Question

Esearchers believed that an increase in lean body mass is associated with an increase in maximal oxygen uptake. A scatterplot of the measurements taken from 18 randomly selected college athletes displayed a strong positive linear relationship between the two variables. A significance test for the null hypothesis that the slope of the regression line is 0 versus the alternative that the slope is greater than 0 yielded a p-value of 0.04.
Which statement is an appropriate conclusion for the test?

(A)The p-value of 0.04 indicates that 4% ofthe variation in maximal oxygen uptake for college athletes can be explained by the amount of lean body mass.
(B) The p-value of0.04 indicates that 16% of the variation in maximal oxygen uptake for college athletes can be explained by the amount of lean body mass.
(C)The strong positive linear relationship displayed in the scatterplot along with a p-value less than 0.05 indicates that college athletes with higher lean body mass tend to have higher maximal oxygen uptake.
(D)The strong positive linear relationship displayed in the scatterplot along with a p-value less than 0.05 indicates that an increase in lean body mass causes an increase in maximal oxygen uptake for college athletes.
(E) A p-value less than 0.05 indicates that the relationship displayed in the scatterplot is likely due to chance, and that there is no statistical evidence ofa relationship between lean body mass and maximal oxygen uptake for college athletes.

Answer 1

Answer:

The correct option is (D).

Step-by-step explanation:

The p-value is well defined as per the probability, [under the null hypothesis (H₀)], of attaining a result equivalent to or more extreme than what was truly observed.

A small p-value (typically ≤ 0.05) specifies solid proof against the null hypothesis (H₀), so you discard H₀.

A large p-value (> 0.05) specifies fragile proof against the H₀, so you fail to discard H₀.

The hypothesis is defined as follows:

H₀: The slope of the regression line is 0, i.e. β = 0.

Hₐ: The slope of the regression line is greater than 0, i.e. β > 0.

A scatter-plot of the measurements taken from 18 randomly selected college athletes displayed a strong positive linear relationship between the two variables; lean body mass and maximal oxygen uptake.

The significance level of the test is, α = 0.05.

The p-value of the test is, p-value = 0.04

As the p-value = 0.04 < α = 0.05, we reject the null hypothesis.

So, it can be concluded that there is a statistical evidence of a relationship between lean body mass and maximal oxygen uptake for college athletes.

Since the scatter-plot shows a strong positive linear relationship, it implies that as the an increase in lean body mass causes an increase in maximal oxygen uptake for college athletes.

Thus, the correct option is (D).