18. the table below shows the time x (in hours) students spend studying for a science test and the grade they earned. make a scatter plot, then answer the questions. a. what is the correlation? (1 pt) b. write the equation of the line that models that relationship between the grade and the hours spent studying. (4 pts) c. explain the meaning of the slope of the line of best fit. (2 pts)

Question

Accepted Answer

# Explanation:
## Step1: Calculate the correlation
- Plot the points $(3,84)$, $(2,77)$, $(5,92)$, $(1,70)$, $(0,60)$, $(4,90)$, $(3,75)$ on the scatter - plot.
- As the number of hours spent studying ($x$) increases, the grade ($y$) also tends to increase. So, there is a positive correlation.

## Step2: Find the line of best - fit
- First, calculate the mean of $x$ values: $\bar{x}=\frac{3 + 2+5+1+0+4+3}{7}=\frac{18}{7}\approx2.57$
- Calculate the mean of $y$ values: $\bar{y}=\frac{84 + 77+92+70+60+90+75}{7}=\frac{548}{7}\approx78.29$
- Calculate the slope $m$:
$$
\begin{align*}
m&=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\
\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})&=(3 - 2.57)(84 - 78.29)+(2 - 2.57)(77 - 78.29)+(5 - 2.57)(92 - 78.29)+(1 - 2.57)(70 - 78.29)+(0 - 2.57)(60 - 78.29)+(4 - 2.57)(90 - 78.29)+(3 - 2.57)(75 - 78.29)\
&=(0.43)(5.71)+(- 0.57)(-1.29)+(2.43)(13.71)+(-1.57)(-8.29)+(-2.57)(-18.29)+(1.43)(11.71)+(0.43)(-3.29)\
&=2.4553 + 0.7353+33.3153 + 13.0153+46.9053+16.7453-1.4147\
&=111.7675
\end{align*}
$$
$$
\begin{align*}
\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}&=(3 - 2.57)^{2}+(2 - 2.57)^{2}+(5 - 2.57)^{2}+(1 - 2.57)^{2}+(0 - 2.57)^{2}+(4 - 2.57)^{2}+(3 - 2.57)^{2}\
&=(0.43)^{2}+(-0.57)^{2}+(2.43)^{2}+(-1.57)^{2}+(-2.57)^{2}+(1.43)^{2}+(0.43)^{2}\
&=0.1849+0.3249 + 5.9049+2.4649+6.6049+2.0449+0.1849\
&=17.7143
\end{align*}
$$
$m=\frac{111.7675}{17.7143}\approx6.31$
- Use the point - slope form $y - y_{1}=m(x - x_{1})$ with the point $(\bar{x},\bar{y})$ (or we can use another method of linear regression). Using the formula $y=mx + b$, substituting $x = 0$ (approximate), when $x = 0$, $y\approx60$, so $b = 60$ (a rough estimate, more accurate calculation can be done using the formula $b=\bar{y}-m\bar{x}$, $b = 78.29-6.31\times2.57=78.29 - 16.21=62.08\approx62$). The equation of the line of best - fit is $y = 6x+62$

## Step3: Interpret the slope
- The slope $m = 6$ in the equation $y=mx + b$ (where $y$ is the grade and $x$ is the number of hours). The slope represents the rate of change. So, for every additional hour a student spends studying, the grade on the science test is expected to increase by approximately $6$ points.

# Answer:
A. Positive correlation.
B. $y = 6x + 62$ (approximate, more accurate calculation of the line of best - fit can be done using more advanced linear regression techniques, but for a basic estimate this is reasonable).
C. For every additional hour a student spends studying, the grade on the science test is expected to increase by approximately $6$ points.

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QUESTION IMAGE

Question

Explanation:

Step1: Calculate the correlation

Step2: Find the line of best - fit

Step3: Interpret the slope

Answer: