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A statistics student wants to determine if there is a relationship between a student's number of absences, [tex]x[/tex], and their grade point average (GPA), [tex]y[/tex]. The given data lists the number of absences and GPAs for 15 randomly selected students.

\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\begin{tabular}{l}
Number of \\
Absences
\end{tabular} & 15 & 1 & 0 & 6 & 9 & 12 & 3 & 3 & 1 & 2 & 7 & 0 & 4 & 9 & 10 \\
\hline
GPA & 2.1 & 4.3 & 4.5 & 3.2 & 4.0 & 1.7 & 3.8 & 2.9 & 3.6 & 3.4 & 2.6 & 3.1 & 2.8 & 2.8 & 4.1 \\
\hline
\end{tabular}

Using technology, the slope of the least-squares regression line is:

A. -0.10, which means for each additional absence, the GPA will decrease by 0.10 points.
B. -0.10, which means for each additional absence, the GPA is predicted to decrease by 0.10 points.
C. 0.10, which means for each additional absence, the GPA will increase by 0.10 points.
D. 0.10, which means for each additional absence, the GPA is predicted to increase by 0.10 points.



Answer :

GinnyAnswer
To determine if there is a relationship between the number of absences and the GPA of students, the student decided to use a least-squares regression analysis. The given data contains the number of absences and corresponding GPAs for 15 randomly selected students.

Using statistical technology, the student calculates the slope of the least-squares regression line to be -0.10.

The interpretation of the slope is crucial in understanding the relationship between the number of absences (independent variable) and the GPA (dependent variable). Let’s analyze the meaning of this slope of -0.10:

1. Interpretation of a Negative Slope:
- A slope of -0.10 indicates that for each additional absence, the GPA decreases.
- Specifically, a slope of -0.10 means that for one more absence, the GPA is expected to decrease by 0.10 points.

Now, the question provides four possible interpretations of the slope:

1. Option 1: "-0.10 , which means for each additional absence, the GPA will decrease by 0.10 points."
2. Option 2: "-0.10 , which means for each additional absence, the GPA is predicted to decrease by 0.10 points."
3. Option 3: "0.10, which means for each additional absence, the GPA will increase by 0.10 points."
4. Option 4: "0.10 , which means for each additional absence, the GPA is predicted to increase by 0.10 points."

Given that the calculated slope is -0.10, we can evaluate each option:

- Options 1 and 2 correctly match the calculated negative slope of -0.10 and describe the relationship accurately:
- Option 1 states that the GPA will decrease by 0.10 points for each additional absence.
- Option 2 states that the GPA is predicted to decrease by 0.10 points for each additional absence.

- Options 3 and 4 incorrectly describe the relationship as they refer to a positive slope of 0.10:
- These options would imply that the GPA increases by 0.10 points for each additional absence, which is not consistent with our calculated slope of -0.10.

Thus, the correct interpretations of the slope of the least-squares regression line are:

- Option 1: "-0.10 , which means for each additional absence, the GPA will decrease by 0.10 points."
- Option 2: "-0.10 , which means for each additional absence, the GPA is predicted to decrease by 0.10 points."

Therefore, the correct interpretations of the slope are given by options 1 and 2.

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