Question

The data show the chest size and weight of several bears. Find the regression​ equation, letting chest size be the independent​ (x) variable. Then find the best predicted weight of a bear with a chest size of

3939

inches. Is the result close to the actual weight of

126126

​pounds? Use a significance level of 0.05.

Chest size​ (inches)	44	41	41	55	51	42
Weight​ (pounds)	213	206	176	309	300	178

n	alphaαequals=0.05	alphaαequals=0.01	​NOTE: To test H0​: rhoρequals=0 against H1​: rhoρnot equals≠​0, reject H0 if the absolute value of r is greater than the critical value in the table.
4	0.950	0.990
5	0.878	0.959
6	0.811	0.917
7	0.754	0.875
8	0.707	0.834
9	0.666	0.798
10	0.632	0.765
11	0.602	0.735
12	0.576	0.708
13	0.553	0.684
14	0.532	0.661
15	0.514	0.641
16	0.497	0.623
17	0.482	0.606
18	0.468	0.590
19	0.456	0.575
20	0.444	0.561
25	0.396	0.505
30	0.361	0.463
35	0.335	0.430
40	0.312	0.402
45	0.294	0.378
50	0.279	0.361
60	0.254	0.330
70	0.236	0.305
80	0.220	0.286
90	0.207	0.269
100	0.196	0.256
n	alphaαequals=0.05	alphaαequals=0.01

PrintDone

What is the regression equation?

Answer 1

We need to find the Least Squares Regression Line equation for the independent variable, i.e., Chest size(inches) and the dependent variable, i.e., Weight(pounds) for bears.

The Least Squares Regression Line is given by

where, x: Chest size , y: Weight

The following data is given for six bears:

n=6, the number of bears in the data

44	213	1936	9372	45369
41	206	1681	8446	42436
41	176	1681	7216	30976
55	309	3025	16995	95481
51	300	2601	15300	90000
42	178	1764	7476	31684

The formula to calculate the intercept (a) and slope (b) of the regression line is as follows:

                        

So, the intercept of the regression line is calculated as

                

So, the slope of the regression line is calculated as

Now the equation of regression line is written as -

where, x: Chest size, predicted Weight of bear for given Chest size

Now the predicted weight of the bear for Chest size i.e., and the actual weight of bear is given as ,

Using the regression equation-

So, the predicted weight of bear for the given Chest size of is

Hence, the Predicted and Actual Weight of bear for Chest size of are not very close, because there is an residual or error associated with it is , which means the Predicted value of weight is higher than the actual value of Weight for

__________________________________________

Hypothesis test for the population correlation:

and

At significance level of we need to test this hypothesis.

r: correlation coefficient for sample data, where as population correlation coefficient

So, the correlation coefficient r is calculated as

Critical value: For the critical value is given as

Decision: and the critical value

Since,

So, there is evidence at significance level of that their is a linear correlation between the Chest size of bear and Weight of bears.