Gm533 week 6 lecture april 2012

Week 6 Checkpoint Help GM533
B Heard
(Don’t copy or post without my
permission, students may
download a copy for personal use)

• Always be on the lookout for typos…..
• Remember I usually do these charts “on the fly”
▫ Do not rely solely on me or these charts to do well
in this class!
▫ The live lectures are only part of what you have
available to you
▫ Good Luck!

13.8 THE REAL ESTATE SALES PRICE
CASE
• A real estate agency collects data concerning y =
the sales price of a house (in thousands of
dollars), and x = the home size (in hundreds of
square feet). The data are given in the table
below. The MINITAB output from fitting a least
squares regression line to the data is on the next
page.

a) By using the formulas
illustrated in Example 13.2
(see page 497) and the data
provided, verify that (within
rounding) b0 = 48.02 and b1
= 5.700, as shown on the
MINITAB output.

I put the data in Excel and
did the math there. The
formulas are provided in the
text.


Size (x) Price (y)

23 180

11 98.1

20 173.1

17 136.5

15 141

21 165.9

24 193.5

13 127.8

19 163.5

25 172.5

Size (x) Price (y) xy x^2
23 180 4140 529
11 98.1 1079.1 121
20 173.1 3462 400
17 136.5 2320.5 289
15 141 2115 225
21 165.9 3483.9 441
24 193.5 4644 576
13 127.8 1661.4 169
19 163.5 3106.5 361
25 172.5 4312.5 625

Sum of x's Sum of y's sum of xy's sum of x^2's (sum of x's)^2
188 1551.9 30324.9 3736 35344
n
10

SSxy SSxx
1149.18 201.6

b1 y bar x bar
5.700298 155.19 18.8

b0
48.0244

y hat = b0 + b1*x
y hat = 48.0244 plus 5.70029762 x

b) Interpret the meanings of b0 and b1. Does the
interpretation of b0 make practical sense?

The b1 is 5.70 which basically is saying for every
100 square feet the average sales price increases
that much
b0 is the y intercept when x is zero. In other
words, it says that a house with zero square feet
would cost about 48 thousand dollars. No, this
doesn’t make sense (I will talk about this).

c) Write the least squares prediction equation.

y hat = 48.02 + 5.7x

y hat = b0 + b1*x
y hat = 48.0244 plus 5.70029762 x

d) Use the least squares line to obtain a point
estimate of the mean sales price of all houses
having 2,000 square feet and a point prediction
of the sales price of an individual house having
2,000 square feet.
Plug and chug (insert 20 for x remember the size
was in 100’s of square feet)
y hat = 48.02 + 5.7(20) = 162.02 (in thousands)
So the predicted price would be $162,020

13.21 THE STARTING SALARY CASE
The MINITAB output of a simple linear regression
analysis of the data set for this case (see Exercise
13.4 on page 501) is given in Figure 13.11. Recall
that a labeled MINITAB regression output is on
page 509.

bo
(Part a)
b1


SSE
(Part b)
s^2

s

sb1
(Part c)
t

t = b1/sb1 (show)

df
(Part d)


(Part d continued)
t.025 = 2.57 compared
to 14.44 ?

Reject because it’s way
out there in the
rejection region 
Reject H0, there is
strong evidence of
something going on
Table from with x and y
http://www.statsoft.com/textbook/distribution-tables/#t
(I just search on Internet, you have one in text)


(Part e)
t.005 = 4.03 compared
to 14.44 ?

Reject because it’s still
way out there in the
rejection region 
Reject H0, there is
strong evidence of
something going on
Table from with x and y (Very
http://www.statsoft.com/textbook/distribution-tables/#t
strong relationship)
(I just search on Internet, you have one in text)

f) p value was .000 agrees with previous two to
reject at all alphas. Very very strong evidence of
an x and y relationship
g) 95% CI Just use what you have found above
The interval is b1 +/- t.025 sb1
h) 99% CI Just use what you have found above
The interval is b1 +/- t.005 sb1

bo
sbo
(Part i)
t

t = b0/sb0 (show)

j) p value was .000, reject at all alphas. Very very
strong evidence of an x and y relationship
k) Use the formulas and the data! (should give you
the same answer you got in part c for sb1 and in part
i for sbo.

13.30 THE FUEL CONSUMPTION CASE
The following partial MINITAB regression output
for the fuel consumption data relates to
predicting the city’s fuel consumption (in MMcf
of natural gas) in a week that has an average
hourly temperature of 40°F.


Part a


Part b

c) Remembering that s = .6542; SSxx = 1,404.355;
n = 8, hand calculate the distance value when x0
= 40. Remembering that the distance value
equals , use s and from the computer output to
calculate (within rounding) the distance value
using this formula. Note that, because MINITAB
rounds sy, the first hand calculation is the more
accurate calculation of the distance value.

Distance Value (dv) = 1/8 + (40-43.98)2 /
1404.355 = 0.1363

And
Distance Value (dv) = (0.241 / 0.6542)2 = 0.1357

d) Remembering that for the fuel consumption
data b0 = 15.84 and b1 = -.1279, calculate (within
rounding) the confidence interval of part a and
the prediction interval of part b.

CI: 15.84 - 0.1279 (40) ± 2.447*0.6542*√(0.1363)
= [10.13299, 11.31501]
For PI, just substitute 1.1363 for 0.1363

e) Remember you are predicting for one day, so
use prediction interval.
Since 9.01 < 9.595 and 12.43 > 11.847 the
city cannot be ____ % confident it won’t pay a
fine. (Fill in the blank)

THE FRESH DETERGENT CASE
In Exercises 13.50 through 13.55, we give MINITAB and Excel outputs of simple linear
regression analyses of the data sets related to six previously discussed case studies.
Using the appropriate computer output,
a Use the explained variation and the unexplained variation as given on the computer
output to calculate (within rounding) the F (model) statistic.
b Utilize the F (model) statistic and the appropriate critical value to test H0 : β1 = 0
versus Hα : α1 ≠ 0 by setting a equal to .05. What do you conclude about the
regression relationship between y and x?
c Utilize the F (model) statistic and the appropriate critical value to test H0 : β1 = 0
versus Hα : β1 ≠ 0 by setting a equal to .01. What do you conclude about the
regression relationship between y and x?
d Find the p -value related to F (model) on the computer output and report its value.
Using the p -value, test the significance of the regression model at the .10, .05, .01,
and .001 levels of significance. What do you conclude?
e Show that the F (model) statistic is (within rounding) the square of the t statistic for
testing H0 : β1 = 0 versus Hα : b1 ≠ 0. Also, show that the F.05 critical value is the
square of the t025 critical value.
Note that in the lower right hand corner of each output we give (in parentheses) the
number of observations, n, used to perform the regression analysis and the t statistic
for testing H0 : β1 = 0 versus Hα : β1 ≠ 0.

Used a table at http://www.statsoft.com/textbook/distribution-tables/#f (I was lazy)

F.05 =4.196, reject H0 (df1 (top) = 1, df2 (left) =
28). Looks like there is STRONG evidence of a
significant relationship between x and y.

c) F.01 =7.636, reject H0 (df1
(top) = 1, df2 (left) = 28). Looks
like there is STRONG (Very
because of .01) evidence of a
significant relationship between x
and y.

Used a table at http://www.statsoft.com/textbook/distribution-tables/#f (I

d) p-value is ______ (smaller than all levels of
significance), reject H0 . Again, pick your “ly”
ending word that means there is definitely
strong evidence of a significant relationship
between x and y.


Part e
Square this number, you should see that it gives you a result
within rounding error.
Then get your table value for t.025 and verify
(t.025)2 = 4.19 = F.05

I will post these in the Stat Cave at
www.facebook.com/statcave

Gm533 week 6 lecture april 2012

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