Question

answer the questions below.
(a)
a piano instructor examined student progress. she noticed that when students spend less time playing video games, their memorization ability tends to increase. what can she conclude?

• there is no correlation between amount of time spent playing video games and memorization ability.
• there is a correlation between amount of time spent playing video games and memorization ability. there may or may not be causation. further studies would have to be done to determine this.
• there is a correlation between amount of time spent playing video games and memorization ability. there is probably also causation. this is because there is an increase in memorization ability with a decrease in the amount of time spent playing video games.

(b)
an environmental study was conducted. it reported that houses with more inhabitants do not necessarily produce more or less trash. what can be determined?

• there is no correlation between number of inhabitants and amount of trash produced.
• there is a correlation between number of inhabitants and amount of trash produced. however, there is no causation. this is because there is probably an increase in the amount of trash produced with an increase in the number of inhabitants.
• there is a correlation between number of inhabitants and amount of trash produced. there may or may not be causation. further studies would have to be done to determine this.

(c)
data was collected at a pool. it showed that being older does not imply that a swimmer will stay for a shorter or longer amount of time. what is likely true?

• there is no correlation between age and length of stay.
• there is a correlation between age and length of stay. there may or may not be causation. further studies would have to be done to determine this.
• there is a correlation between age and length of stay. however, there is no causation. this is because there is likely an increase in the length of stay with a decrease in age.

Explanation:

Part (a)

Step1: Analyze Correlation Direction

The instructor notices less video - game time (let's call this variable \(x\)) is associated with increased memorization ability (variable \(y\)). So, as \(x\) decreases, \(y\) increases, which means there is a negative correlation between \(x\) and \(y\) (or a positive correlation between the "less video - game time" and "increased memorization ability"). But correlation does not imply causation. Just because two variables are correlated, we can't say that one causes the other. We need further studies to check for causation.
The first option says there is no correlation, which is wrong. The third option says there is a correlation and also causation, which is wrong because correlation doesn't mean causation. The fourth option has the wrong direction of correlation (it says increase in memorization with decrease in video - game time, but the option's description of the correlation direction is incorrect as it mixes up the variables). The second option correctly states that there is a correlation between the amount of time spent playing video games and memorization ability, and that there may or may not be causation (so further studies would be needed to determine causation).

Step2: Select the Correct Option

Based on the analysis of correlation and causation, the correct option is the second one: "There is a correlation between amount of time spent playing video games and memorization ability. There may or may not be causation. Further studies would have to be done to determine this."

Step1: Analyze the Environmental Study

The study says that houses with more inhabitants do not necessarily produce more or less trash. So, we need to see the relationship between the number of inhabitants and the amount of trash produced. If more inhabitants don't necessarily mean more or less trash, it means that there is a correlation (because as the number of inhabitants changes, the amount of trash produced changes in a way that is not a simple direct or inverse relationship, but there is still a relationship) but no causation. Also, if more inhabitants don't necessarily produce more trash, it is probable that as the number of inhabitants increases, the amount of trash produced increases (since more people would likely produce more trash on average, but the study says it's not necessary, so it's a probable increase).
The first option says there is no correlation, which is wrong. The third option says there is a correlation and also that there is no causation, but it says "probably an increase" which is more accurate than the third option's description. The fourth option is wrong as we can't say that we need further studies to determine the correlation (the study already gives us some information about the relationship). The second option correctly states that there is a correlation between the number of inhabitants and the amount of trash produced, that there is probably an increase in the amount of trash with an increase in the number of inhabitants, and that there may or may not be causation (so further studies would be needed to determine causation).

Step2: Select the Correct Option

Based on the analysis of the relationship between the number of inhabitants and trash production, the correct option is the second one: "There is a correlation between number of inhabitants and amount of trash produced. However, there is no causation. This is probably an increase in the amount of trash produced with an increase in the number of inhabitants."

Step1: Analyze the Pool Data

The data shows a correlation between age and length of stay. But correlation does not imply causation. Just because older people have a certain length of stay, we can't say that age causes the length of stay to be shorter or longer. Also, the data shows a correlation, so the first option (no correlation) is wrong. The third option says there is a correlation and also causation, which is wrong. The fourth option has the wrong direction of correlation (it says increase in length of stay with decrease in age, but the analysis is about age and length of stay in general). The second option correctly states that there is a correlation between age and length of stay, and that there may or may not be causation (so further studies would be needed to determine causation).

Step2: Select the Correct Option

Based on the analysis of correlation and causation in the pool data, the correct option is the second one: "There is a correlation between age and length of stay. There may or may not be causation. Further studies would have to be done to determine this."

Answer:

There is a correlation between amount of time spent playing video games and memorization ability. There may or may not be causation. Further studies would have to be done to determine this. (The option corresponding to this statement in the original question's (a) part)

Part (b)