are the number of bottles sold or left on the grocery store shelf last thursday.

sales of juice bottles

	sold	left on the shelf
old label	r	s

comparing which two values will determine whether the new label had an impact on sales?

• p and q
• p and r
• \\(\frac{p}{p + r}\\) and \\(\frac{q}{q + s}\\)
• \\(\frac{p}{p + q}\\) and \\(\frac{r}{r + s}\\)

Explanation:

Step1: Identify relevant proportions

To test if the new label impacts sales, we need to compare the sales rates (proportion of bottles sold) for new vs. old labels.
For new labels: Total bottles = $p+q$, so sales rate is $\frac{p}{p+q}$.
For old labels: Total bottles = $r+s$, so sales rate is $\frac{r}{r+s}$.

Step2: Evaluate comparison logic

Comparing these two proportions tells us if the new label changes the likelihood of a bottle being sold. Other options compare raw counts or unrelated proportions, which do not isolate the label's impact on sales rate.

Answer:

$\boldsymbol{\frac{p}{p+q}}$ and $\boldsymbol{\frac{r}{r+s}}$ (the fourth option)