Question

The following data represents the number of days absent and the final grade for a sample of college students in a general education course at a large state university. 0 1 2 3 4 5 6 7 8 9 No. of absences Final Grade 89.2 86.4 83.5 81.1 78.2 73.9 64.3 71.8 65.5 66.2 a) Which variable is the explanatory variable? b) Draw a scatter plot and describe your scatter plot (Direction, Strength, Form). c) Compute the correlation coefficient d) Does a linear relation exist between the number of absences and the final grade? Justify your answer. e) Write out the least-squares regression line equation. f) Compute and draw the residual (on your scatter plot) for a student who misses 5 class meetings. g) Explain the slope in context. h) Is the y-intercept meaningful in this situation? Explain. i) Compute and interpret the coefficient of determination. j) Construct a residual plot to verify the requirements of the least-squares regression model.

Answer 1

a)

here No of absences is explanatory variable and Final grade is dependent variable.

b)

e following scatter plot is obtained based on the data provided: Scatter Plot of X and Y 90.57 85.57 80.57 > 75.57 70.57 65.57 60.57 -1.35 3.65 8.65 X

direction is top left to bottom right.

two variables are negatively correlated.

c)

X X Y X2 y2 0 89.2 0 0 7956.64 1 86.4 86.4 1 7464.96 2 83.5 167 4 6972.25 3 81.1 243.3 9 6577.21 4 78.2 312.8 16 6115.24 5 73.9 369.5 25 5461.21 6 64.3 385.8 36 4134.49 7 71.8 502.6 49 5155.24 8 65.5 524 64 4290.25 9 66.2 595.8 81 4382.44 Sum = 45 760.1 3187.2 285 58509.93 The correlation coefficient r is computed using the following expression: r= SSXY SSxx SSyy

where SSxy = x;Y Exx-1(2-) (8) SSxx = {x? -1 (3x 2 i=1 2 72 n 1 SSyy=y? Yi n i=1 In this case, based on the data provided, we get that SSxy = 3187.2 - 1 (45 x 760.1) = -233.25 10 1 SSxx = 285 – 1o (45)2 = 82.5 SSyy = 58509.93 - 1 (760.1)= 734.729 10 Therefore, based on this information, the sample correlation coefficient is computed as follows r = SSXY SSxx SSyy -233.25 82.5 x 734.729 -0.947

d)

assume significance level as 0.05

The following needs to be tested: Ho: p=0 HA: p=0 where p corresponds to the population correlation. The sample size is n=10, so then the number of degrees of freedom is df = n - 2 = 10 – 2 = 8 The corresponding critical correlation value re for a significance level of a = 0.05, for a two-tailed test is: rc = 0.632 Observe that in this case, the null hypothesis H : p=0 is rejected if |r| > rc = 0.632. Based on the sample correlation provided, we have that r = 0.947 > rc = 0.632, from which is concluded that the null hypothesis is rejected.

hence we cal conclude that there is a linear relation ship between two variables because $\rho$ is different than zero.

we are allowed to solve four sub parts only.