Question

The table below gives the number of hours seven randomly selected students spent studying and their corresponding midterm exam grades. Using this data, consider the equation of the regression line, yˆ=b0+b1x, for predicting the midterm exam grade that a student will earn based on the number of hours spent studying. Keep in mind, the correlation coefficient may or may not be statistically significant for the data given. Remember, in practice, it would not be appropriate to use the regression line to make a prediction if the correlation coefficient is not statistically significant. Hours Studying 1.5 2 2.5 3.5 4 4.5 5 Midterm Grades 60 68 71 75 81 86 96 Table Step 5 of 6: Substitute the values you found in steps 1 and 2 into the equation for the regression line to find the estimated linear model. According to this model, if the value of the independent variable is increased by one unit, then find the change in the dependent variable yˆ.

Answer 1

Here by the problem,

The table below gives the number of hours n=7 randomly selected students spent studying and their corresponding midterm exam grades. Using this data, we consider the equation of the regression line,

for predicting the midterm exam grade that a student will earn (as y) based on the number of hours spent studying (as x). Now in order to be sure there lies a lineear relationship between x and y, we will test whetherthe value of correlation coefficient is significant enough( that is not equal to 0). SO we calculate the correlation coefficient as below,

	Hours studying (X)	Midterm grades (Y)	c=X-mean(X)	d=Y-mean(Y)	c^2	d^2	cd
	1.5	60	-1.79	-14.71	3.2041	216.3841	26.3309
	2	68	-1.29	-6.71	1.6641	45.0241	8.6559
	2.5	71	-0.79	-3.71	0.6241	13.7641	2.9309
	3.5	75	0.21	0.29	0.0441	0.0841	0.0609
	4	81	0.71	6.29	0.5041	39.5641	4.4659
	4.5	86	1.21	11.29	1.4641	127.4641	13.6609
	5	96	1.71	21.29	2.9241	453.2641	36.4059
Total	23	537			10.4287	895.5487	92.5113
Average	3.29	74.71

So the correlation coefficient be,

Note that in order to test whether the correlation coefficent is significant or not our test statistic be,

where under Ho,

Now putting the values we get,

We can note that

which is very very low. Hence we can say here that the null hypothesis (that the population correlation coefficient is not significant or equal to 0)is rejected implying the correlation coefficient (as well as regression coefficient is a non zero value)

So the intercept and the regression coefficient be,

where

SO putting the values we get,

And

So the equation for the regression line to find the estimated linear model be

So according to this model, if the value of the independent variable is increased by one unit, then the change in the dependent variable be equal to the value of regression coefficient 8.871

Hence the answer...............

Thank you...........