Question

In this discussion, you will apply the statistical concepts and techniques covered in this week's reading about correlation coefficient and simple linear regression. A car rental company wants to evaluate the premise that heavier cars are less fuel efficient than lighter cars. In other words, the company expects that fuel efficiency (miles per gallon) and weight of the car (often measured in thousands of pounds) are correlated. Performing this analysis will help the company optimize its business model and charge its customers appropriately.

In this discussion, you will work with a cars data set that includes two variables:

• Miles per gallon (coded as mpg in the data set)
• Weight of the car (coded as wt in the data set)

The random sample will be drawn from a CSV file. This data will be unique to you, and therefore your answers will be unique as well. Run Step 1 in the Python script to generate your unique sample data.

In your initial post, address the following items:

1. You created a scatterplot of miles per gallon against weight; check to make sure it was included in your attachment. Does the graph show any trend? If yes, is the trend what you expected? Why or why not? See Step 2 in the Python script.
2. What is the coefficient of correlation between miles per gallon and weight? What is the sign of the correlation coefficient? Does the coefficient of correlation indicate a strong correlation, weak correlation, or no correlation between the two variables? How do you know? See Step 3 in the Python script.
3. Write the simple linear regression equation for miles per gallon as the response variable and weight as the predictor variable. How might the car rental company use this model? See Step 4 in the Python script.
4. What is the slope coefficient? Is this coefficient significant at a 5% level of significance (alpha=0.05)? (Hint: Check the P-value, , for weight in the Python output.) See Step 4 in the Python script.

<Figure size 640x480 with 1 Axes>

    mpg        wt
mpg  1.000000 -0.863527
wt  -0.863527  1.000000

 OLS Regression Results                            
==============================================================================
Dep. Variable:                    mpg   R-squared:                       0.746
Model:                            OLS   Adj. R-squared:                  0.737
Method:                 Least Squares   F-statistic:                     82.10
Date:                Fri, 02 Oct 2020   Prob (F-statistic):           8.10e-10
Time:                        12:39:57   Log-Likelihood:                -75.289
No. Observations:                  30   AIC:                             154.6
Df Residuals:                      28   BIC:                             157.4
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
Intercept     37.1346      1.960     18.944      0.000      33.119      41.150
wt            -5.2638      0.581     -9.061      0.000      -6.454      -4.074
==============================================================================
Omnibus:                        2.644   Durbin-Watson:                   2.405
Prob(Omnibus):                  0.267   Jarque-Bera (JB):                2.104
Skew:                           0.643   Prob(JB):                        0.349
Kurtosis:                       2.832   Cond. No.                         12.7
==============================================================================

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Answer 1

Does the graph show any trend? If yes, is the trend what you expected?

Ans:From the scatterplot we see that there is negative trend between mpg and weight of the car. this means that if weight of car increases then the milege of the car will decreses. Hence i would expect that if i choose a heavy car than it will give less milege.

What is the coefficient of correlation between miles per gallon and weight? What is the sign of the correlation coefficient? Does the coefficient of correlation indicate a strong correlation, weak correlation, or no correlation between the two variables?

Ans: The coefficient of correlation between miles per gallon and weight is -0.8592 . The sign of the correlation coefficient is negative. The coefficient of correlation indicate a strong correlation because correlation coefficient lies between -1 and 1 and if coefficient of correlation is close to -1 or 1 than there is a strong correlation between the variables. Here the correlation coefficient is -0.8592 which is close to -1, so we can say that there is strong correlation between the variables.

Write the simple linear regression equation for miles per gallon as the response variable and weight as the predictor variable. How might the car rental company use this model?

Ans:

The simple linear regression equation for miles per gallon as the response variable and weight as the predictor variable is written as

mpg = 37.2757 - 5.3542*wt

The car rental company uses this model to decide the miles per gallon of car according to the weight of the car. the company can make more profit by using lighter cars

What is the slope coefficient? Is this coefficient significant at a 5% level of significance (alpha=0.05)? (Hint: Check the P-value, , for weight in the Python output.)

Ans: From the regression output, we see that the slope coefficient is -5.3542 and the p-value is 0.000 which is less than 0.05, so the slope coefficient is significant at 5% level of significant.

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Happy Learning :)