Question

researchers carried out a survey of fourth -, fifth -, and sixth - grade students in michigan. students were asked if good grades,
athletic ability, or being popular was most important to them. the two - way table summarizes the survey data.

suppose we select one of these students at random. whats the probability that:
(a) the student is a sixth grader or a student who rated good grades as important?
(round to 3 decimal places. leave your answer in decimal form.)

(b) the student is not a sixth grader and did not rate good grades as important?
(round to 3 decimal places. leave your answer in decimal form.)

Explanation:

Step1: Calculate the total number of students

The total number of students is \(335\) (from the table).

Step2: Calculate the number of students who are sixth - graders or rated good grades as important

The number of sixth - graders is \(135\). The number of students who rated good grades as important: \(49 + 50+69=168\). But we have double - counted the sixth - graders who rated good grades as important (\(69\)). So the number of students who are sixth - graders or rated good grades as important is \(135 + 168-69=234\).
The probability \(P=\frac{234}{335}\approx0.699\)

Step3: Calculate the number of students who are not sixth - graders and did not rate good grades as important

The number of students who are not sixth - graders is \(335 - 135 = 200\). The number of students who did not rate good grades as important is \(335-168 = 167\). The number of non - sixth - graders who did not rate good grades as important: \(19+24+22 + 36=101\) (from the table, sum of non - sixth - graders in 'Popular' and 'Athletic' columns). The probability \(P=\frac{101}{335}\approx0.302\)

Answer:

(a) \(0.699\)
(b) \(0.302\)