Question

a combined group of middle and high school teachers complete a survey to see when each teacher prefers to go on vacation. the two - way table summarizes the results.
based on the results of the survey, which statement about the surveyed teachers is true?
a greater percentage of high school teachers prefer to vacation in june than july.
a smaller percentage of middle school teachers prefer to vacation in july than august.
it is more likely that a high school teacher prefers to vacation in july than june.
it is more likely that a teacher surveyed is a middle school teacher than a high school teacher

Explanation:

Step1: Calculate the percentage of high - school teachers preferring June and July

The number of high - school teachers is \(n = 84\).
The percentage of high - school teachers preferring June is \(P_{June}=\frac{31}{84}\approx0.369\).
The percentage of high - school teachers preferring July is \(P_{July}=\frac{48}{84}\approx0.571\).
Since \(0.369<0.571\), option A is wrong.

Step2: Calculate the percentage of middle - school teachers preferring July and August

The number of middle - school teachers is \(n = 96\).
The percentage of middle - school teachers preferring July is \(P_{July}=\frac{32}{96}\approx0.333\).
The percentage of middle - school teachers preferring August is \(P_{August}=\frac{31}{96}\approx0.323\).
Since \(0.333 > 0.323\), option B is wrong.

Step3: Calculate the probability of a high - school teacher preferring July and June

The probability that a high - school teacher prefers July is \(P_{July}=\frac{48}{84}\), and the probability that a high - school teacher prefers June is \(P_{June}=\frac{31}{84}\).
Since \(\frac{48}{84}>\frac{31}{84}\), it is more likely that a high - school teacher prefers to vacation in July than June.

Step4: Calculate the probability of a teacher being a middle - school teacher and a high - school teacher

The probability that a teacher is a middle - school teacher is \(P_{middle}=\frac{96}{180}\approx0.533\).
The probability that a teacher is a high - school teacher is \(P_{high}=\frac{84}{180}\approx0.467\).
Since \(0.533>0.467\), option D is wrong.

Answer:

It is more likely that a high - school teacher prefers to vacation in July than June.