Question

A researcher wanted to know if there is an association between exercise and sleep for the population of $16$-year-olds in the United States. She obtained survey responses from a random sample of $2000$ United States $16$-year-olds and found convincing evidence of a positive association between exercise and sleep. Which of the following conclusions is well supported by the data?
A. There is a positive association between exercise and sleep for $16$-year-olds in the United States.
B. There is a positive association between exercise and sleep for $16$-year-olds in the world.
C. Using exercise and sleep as defined by the study, an increase in sleep is caused by an increase of exercise for $16$-year-olds in the United States.
D. Using exercise and sleep as defined by the study, an increase in sleep is caused by an increase of exercise for $16$-year-olds in the world.

Answer 1

From the question, we need to analyse the formula of correlation because correlation helps to analyse the relationship between two variables. Here a researcher is interested to know the type of association between two variables. So, correlation is the method of analysing the relationship.

Formula used: We will use the conclusions obtained from the values of correlation coefficient. It can be of three types- Positive, Negative and zero. A positive correlation stands for a positive relationship between the variables. A negative correlation stands for negative relationship between the variables. A zero correlation stands for no relationship between the variables.

Complete step-by-step solution:
Here, a researcher wanted to know if there is an association between exercise and sleep for the population of $16$-year-olds in the United States. For her study, She obtained survey responses from a random sample of $2000$ United States $16$-year-olds.
She found convincing evidence of a positive association between exercise and sleep, that is, exercise and sleep are related to each other and that too positively. A positive relationship means the increase in one variable would also increase the other. Here, the increase in exercise would result in an increase in sleep.
The study is for the $16$-year-olds in the United States, so, the results will obtain a conclusion for $16$-year-olds in the United States. We can conclude, there is a positive association between exercise and sleep for $16$-year-olds in the United States.
As the data provided, uses exercise and sleep as variables, we can also say, Using exercise and sleep as defined by the study, an increase in sleep is caused by an increase of exercise for $16$-year-olds in the United States.

Thus, Option A and C are correct. But option C is more précised.

Note: Correlation is the relationship between two variables. It helps to study the behaviour of the related variables. Though it provides an insight of the relationship, it cannot conclude a quantitative conclusion.