Question

question 1 (multiple choice worth 5 points) (02.01h mc) the two - way frequency table given shows the results from a survey of students who attend the afterschool program.

	plays a sport	doesnt play a sport	total
doesnt take art class
total			225

does the data show an association between taking an art class and playing a sport?
there is a strong, positive association.
there is a strong, negative association.
there is a weak, positive association.
there is a weak, negative association.

Explanation:

Step1: Complete the two - way table

First, find the number of students who play a sport and don't take an art class. The total number of students who play a sport is 140, and the number of students who play a sport and take an art class is 45. So the number of students who play a sport and don't take an art class is \(140 - 45=95\).

Next, find the number of students who don't play a sport and don't take an art class. The total number of students is 225, and the total number of students who take an art class is \(45 + 45 = 90\). So the number of students who don't take an art class is \(225-90 = 135\). The number of students who don't play a sport and take an art class is 45, so the number of students who don't play a sport and don't take an art class is \(135 - 45=90\)? Wait, no, let's do it another way. The total number of students who don't play a sport is \(225 - 140=85\). The number of students who don't play a sport and take an art class is 45, so the number of students who don't play a sport and don't take an art class is \(85 - 45 = 40\). And the number of students who don't take an art class is \(40+95 = 135\) (since 95 play a sport and don't take art, 40 don't play sport and don't take art).

Now we have the completed table:

	Takes Art Class	Doesn't Take Art Class	Total
Doesn't Play a Sport	45	40	85
Total	90	135	225

Step2: Calculate the conditional probabilities

The probability that a student plays a sport given that they take an art class: \(P(\text{Play Sport}|\text{Take Art})=\frac{45}{90} = 0.5\)

The probability that a student plays a sport given that they don't take an art class: \(P(\text{Play Sport}|\text{Don't Take Art})=\frac{95}{135}\approx0.7037\)

The difference between these two probabilities is \(|0.5 - 0.7037|= 0.2037\). Since the difference is relatively small (less than 0.5), we consider the association to be weak. Also, since \(P(\text{Play Sport}|\text{Take Art})<P(\text{Play Sport}|\text{Don't Take Art})\), as the probability of playing a sport is lower for those who take art class, the association is negative. So there is a weak, negative association.

Answer:

There is a weak, negative association.