Question

Run a regression analysis on the following bivariate set of data with y as the response variable.

x	y
6.7	-14.1
29.8	23.2
61.4	85.8
23.6	11.5
1.4	-54.8
29.5	22
54.2	30.9
35.1	7.9
23.3	13.2
29.5	6.7
27.3	29.2
18.6	-15.7

Find the correlation coefficient and report it accurate to three decimal places.
r =

What proportion of the variation in y can be explained by the variation in the values of x? Report answer as a percentage accurate to one decimal place. (If the answer is 0.84471, then it would be 84.5%...you would enter 84.5 without the percent symbol.)
r² = %

Based on the data, calculate the regression line (each value to three decimal places)

y =  x +

Predict what value (on average) for the response variable will be obtained from a value of 40.7 as the explanatory variable. Use a significance level of α=0.05α=0.05 to assess the strength of the linear correlation.

What is the predicted response value? (Report answer accurate to one decimal place.)
y =

Answer 1

x	y
6.7	-14.1	44.89	−94.47	-21.67	−26.25	568.838	469.589	689.063
29.8	23.2	888.04	691.36	1.43	11.05	15.802	2.045	122.103
61.4	85.8	3769.96	5268.12	33.03	73.65	2432.660	1090.981	5424.323
23.6	11.5	556.96	271.4	-4.77	−0.65	3.101	22.753	0.423
1.4	-54.8	1.96	−76.72	-26.97	−66.95	1805.642	727.381	4482.303
29.5	22	870.25	649	1.13	9.85	11.131	1.277	97.023
54.2	30.9	2937.64	1674.78	25.83	18.75	484.313	667.18	351.563
35.1	7.9	1232.01	277.29	6.73	−4.25	−28.603	45.293	18.063
23.3	13.2	542.89	307.56	−5.07	1.05	−5.324	25.705	1.103
29.5	6.7	870.25	197.65	1.13	−5.45	−6.159	1.277	29.703
27.3	29.2	745.29	797.16	−1.07	17.05	−18.244	1.145	290.703
18.6	-15.7	345.96	−292.02	-9.77	−27.85	272.094	95.453	775.623

The formula to calculate regression line is :

where,

is the slope of the regression line and is the y intercept.

so, the equation for regression line is,

=>

So to find the estimated value of at x=40.7, we just need to put the value of x in the regression line.

The formula for correlation coefficient(r) :

And using the value from the above table we can calculate r.

So, r= 0.8899 we then have or its 79.19 %.

i.e.,79.19% of the total variation in y is explained due to its linear relationship with x, and approximately 20.81% remains unexplained.

Now to check the strength of correlation cofficient, we use the t-value, Since nowhere in question mentioned tha whether its a one tailed or two tailed testing, so we just mention both the values, at significance level,.

r=0.889, , df= N-2=12-2=10

so, this is our t-calculated, and now from t-table we can find the t-critical value.

, two-tailed critical value at

, one-tailed critical value at

Since, , both the one-tailed and two-tailed and our is also positivem which indiacate that the relationship is positive.