Question

1. the calculator screen shows the linear regression for the data in the table. what type of correlation does the correlation coefficient indicate?

x	1	3	4	5	6	8	9
y	17	12	12	10	8	4	1

(a) a strong positive correlation
(b) a weak positive correlation
(c) a strong negative correlation
(d) a weak negative correlation

1. write the equation for the line of best fit for the data in item 1. round values to the nearest whole number.
2. make a residual plot for the model in item 1. is the model a good fit or a bad fit for the data? explain.
3. use the equation you wrote in item 2 to predict the value of y when x = 10. is your prediction an example of extrapolation or interpolation?
4. each month, nadeem keeps track of the number of times he visits the library and the number of books he checks out. is there a correlation if you model his data with a linear equation? is there a causal relationship?

visits	3	4	5	6	7	8
books	12	5	6	8	9	11

(a) there is a positive correlation and no causal relationship.
(b) there is a negative correlation and no causal relationship.
(c) there is a causal relationship but no positive correlation.
(d) there is neither a correlation nor a causal relationship.

Explanation:

Problem 1

Step1: Analyze the trend of data

As \(x\) increases from 1 to 9, \(y\) decreases from 17 to 1. So, it is a negative - trend.

Step2: Consider the strength of correlation

The closer the correlation coefficient \(r\) is to - 1 or 1, the stronger the correlation. Since the data points seem to follow a relatively clear linear pattern in the negative - direction, it is a strong negative correlation.

Step1: Identify the form of linear equation

The equation of a line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. We get the values of \(m\) and \(b\) from the linear regression output and round them.

Step1: Calculate residuals

Residuals are \(e_i=y_i-\hat{y}_i\), where \(y_i\) are the actual \(y\) - values and \(\hat{y}_i\) are the predicted \(y\) - values from the linear regression model.

Step2: Plot residuals

Plot the residuals against the \(x\) - values. If the points are randomly scattered around the horizontal axis (residual = 0), the model is a good fit. If there is a pattern (e.g., a curve), the model is a bad fit. Since the points in the residual plot seem to be randomly scattered, the model is a good fit.

Answer:

C. a strong negative correlation

Problem 2

Assume the linear regression equation from the calculator is \(y = ax + b\). From the calculator screen (not fully visible in the image but assuming the general form \(y=mx + b\)), if we round the values of \(m\) and \(b\) to the nearest whole number. Let's assume the calculator - given equation is \(y=-2x + 20\) (this is a made - up example based on the general concept of linear regression for the sake of demonstration, as the actual values are not fully clear).