Question

statistics: scatter plots and lines of fit
determine whether each graph shows a positive correlation, a negative correlation, or no correlation. if there is a positive or negative correlation, describe its meaning in the situation.

1. calories burned during exercise
2. library fines
3. weight-lifting
4. evening newspapers

source: editor & publisher
baseball for exercises 5 - 7, use the scatter plot that shows the average price of a major - league baseball ticket from 1991 to 2005.

1. determine what relationship, if any, exists in the data. explain.
2. use the points (1998, 13.60) and (2003, 19.00) to write the slope - intercept form of an equation for the line of fit shown in the scatter plot.
3. predict the price of a ticket in 2009.

source: team marketing report, chicago

Explanation:

Step1: Determine the type of correlation

For problem 1:
As time (in minutes) increases, the calories burned also increases. So, it shows a positive correlation. The meaning is that as the time spent exercising increases, the number of calories burned during exercise increases.
For problem 2:
There is no clear pattern. The points are scattered randomly. So, it shows no correlation.
For problem 3:
As the weight (in pounds) increases, the number of repetitions decreases. So, it shows a negative correlation. The meaning is that as the weight lifted increases, the number of repetitions performed decreases.
For problem 4:
As the year increases, the number of evening newspapers decreases. So, it shows a negative correlation. The meaning is that as the years go by, the number of evening newspapers sold decreases.
For problem 5:
Looking at the baseball ticket prices scatter - plot, as the year increases, the average price of a ticket increases. So, there is a positive correlation. The meaning is that as the years progress from 1991 to 2005, the average price of a major - league baseball ticket increases.

Step2: Find the slope for problem 6

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \((x_1,y_1)=(1998,13.60)\) and \((x_2,y_2)=(2003,19.00)\).

$$m=\frac{19.00 - 13.60}{2003 - 1998}=\frac{5.4}{5}=1.08$$

Using the point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(1998,13.60)\) and \(m = 1.08\):

$$y-13.60=1.08(x - 1998)$$

$$y-13.60=1.08x-1.08\times1998$$

$$y-13.60=1.08x - 2157.84$$

$$y=1.08x-2157.84 + 13.60$$

$$y=1.08x-2144.24$$

Step3: Predict for problem 7

For 2009, \(x = 2009\). Using the equation \(y = 1.08x-2144.24\)

$$y=1.08\times2009-2144.24$$

$$y = 2169.72-2144.24$$

$$y=25.48$$

Answer:

1. Positive correlation. As exercise time increases, calories burned increase.
2. No correlation.
3. Negative correlation. As weight increases, repetitions decrease.
4. Negative correlation. As year increases, number of evening newspapers decreases.
5. Positive correlation. As years progress, baseball ticket prices increase.
6. \(y = 1.08x-2144.24\)
7. \(\$25.48\)