10 undergraduate Statistics Students are enrolled in class. If its known that the
probability of passing his course (with a B or above) is 70%; calculate the mean and the
standard deviation for this binominal distribution.
2.1  1.449
2. A national trend predicts that woman will account for half of all business travellers in
2007. To attract these women hotels are offering amenities that woman particularly
like. A recent survey of American Hotels found that 70% offers hairdryers in
bathrooms. Consider a random and independant sample of 20 hotels. Find the
probability that more than 7 but less than 13 of the hotels in the sample offered
hairdryers in the bathrooms
hotels in the sample offered hairdryers in the bathrooms. Then X follows a binomial
distribution with n=20 and p=70%. Hence, the probability that more than 7 but less than
13 of the hotels in the sample offered hairdryers in the bathrooms is
P(7  X  13)  P( X  8)  P( X  9)  P( X  10)  P( X  11)  P( X  12)
 20 
 20 
 20 
 20 
 20 
  0.7 8 0.312   0.7 9 0.311   0.710 0.310   0.7110.39   0.712 0.38
8 
10 
12 
9 
11 
=0.22645
3. The Marine Corps Height Requirements for Men:
The US Marine Corps requires that men have heights between 64" and 78" (The
National Health Survey shows that heights of men are normally distributed with a
mean of 69.0" and a standard deviation of 2.8"). Find the following:
4. Find the percentage of men meeting those height requirements. Are too many men
denied the opportunity to join the Marines because they are too short or too tall?
with a mean of 69.0" and a standard deviation of 2.8". Now we can compute
P(64  X  78)  P( 642.869 

X 69
7869
2.8
2.8
X 69
2.8
 P(1.786 
=0.9993-0.037
=0.9623
)
 3.214)
So, 96.23% of men meet those height requirements.
As this percentage is very big, not too many men are denied the opportunity to join the
Marines because they are too short or too tall.
5. If you are appointed to be the Secretary of Defense and you want to change the
requirement so that only the shortest 2% and the tallest 2% of all men are rejected,
what are the new minimum and maximum height requirements?
with a mean of 69.0" and a standard deviation of 2.8". Now we want to find A and B so
that
P( X  A)  0.02 and P( X  B)  0.02
As P( X  A)  P( X2.869  A2.869 )  0.02 and P( X  B)  P( X2.869 
up the z-table, we have
A69
and B2.869  2.054
2.8  2.054
B 69
2.8
)  0.02 , by looking
Hence, A=63.25" and B=74.74"
So, the new minimum and maximum height requirements are 63.25" and 74.74",
respectively.
6. If 64 men are randomly selected, find the probability that their mean height is greater
than 68.0"
with a mean of 69.0" and a standard deviation of 0.35". Now we can compute
P(Y  68)  P( Y0.3569 
6869
0.35
)
 P(
 2.857)
=0.9979
Y 69
0.35
So, if 64 men are randomly selected, the probability that their mean height is greater than
68.0" is 0.9979.
Correlation/Regression
7. Part III
Scenario
Is there a correlation between ones hair length and ones level of intelligences (I.Q.)?
Samples of 5 individuals were recorded measuring hair length to I.Q.
Hair Length--Inches (X)
36
5
23
18
0 (Bald)
24
I.Q. (Y)
176
120
100
88
145
134
After determining the correlation, please do the following
a)
Draw the scatter diagram
b)
Calculate the Regression Slope and Y-Intercept
c)
Draw the Regression Line on the Scatter Diagram
d)
Predict the possible IQ with a person who has Approx. 15th inches of
hair.
Solution.
a) Use Excel, we get the scatter diagram as follows.
Y
200
180
160
140
120
100
80
60
40
20
0
Y
0
5
10
15
20
25
b) Use Excel, we get the following table (see sheet 4).
30
35
40
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.263711
R Square
0.069543
Adjusted R
Square
-0.16307
Standard Error
34.35132
Observations
6
ANOVA
df
1
4
5
SS
352.7813
4720.052
5072.833
Coefficients
115.9639
0.634119
Standard
Error
24.82858
1.159741
Regression
Residual
Total
Intercept
X Variable 1
MS
352.7813
1180.013
F
0.298964
Significance
F
0.613604
t Stat
4.670582
0.546776
P-value
0.009514
0.613604
Lower 95%
47.02873
-2.58584
Upper
95%
184.8991
3.854076
Lower
95.0%
47.02873
-2.58584
From the above table, we know that the Regression Slope is 0.634119 and Y-Intercept
is 115.9639.
c)
Draw the Regression Line on the Scatter Diagram
Use Excel, we can draw the Regression Line on the Scatter Diagram and get the
following.
Chart Title
200
180
160
140
120
100
80
60
40
20
0
Y
Linear (Y)
0
10
20
30
40
Upper
95.0%
184.8991
3.854076
d) Predict the possible IQ with a person who has Approx. 15th inches of hair.
By the table in part b), we know that the equation of the Regression Line is
Yˆ  115.9639  0.634119 X
Now, we let X=15, by the equation, we have
Yˆ  115.9639  0.634119 *15  125.48
So, the prediction of the possible IQ with a person who has Approx. 15th inches of hair is
125.48.