We can use these data to understand the idea of correlation, linear regression and outliers.

Below is a Scatter-Plot with Least Squares Regression Lines, SPSS was used.

The dots on the graph denote cities.

Answer the following:

Below is the model information and correlation if we use all 43 cities.

Model Summary

Model

R

R Square

Adjusted R Square

Std. Error of the Estimate

1

.139^a

.019

-.005

138.5738

a. Predictors: (Constant), Public Health Response Time (days)

Coefficients^a

Model

Unstandardized Coefficients

Standardized Coefficients

t

Sig.

B

Std. Error

Beta

1

(Constant)

482.263

33.445

14.420

.000

Public Health Response Time (days)

2.804

3.131

.139

.895

.376

a. Dependent Variable: Excess Deaths per 100,000 population

1.a. What is the explanatory (independent) variable?__ Describe how the number of cases are changing in the plot.

_________________________

b.What is the response (dependent) variable?______________________________

Questions 2 and 3 below are about the full dataset with all 43 cities:

2.Describe the following properties of the scatterplot:

a.Pattern:Is it linear (straight), curved, clustered, or none of these?__________________

b.Strength:How close are the points to the pattern?(Strong, medium, weak?) __________

c.Direction:Positive (upward) or negative (downward) trend? __________________

d.Outliers ? Are there any and what are they?

Outliers in a Scatterplot:Outliers can really impact the value of r and the linear regression line.

The point can be an outlier in the x-coordinate.This occurs if the point's x value is atypically low or high compared to the rest of the x-values.The point can be an outlier in the y-coordinate.This occurs if the point's y value is atypically low or high compared to the rest of the y-values.The point can be an outlier because it does not fit the overall pattern of the data.

3.What is the correlation coefficient?

4.Is it an appropriate statistic to use in the presence of outliers?

Given there were three points that were very unlike the rest, I excluded those points and refit the linear regression model.What do you notice about the correlation coefficient for the reduced data compared to the full data?

Model Summary^b,c

Model

R

R Square

Adjusted R Square

Std. Error of the Estimate

Outlier_labels =(Selected)

Outlier_labels ~=(Unselected)

1

.415^a

.

.172

.150

115.8574

a. Predictors: (Constant), Public Health Response Time (days)

b. Unless noted otherwise, statistics are based only on cases for which Outlier_labels =.

c. Dependent Variable: Excess Deaths per 100,000 population

Coefficients^a,b

Model

Unstandardized Coefficients

Standardized Coefficients

t

Sig.

B

Std. Error

Beta

1

(Constant)

435.913

31.400

13.882

.000

Public Health Response Time (days)

9.658

3.435

.415

2.812

.008

a. Dependent Variable: Excess Deaths per 100,000 population

b. Selecting only cases for which Outlier_labels =

5.Write the equation of the least squares regression line for this data set:

6.Suppose a city had a public response time of 10 days.What would be the predicted number of excess deaths per 100,000 population?

7.What is the value of the y-intercept? __________Interpret this value in terms of the application?

8.What is the value of the slope? ___________Interpret this value in terms of the application.

9.What percentage of the variation in excess flu deaths per 100,000 can be accounted for by the linear relationship with public health response time?

10. Here's a graph of early data of cumulative Covid 19 cases in March for Michigan:focus on the red line

a. Is the plot linear?

b. Describe how the number of cases are changing in the plot.

c. What do you think the graph will look like in the next two months?