514
Part VII
Inference When Variables Are Related
Review of Part VII – Inference When Variables Are Related
1. Genetics.
H
0
: The proportions of traits are as specified by the ratio 1:3:3:9.
H
A
: The proportions of traits are not as specified.
Counted data condition:
The data are counts.
Randomization condition:
Assume that these students are representative of all people.
Expected cell frequency condition:
The expected counts (shown in the table) are all greater
than 5.
Under these conditions, the sampling distribution of the test statistic is
χ
2
on 4 – 1 = 3
degrees of freedom.
We will use a chisquare goodnessoffit test.
Trait
Observed
Expected
Residual =
(
Obs – Exp
)
()
Obs
Exp
−
2
Component =
Obs
Exp
Exp
−
2
Attached,
noncurling
10
7.625
2.375
5.6406
0.73975
Attached,
curling
22
22.875
– 0.875
0.7656
0.03347
Free,
noncurling
31
22.875
8.125
66.0156
2.8859
Free,
curling
59
68.625
– 9.625
92.6406
1.35
≈
∑
5.01
2
501
=
.
.
Since the
P
value = 0.1711 is high, we fail to reject
the null hypothesis.
There is no evidence that the proportions of traits are
anything other than 1:3:3:9.
2. Tableware.
a)
Since there are 57 degrees of freedom, there were 59 different products in the analysis.
b)
84.5% of the variation in retail price is explained by the polishing time.
c)
Assuming the conditions have been met, the sampling distribution of the regression slope
can be modeled by a Student’s
t
model with (59 – 2) = 57 degrees of freedom.
We will use a
regression slope
t
interval.
For 95% confidence, use
t
57
2 0025
∗
≈
.
, or estimate from the table
t
50
2 009
∗
≈
.
.
b
t
SE b
n
12
1
2 49244
2 0025
0 1416
2 21 2 78
±×
=±×
≈
−
∗
.(
.
)
.
(
.
,
.
)
d)
We are 95% confident that the average price increases between $2.21 and $2.78 for each
additional minute of polishing time.
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515
3. Hard water.
a)
H
0
: There is no linear relationship between calcium concentration in water and mortality
rates for males.
β
1
0
=
()
H
A
: There is a linear relationship between calcium concentration in water and mortality
rates for males.
1
0
≠
b)
Assuming the conditions have been satisfied, the sampling distribution of the regression
slope can be modeled by a Student’s
t
model with (61 – 2) = 59 degrees of freedom.
We
will use a regression slope
t
test.
The equation of the line of best fit for these data points is:
Mortality
Calcium
ˆ
.(
)
=−
1676
3 23
, where
mortality
is measured in deaths per 100,000, and
calcium concentration is measured in parts per million.
The value of
t
=
– 6.73.
The
P
value of less than 0.0001 means that the
association we see in the data is unlikely to occur by chance.
We reject the
null hypothesis, and conclude that there is strong evidence of a linear
relationship between calcium concentration and mortality.
Towns with
higher calcium concentrations tend to have lower mortality rates.
c)
For 95% confidence, use
t
59
2 001
∗
≈
.
, or estimate from the table
t
50
2 009
∗
≈
.
.
b
t
SE b
n
12
1
323
2001
048
419
227
±×
±
×
≈ −
−
−
∗
.
)
.
,
.
)
d)
We are 95% confident that the average mortality rate decreases by between 2.27 and 4.19
deaths per 100,000 for each additional part per million of calcium in drinking water.
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 Spring '08
 WEINSTEIN
 Normal Distribution, Null hypothesis, representative

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