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)} | 4 | 3 | 5 | 6 | 1 | 5 | 9 | 10 | 10 | 3 |
y _{(Salary in $1000)} | 38 | 34 | 31 | 32 | 32 | 38 | 43 | 37 | 40 | 33 |
In this setting we have ?x = 56, ?y = 358, ?x^{2} = 402, ?y^{2} = 12,960, and ?xy = 2079.
(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 | = | |
? | = | + x |
(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).
Submission Data |
(c) Find the sample correlation coefficient r and the
coefficient of determination. (Round your answers to three decimal
places.)
r = | |
r^{2} = |
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
? is positive at the 5% level of significance. (Round your
test statistic to three decimal places and your P-value to
four decimal places.)
t = | |
P-value = |
Conclusion
Reject the null hypothesis. There is sufficient evidence that ? > 0.Reject the null hypothesis. There is insufficient evidence that ? > 0. Fail to reject the null hypothesis. There is sufficient evidence that ? > 0.Fail to reject the null hypothesis. There is insufficient evidence that ? > 0.
(e) If someone had x = 8 job changes, what does the
least-squares line predict for y, the annual salary?
(Round your answer to two decimal places.)
thousand dollars
(f) Find S_{e}. (Round your answer to two decimal
places.)
S_{e} =
(g) Find a 95% confidence interval for the annual salary of an
individual with x = 8 job changes. (Round your answers to
two decimal places.)
lower limit | thousand dollars |
upper limit | thousand dollars |
(h) Test the claim that the slope ? of the population
least-squares line is positive at the 5% level of significance.
(Round your test statistic to three decimal places and your
P-value to four decimal places.)
t = | |
P-value = |
SUMMARY OUTPUT | |||||
Regression Statistics | |||||
Multiple R | 0.658567891 | ||||
R Square | 0.433711668 | ||||
Adjusted R Square | 0.362925626 | ||||
Standard Error | 3.188240199 | ||||
Observations | 10 | ||||
ANOVA | |||||
df | SS | MS | F | Significance F | |
Regression | 1 | 62.28099548 | 62.28099548 | 6.127078986 | 0.03839229 |
Residual | 8 | 81.31900452 | 10.16487557 | ||
Total | 9 | 143.6 | |||
Coefficients | Standard Error | t Stat | P-value | Lower 95% | |
Intercept | 31.09954751 | 2.149997363 | 14.46492356 | 5.10513E-07 | 26.1416447 |
x | 0.839366516 | 0.339097744 | 2.475293717 | 0.03839229 | 0.057405715 |
a)
xbar =5.6
ybar = 35.8
b = 0.839366516 = 0.8394
y = 31.0995 + 0.8394 *x
b)
c)
r = 0.65856 = 0.6586
r^2 = 0.4337
hence
100* r^2 % = 43.37 % is explained
d)
t = 2.475
p-value= 0.03839/2 = 0.019195 = 0.0192
e)
y = 31.0995 + 0.8394 *x
= 31.0995 + 0.8394 *8 = 37.8147
f)
se = 3.18824 = 3.19
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