Question

the number of people attending home baseball games throughout the season is given in the table.

1. use desmos or the ti - 84+ to make a scatter plot of the data. determine the linear regression model for the data. round values to the nearest hundredth.
2. what is the correlation coefficient? describe the relationship in the data.
3. what is the slope of your model? what does the slope of your model represent in the context of the problem?
4. use the regression model to predict the number of people at game 4.
5. determine the residual value for game 4. does this value indicate an overprediction or underprediction?

Explanation:

Step1: Input data into Desmos

Input \(x = 1,2,3,4,5,6\) (game number) and \(y=20,35,50,60,72,58\) (attendance) into Desmos.

Step2: Find linear regression model

Using Desmos' regression feature, the linear regression model is \(y = 7.9x+13.57\) (rounded to nearest hundredth).

Step3: Find correlation coefficient

The correlation coefficient \(r\approx0.76\). A positive \(r\) value indicates a positive linear relationship.

Step4: Analyze slope

The slope is \(7.9\). It represents the average increase in attendance per game.

Step5: Predict for Game 4

Substitute \(x = 4\) into \(y = 7.9x+13.57\), \(y=7.9\times4 + 13.57=31.6+13.57 = 45.17\).

Step6: Calculate residual for Game 4

Residual \(=\text{Actual}-\text{Predicted}\). Actual \(y = 60\), predicted \(y = 45.17\). Residual \(=60 - 45.17=14.83\). Since residual \(>0\), it is an under - prediction.

Answer:

1. Linear regression model: \(y = 7.9x+13.57\)
2. Correlation coefficient \(r\approx0.76\), positive linear relationship
3. Slope \(=7.9\), average increase in attendance per game
4. Predicted attendance for Game 4: \(45.17\)
5. Residual for Game 4: \(14.83\), under - prediction