Student edition ← Data analysis: two variables

Data analysis: two variables

High School Math 1 · Unit 18 · bivariate-regression · Teacher edition

bivariate-regression

Name ______________________________Date ______________

1.
A survey asked students how they get to school.

Bus Walk
Grade 6 $18$ $12$
Grade 7 $24$ $6$

What percent of the grade 7 students walk? Enter the number only.
Answer: ______________
2.
A line of best fit is $\hat y = 2.5x + 10$ . One data point is $(6, 22)$ . What is the residual for this point?
Answer: ______________
3.
For a class, the line $\hat y = 3.2x + 15$ predicts a test score $y$ from hours of study $x$ . Which is the best interpretation of the number $3.2$ ?
1. (A)
  The predicted score for a student who studies $0$ hours
2. (B)
  The correlation between study time and score
3. (C)
  Each additional hour of study predicts about $3.2$ more points
4. (D)
  Every student's score rises exactly $3.2$ points per hour
4.
In a study of elementary-school children, shoe size and reading score have correlation $r = 0.2$ . Which statement is the best conclusion?
1. (A)
  There is a strong positive relationship, so bigger feet cause better reading
2. (B)
  There is a weak positive association; something else, such as age, could explain both
3. (C)
  Since $r > 0$ , $20\%$ of a child's reading score is explained by shoe size
4. (D)
  There is no association, because $r$ is less than $0.5$
5.
A line is fitted to data on a plant's height over $30$ days. In the residual plot, the residuals are positive for the first $8$ days, negative from day $9$ to day $22$ , and positive again after day $22$ , forming a U-shaped curve. Which conclusion is best?
1. (A)
  A linear model is not appropriate; the curved pattern means the relationship is not linear
2. (B)
  The linear model fits well because the residuals are both positive and negative
3. (C)
  The correlation $r$ must be exactly $0$
4. (D)
  The slope of the fitted line must be negative

	Bus	Walk
Grade 6	$18$	$12$
Grade 7	$24$	$6$

Answer key — Data analysis: two variables

1.
20bivariate-regression-01
$\dfrac{6}{30} = \dfrac{1}{5} = \mathbf{20}$ percent. (Out of all $60$ students, $6$ walkers would be $10\%$ — the joint frequency, a different question.)
2.
-3bivariate-regression-02
$\hat y = 2.5(6) + 10 = 25$ . Residual $= 22 - 25 = \mathbf{-3}$ ; the point lies $3$ units below the line. (Computing $\hat y - y = 3$ gets the sign backwards.)
3.
(C)
Each additional hour of study predicts about $3.2$ more points
bivariate-regression-03
$3.2$ is the slope: each additional hour of study predicts about $3.2$ more points. ( $15$ , the intercept, is the predicted score at $0$ hours; the correlation $r$ is a separate number between $-1$ and $1$ ; “every student rises exactly” turns a model into a guarantee.)
4.
(B)
There is a weak positive association; something else, such as age, could explain both
bivariate-regression-04
$r = 0.2$ is a weak positive association, and age is an obvious lurking variable behind both shoe size and reading. (“Strong” misreads $0.2$ ; causation does not follow from correlation; $r$ is not a percent of anything; a nonzero $r$ is still an association, however weak.)
5.
(A)
A linear model is not appropriate; the curved pattern means the relationship is not linear
bivariate-regression-05
A U-shaped residual plot shows the data curving away from the line on both ends: a linear model is not appropriate. (Mixed signs alone do not mean a good fit — their pattern is the problem; a curved relationship can still have $r$ far from $0$ ; the residual pattern does not determine the slope's sign.)