Question

A sociologist is interested in the relation between x = number of job changes and γ = annual salary (in thousands of dollars) for people living in the Nashville area. A random sample of 10 people employed in Nashville provided the following information. 9 10 10 3 34 36 32 32 32 3843 37 40 33 5 76 x (number of job changes) Y (Salary in $1000 In this setting we have Σχ = 62, Ху-357, ΣΧ2-462, Xy2-12,875, and ΣΧγ = 2278 (a) Find x, y, b, and the equation of the least-squares line. (Round your answers for x and y to two decimal places. Round your least-squares estimates to four decimal places.) (b) Draw a scatter diagram displaying the data. Graph the least-squares line on your scatter diagram. Be sure to plot the point (x, y} 42 42 40 40 38 38 36 36 34 34 32 32 30 30 10 42 42 40 40 38 36 34 34 32 32F 30 10 30 0 10 8 0 (c) Find the sample correlation coefficient r and the coefficient of determination. (Round your answers to three decimal places.) What percentage of variation in y is explained by the least-squares model? (Round your answer to one decimal place.) (d) Test the claim that the population correlation coefficient ρ is positive at the 5% level of significance. (Round your test statistic to three decimal places.) Find or estimate the P-value of the test statistic. P-value0.250 0.125

Answer 1

(a)

Mean:

$\bar{x}=\frac{\sum x}{n}=\frac{62}{10}=6.2$
$\bar{y}=\frac{\sum y}{n}=\frac{357}{10}=35.7$

Slope of the regression equation is

$b=\frac{S_{xy}}{S_{xx}}=0.8325$

and intercept of the equation will be

$b=\frac{1}{n}(\sum y - b_{1} \sum x)=30.5387$

So the regression equation will be

y'=30.5387+0.8325*x

(b)

42 40 38 36 34 32 30 4 2 10 0 1.2 2

(c)

The coefficient of correlation is :

$r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}=0.643$

The coefficient of determination is:

$r^{2}=0.643^{2}=0.413449\approx 41.3\%$

That is 41.3% variation is explained by y.

(d)

Test statistics:

$t=\frac{r}{\sqrt{\frac{1}{n-2}(1-r^{2})}}=\frac{0.643}{\sqrt{\frac{(1-0.643^{2})}{10-2}}}= 2.375$

Degree of freedom: df=n-2=8

The p-value from t table is:

0.010 < p-value < 0.025

First option is correct.