Chapter 10 Solutions:
Section 10.1:
1.)
x = length of metacarpal
y = height of person
Heigth of Person (cm)
Height verse Metacarpal
Length
190
180
170
160
150
0
10
20
30
40
Length of Metacarpal (cm)
50
60
ŷ = 1.719x + 93.709
x = 44 cm ŷ = 1.719 ( 44 ) + 93.709 = 169.3 cm
x = 55 cm ŷ = 1.719 ( 55 ) + 93.709 = 188.3 cm
Life Expectancy vs Fertility
Rate
Life Expectancy
2.)
3.)
The height for the 44 cm is probably closer to the true height than the one for 55
cm because it is interpolation, while the one for 55 cm is extrapolation.
x = fertility rate of a country
y = life expectancy of a country
100
80
60
40
20
0
0
2
4
Fertility Rate
6
8
ŷ = -4.706x + 84.873
x = 2.7 ŷ = -4.706 ( 2.7 ) + 84.873 = 72.17
x = 8.1 ŷ = -4.706 ( 8.1) + 84.873 = 46.75
The life expectancy for an x of 2.7 is probably closer to the true value than the one
for 8.1 because it is interpolation, while the one for 8.1 is extrapolation.
4.)
5.)
x = height (inches)
y = weight (pounds)
Weight (pounds)
Weight vs Height
300
250
200
150
100
50
0
68
70
72
74
76
Height (inches)
78
80
ŷ = 5.85883x - 230.94
x = 75 inches ŷ = 5.85883 ( 75 ) - 230.94 » 208.5 pounds
x = 68 inches ŷ = 5.85883 ( 68 ) - 230.94 » 167.5 pounds
6.)
7.)
The weight from the height of 75 inches is probably closer to the true value since
it is interpolation. The weight from the height of 68 inches is extrapolation.
x = amount of calories in beef hotdogs
y = amount of sodium (mg) in beef hotdogs
Sodium vs Calories
700
Sodium (mg)
600
500
400
300
200
100
0
0
50
100
Calories
150
200
ŷ = 4.01327x - 228.33
x = 170 ŷ = 4.01327 (170 ) - 228.33 » 453.9 mg
x = 120 ŷ = 4.01327 (120 ) - 228.33 » 253.3 mg
8.)
9.)
The sodium for 170 calories is probably closer to the true value since it is
interpolation. The sodium for 120 calories is extrapolation.
x = number of cigarette sales (per capita)
y = number of deaths per one hundred thousands from bladder cancer
Number of Bladder Cancer
Deaths (per 100 Thousand)
Bladder Cancer Deaths vs
Cigarette Sales
8
6
4
2
0
0
10
20
30
40
Number of Cigarette Sales (per Capita)
50
ŷ = 0.121821x +1.0861
x = 20 ŷ = 0.121821( 20 ) +1.0861 = 3.52 hundred thousand
x = 6 ŷ = 0.121821( 6 ) +1.0861 = 1.82 hundred thousand
10.)
The number of bladder cancer deaths for 20 cigarette sales per capita is probably
closer to the true value since it is interpolation. The number of bladder cancer
deaths for 6 cigarette sales per capita is extrapolation.
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Section 10.2:
1.)
2.)
3.)
4.)
5.)
6.)
7.)
x = length of metacarpal
y = height of person
correlation coefficient: r » 0.8578 , which is a strong, positive correlation
coefficient of determination: r 2 » 0.7357 , 73.8% of the variability in the height of
a person is accounted for by the length of the metacarpal
x = fertility rate of a country
y = life expectancy of a country
correlation coefficient: r » -0.9313 , which is a strong, negative correlation
coefficient of determination: r 2 » 0.8674 , 86.7% of the variability in the life
expectancy of a country is accounted for by the fertility rate of the country
x = height (inches)
y = weight (pounds)
correlation coefficient: r » 0.6605 , moderate, positive correlation
coefficient of determination: r 2 » 0.4362 , 43.6% of the variability in the weight
of a baseball player is accounted for by the height of the player
x = amount of calories in beef hotdogs
y = amount of sodium (mg) in beef hotdogs
correlation coefficient: r » 0.8871, strong, positive correlation
8.)
9.)
10.)
11.)
12.)
13.)
14.)
coefficient of determination: r 2 » 0.7869 , 78.7% of the variability in the amount
of sodium in beef hotdogs is accounted for by the number of calories
x = number of cigarette sales (per capita)
y = number of deaths per one hundred thousands from bladder cancer
correlation coefficient: r » 0.7036 , moderate, positive correlation
coefficient of determination: r 2 » 0.4951, 49.5% of the variability in the number
of deaths per one hundred thousands from bladder cancer is accounted for by the
number of cigarette sales per capita
No, just because there is a correlation between police expenditure and crime rate,
doesn’t mean that one caused the other. Both could be changed because the area
was more affluent, so it could spend more on police and there was less crime
because people were better off.
No, a positive correlation just means that the two go up at the same time, but that
does not mean that higher temperatures cause women to die from breast cancer.
These two seem completely unrelated, and probably are just a coincidence.
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Section 10.3:
1.)
x = length of metacarpal
y = height of person
a.)
1. State the null and alternative hypotheses and the level of significance
Ho : r = 0
H1 : r > 0
a = 0.01
2. State and check the assumptions for the hypothesis test
a. A random sample of length of metacarpal bone and height of a person was
taken. This is not stated, so it is not sure if this is true.
b. The distribution for each height of a person value is normally distributed for
every length of metacarpal bone.
i.
Look at the scatter plot of length of metacarpal bone versus height of a
person. It looks fairly linear.
ii.
There are no points that appear to be outliers.
iii.
The residual plot for height versus length appears to be fairly random.
It appears that there is a normal distribution.
Residuals
10
5
0
30
35
40
45
50
55
-5
3. Find the test statistic and p-value
Test statistic:
t » 4.087
p-value:
2.)
3.)
p » 0.0032
4. Conclusion
Reject H o since the p-value is less than 0.01.
5. Interpretation
There is enough evidence to show that there is a positive correlation between
length of metacarpal bone one and height of a person.
b.) se » 4.559
c.) x = 44 cm.
y = 169.351 cm ± 18.0349 cm
or
151.3161 cm < y < 187.3859 cm
x = fertility rate of a country
y = life expectancy of a country
a.)
1. State the null and alternative hypotheses and the level of significance
Ho : r = 0
H1 : r < 0
a = 0.01
2. State and check the assumptions for the hypothesis test
a. A random sample of fertility rate and life expectancy was taken. This is stated,
so it is safe to assume this.
b. The distribution for each life expectance value is normally distributed for
every fertility rate.
i.
Look at the scatter plot of fertility rate versus life expectancy. It looks
fairly linear.
ii.
There are no points that appear to be outliers.
iii.
The residual plot for fertility rate versus life expectancy appears to be
fairly random.
It appears that there is a normal distribution.
Residuals
8
6
4
2
0
-2 0
-4
-6
-8
2
4
6
8
3. Find the test statistic and p-value
Test statistic:
t » -12
p-value:
4.)
5.)
p < 0.0001
4. Conclusion
Reject H o since the p-value is less than 0.01.
5. Interpretation
There is enough evidence to show that there is a negative correlation between
fertility rate and life expectancy.
b.) se » 3.133
c.) x = 2.7%.
y = 72.168 years ± 9.223 years
or
62.945 years < y < 81.391 years
x = height (inches)
y = weight (pounds)
a.)
1. State the null and alternative hypotheses and the level of significance
Ho : r = 0
H1 : r > 0
a = 0.05
2. State and check the assumptions for the hypothesis test
a. A random sample of height and weight of baseball players was taken.
This is not stated, so it is not sure if this is true.
b. The distribution for each weight of a baseball players value is normally
distributed for every height.
i.
Look at the scatter plot of weight versus height of a baseball player.
It looks fairly linear.
ii.
There are no points that appear to be outliers.
iii.
The residual plot for weight versus height appears to be fairly
random.
It appears that there is a normal distribution.
Residuals
30
20
10
0
-10 68
70
72
74
76
78
80
-20
-30
-40
3. Find the test statistic and p-value
Test statistic:
t » 4.218
p-value:
6.)
7.)
p » 0.00016
4. Conclusion
Reject H o since the p-value is less than 0.05.
5. Interpretation
There is enough evidence to show that there is a positive correlation between
height and weight of baseball players.
b.) se » 15.331
c.) x = 75 inches.
y = 208.47 inches ± 32.45 inches
or
176.02 inches < y < 240.92 inches
x = amount of calories in beef hotdogs
y = amount of sodium (mg) in beef hotdogs
a.)
1. State the null and alternative hypotheses and the level of significance
Ho : r = 0
H1 : r ¹ 0
a = 0.05
2. State and check the assumptions for the hypothesis test
a. A random sample of amount of calories and amount of sodium was taken.
This is not stated, so it is not sure if this is true.
b. The distribution for each amount of sodium value is normally distributed
for every amount of calories.
i.
Look at the scatter plot of amount of calories versus amount of
sodium for a beef hotdog. It looks fairly linear.
ii.
iii.
There are no points that appear to be outliers.
The residual plot for scatter plot of amount of calories versus
amount of sodium for a beef hotdog appears to be fairly random.
It appears that there is a normal distribution.
Residuals
150
100
50
0
-50
0
50
100
150
200
-100
3. Find the test statistic and p-value
Test statistic:
t » 8.153
p-value:
8.)
9.)
p < 0.0001
4. Conclusion
Reject H o since the p-value is less than 0.05.
5. Interpretation
There is enough evidence to show that there is a correlation between amount of
calories and amount of sodium.
b.) se » 48.580
c.) x = 170 calories.
y = 453.92 mg ± 105.46 mg
or
348.46 mg < y < 559.38 mg
x = number of cigarette sales (per capita)
y = number of deaths per one hundred thousands from bladder cancer
a.)
1. State the null and alternative hypotheses and the level of significance
Ho : r = 0
H1 : r > 0
a = 0.01
2. State and check the assumptions for the hypothesis test
a. A random sample of number of deaths per one hundred thousand from
bladder cancer and the number of cigarettes sold per capita was taken.
This is not stated, so it is not sure if this is true.
b. The distribution for each number of deaths per one hundred thousand from
bladder cancer is normally distributed for every number of cigarettes sold
per capita.
i.
Look at the scatter plot of number of deaths per one hundred
thousand from bladder cancer versus number of cigarettes sold per
capita. It looks fairly linear.
ii.
There are no points that appear to be outliers.
iii.
The residual plot for number of deaths per one hundred thousand
from bladder cancer versus number of cigarettes sold per capita
appears to be fairly random.
It appears that there is a normal distribution.
Residuals
2
1.5
1
0.5
0
-0.5
0
10
20
30
40
50
-1
-1.5
3. Find the test statistic and p-value
Test statistic:
t » 6.417
p-value:
10.)
p < 0.0001
4. Conclusion
Reject H o since the p-value is less than 0.01.
5. Interpretation
There is enough evidence to show that there is a positive correlation between
cigarette smoking and deaths of bladder cancer.
b.) se » 0.6857
c.) x = 20 sales.
y = 3.523 hundred thousand ± 1.910 hundred thousand
or
1.613 hundred thousand < y < 5.433 hundred thousand
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