Question

3-5 enrichment
scatter plots and lines of fit
a group of students is preparing for a regional scholar bowl, a competition quiz game. the students start by taking a general knowledge exam. the exams are scored and students start work with their faculty advisor, mr. rodriguez, to improve their performance. after two weeks of study and practice, the students test again using a different exam. the scores are provided in the table below.

1st exam	92	72	90	97	60	99	67	87	96	80	62
2nd exam	74	85	73	82	81	78	95	74	76	89	88

1. make a scatter plot of the data. draw a trend line and write its equation.
2. describe the relationship represented by your graph. is the relationship what you expected?

Explanation:

Step1: Identify data points

Let the scores of the 1st exam be $x$ - values and the scores of the 2nd exam be $y$ - values. We have pairs like $(92,74),(72,85),(90,73),\cdots,(62,88)$.

Step2: Plot the scatter - plot

On a coordinate plane, mark each pair $(x,y)$ as a point.

Step3: Draw the trend line

Visually estimate a line that best fits the data points. To find the equation of the line in the form $y = mx + b$ (where $m$ is the slope and $b$ is the y - intercept), we can choose two points on the line. Let's assume two points on the trend - line are $(60,81)$ and $(90,75)$.
The slope $m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{75 - 81}{90 - 60}=\frac{-6}{30}=-0.2$.
Using the point - slope form $y - y_1=m(x - x_1)$ with the point $(60,81)$:
$y-81=-0.2(x - 60)$
$y-81=-0.2x + 12$
$y=-0.2x+93$.

Step4: Analyze the relationship

The slope of the trend line is negative, so there is a negative linear relationship between the 1st exam scores and the 2nd exam scores. This might be unexpected as one might assume that students who did well on the first exam would also do well on the second exam after studying, but it could be due to various factors like the different nature of the two exams.

Answer:

1. Scatter - plot is made by plotting the given data points. The equation of the trend line is $y=-0.2x + 93$.
2. There is a negative linear relationship between the scores of the 1st and 2nd exams. Whether it is expected or not can vary depending on assumptions; one might expect a positive relationship but factors like exam differences can cause a negative one.