Question

A sociologist is interested in the relation between x = number of job changes and y = 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.

x (number of job changes)	6	7	3	6	1	5	9	10	10	3
y (Salary in $1000)	34	33	35	32	32	38	43	37	40	33

In this setting we have Σx = 60, Σy = 357, Σx² = 446, Σy² = 12,869, and Σxy = 2210.

(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.)

x	= _____
y	= _____
b	= _____
ŷ	=______	+ ___ x

(b) Find the sample correlation coefficient r and the coefficient of determination. (Round your answers to three decimal places.)

r = _____
r2 = _____

What percentage of variation in y is explained by the least-squares model? (Round your answer to one decimal place.)

(c) 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.)

t =

What is P value of test statistic

P value =

(d) If someone had x = 3 job changes, what does the least-squares line predict for y, the annual salary? (Round your answer to two decimal places.)
_______thousand dollars

(e) Find S_e. (Round your answer to two decimal places.)
S_e =

(f) Find a 90% confidence interval for the annual salary of an individual with x = 3 job changes. (Round your answers to two decimal places.)

lower limit	___________ thousand dollars
upper limit	___________ thousand dollars

(g) Test the claim that the slope β of the population least-squares line is positive at the 5% level of significance. (Round your test statistic to three decimal places.)

t =

Find the P value of test statistic =

(h) Find a 90% confidence interval for β and interpret its meaning. (Round your answers to three decimal places.)

lower limit ______
upper limit ______

Answer 1

Solution: We can use the excel regression data analysis tool to find the answer to the given questions. The excel output is given below:

Regression Analysis
Regression Statistics
Multiple R	0.658
R Square	0.433
Adjusted R Square	0.362
Standard Error	2.965
Observations	10
ANOVA
	df	SS	MS	F	Significance F
Regression	1	53.767	53.767	6.116	0.039
Residual	8	70.333	8.792
Total	9	124.1
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 90.0%	Upper 90.0%
Intercept	30.9558	2.135	14.497	0.000	26.032	35.880	26.985	34.926
x (number of job changes)	0.7907	0.320	2.473	0.039	0.053	1.528	0.196	1.385
For Average Y
Interval Half Width	2.494
Confidence Interval Lower Limit	30.834
Confidence Interval Upper Limit	35.822

(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) 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.)

Answer: 43.3%

(c) 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.)

What is P value of test statistic

(d) If someone had x = 3 job changes, what does the least-squares line predict for y, the annual salary? (Round your answer to two decimal places.)

33.33 thousand dollars

(e) Find S_e. (Round your answer to two decimal places.)

(f) Find a 90% confidence interval for the annual salary of an individual with x = 3 job changes. (Round your answers to two decimal places.)

lower limit	30.83 thousand dollars
upper limit	35.82 thousand dollars

(g) Test the claim that the slope β of the population least-squares line is positive at the 5% level of significance. (Round your test statistic to three decimal places.)

t = 2.473

Find the P value of test statistic = 0.039

(h) Find a 90% confidence interval for β and interpret its meaning. (Round your answers to three decimal places.)