Question

1. consider the information in the table which shows how long a student studied for a final exam and the grade received.

time(min) grade
20 60
55 75
60 73
50 75
30 82
40 80
120 95
65 85
110 91
100 86
10 70
35 87
which equation represents the relationship between the variables?
a. y = -157.39x + 2.69
b. y = 67.96x + 0.21
c. y = 2.69x - 157.39
d. y = 0.21x + 67.96

1. which approximates the correlation coefficient?

a. 0.21 b. 0.56 c. 0.75 d. 0.96

Explanation:

Step1: Recall the form of a linear - regression equation

The general form of a simple linear - regression equation is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. We can use a graphing calculator or software to perform linear regression on the data (where $x$ is the time studied and $y$ is the grade). However, we can also make some rough estimates. As the time studied ($x$) increases, the grade ($y$) generally increases, so the slope $m>0$.

Step2: Analyze the options

Option A has a negative slope ($m=-157.39$), so we can eliminate it. To further narrow down, we can use the fact that when $x = 0$, the grade should be a non - negative value. We can assume that if a student doesn't study ($x = 0$), they still get some non - zero grade.
We know that the y - intercept $b$ represents the value of $y$ when $x = 0$.
For option B: $y = 67.96x+0.21$, when $x = 0$, $y = 0.21$ which is a very small non - zero value.
For option C: $y = 2.69x−157.39$, when $x = 0$, $y=-157.39$ which is not possible for a grade.
For option D: $y = 0.21x + 67.96$, when $x = 0$, $y = 67.96$ which is a reasonable grade for a student who doesn't study (a base - level knowledge grade). Also, the slope $m = 0.21$ indicates a slow but positive increase in grade as study time increases.

For the correlation coefficient:

Step3: Understand the nature of the correlation

Since there is a positive relationship between the time studied and the grade received (as study time increases, grade generally increases), the correlation coefficient $r>0$. A value of $r$ close to 1 indicates a strong positive linear relationship. Looking at the data, as the time studied increases, the grade shows a fairly consistent upward trend. A correlation coefficient of 0.21 is a weak positive correlation, 0.56 is a moderate positive correlation, 0.75 is a relatively strong positive correlation, and 0.96 is a very strong positive correlation. The data seems to have a relatively strong positive linear relationship, so a correlation coefficient of 0.75 is a reasonable approximation.

Answer:

1. D. $y = 0.21x + 67.96$
2. C. 0.75