Download An Introduction to Statistics Course (ECOE 1302)

The Islamic University of Gaza
Faculty of Commerce
Department of Economics and Political Sciences
An Introduction to Statistics Course (ECOE 1302)
Spring Semester 2012
Final Examination
Date :
28/5/2012
Name:_____________________________________________
Time: Two hour’s
ID:_____________
Instructor: Mr. Ibrahim Abed
DON'T WRITE ON THIS TABLE
QUESTION
POINTS
#1
#2
#3
#4
#5
#6
#7
TOTAL
Question #1: [15 Points]
For each question in this section, circle the correct answer. Each problem is worth 1 point.
1- If we wish to determine whether there is evidence that the proportion of items of interest is the same in
group 1 as in group 2, the appropriate test to use is
a)
the Z test
b)
the  test
2
c)
Both a) and b)
d)
Neither of a) nor b)
2- When testing for independence in a contingency table with 3 rows and 4 columns, there are ________
degrees of freedom.
a)
5
b)
6
c)
7
d)
12
3-Asummary measure that is computed to describe a characteristic of an entire population is called
a)
a parameter
b)
a census
c)
a statistics
d)
The scientific method
4- According to the chebyshev rule , at least 93.75% of all observations in any data set are
contained within a distance of how many standard deviations around the mean
a)
1
b)
2
c)
3
d)
4
5- The mean age of five people in a room is 30 years . One of the people whose age is 50 years
leaves the room . the mean age of the remaining four people in the room is
a)
40
b)
30
c)
25
1
d)
Not able to be determined
from the information given
6- For some value of Z , the probability that a standard variable is below Z is 0.791.The value of z
is
a)
0.81
b)
-0.31
c)
0.31
d)
1.96
d)
Neither a nor b
7- Sampling distributions describe the distribution of
a)
parameters
b)
Statistics
c)
Both a and b
8- The P- value for a Z test of H 0 :   0.5 vs H a :   0.5 where Z stat   2.36 is
a)
PZ  2.36
b)
PZ  2.36 c)
2 PZ  2.36
2 PZ  2.36
d)
9- The symbol for the power of a statistical test is
a)

b)
1- 
10- A type II error is committed when
a) we reject H 0 that is true.
c) we don’t reject H 0 that is true.
c)

d)
1- 
b) we reject H 0 that is false.
d) we don’t reject H 0 that is false
11- If we are testing for the difference between the means of 2 related populations with samples of
n1  20 and n2  20 then the number of degrees of freedom is equal to
a)
39
b)
38
c)
19
d)
18
12- Assuming a linear relationship between X and Y , if the coefficient of correlation ( r )
equals -0.30 , it means
a)
There is no
correlation
b)
c)
The lope ( b1 ) is
negative
Variable X is larger
than variable Y
d)
the variance of
X is negative
13- In a simple linear regression problems, r and b1
a) May have opposite sign .
c) Must have opposite sign.
b) Must have the same sign
d) Are equal
14- If the P – value is less than  in a two- tailed test,
a) The null hypothesis should not be rejected
c) A one tailed test should be used
b) The null hypothesis should be rejected
d) No conclusion should be used
15- A university dean is interested in determining the proportion of students who receive some sort
of financial aid. Rather than examine the records for all students , the dean randomly selects 200
students and finds that 118 of them are receiving financial aid . If the dean wanted to estimate the
proportion of all students receiving financial aid to within 3% with 99% reliability .how many
students would need to be sampled
a)
1844
b)
1784
c)
1503
2
d)
1435
Name:_____________________________________________
ID:_____________
Question #2: [15 Points]
For each question in this section , indicate whether the sentence is true
or false. Each problem is worth 1 point.
1- (
) A statistic is usually unobservable while a parameter is usually observable
2- (
) As the sample size increases , the standard error of the mean increases
3- (
) Other things being equal , as the confidence level for a confidence interval increases , the width
of the interval increases
4- (
) The t distribution is used to develop a confidence interval estimate of the population mean when
the population standard deviation is unknown
5- (
) When testing for differences between the means of two related populations , we can use either a
one –tailed or two tailed test
6- (
) In testing the difference between two proportions using the normal distribution , we may use
either a one – tailed chi-square test or two-tailed Z test
7- (
) The mean of the sampling distribution of a sample mean is the population mean 
8- (
) The quality ("terrible" , "poor" , "fair" , " acceptable" , "very good" , and " excellent" ) of a day
care center is an example of a numerical variable
9- (
) The coefficient of variation measures variability in a data set relative to the size of the
arithmetic mean
10- (
) A box plot is a graphical representation of a 5 _ number summary
11- (
) The probability that a standard normal random variable Z is below 1.96 is 0.4750
12- (
) If we reject a null hypothesis at   5% then we must reject it at   3%
13- (
) A 95% confidence interval for  will be wider than a 96% confidence interval for 
14- (
) In testing for differences between the means of two independent populations, the null
hypotheses is H 0 : 1  2  0
15- (
) The a mount of water consumed by a person per week is an example of a continuous variable
3
Question #3: [18 Points]
The average grades of 8 students in statistics and the number of absences they had during the semester are
shown below
Number of absences(X) 1
2
2
1
3
4
8
3
Average Grade (Y)
94
78
70
88
68
40
30
60
8
x
i 1
a)
(6 Points)
b)
(8 Points)
c)
(2 Points)
d)
(2Points)
i
 24 ,
8
y
i 1
i
 528 ,
8
x
i 1
2
i
 108 ,
8
y
i 1
2
i
 38288 ,
8
x y
i 1
i
i
 1262
Compute the value of the coefficient of correlation. Interpret
Find the estimated regression equation of average grade on number of absences and interpret
their coefficients?
If a student missed 7 classes , what is the estimated grade for him?
Compute the value of the coefficient of determination. Interpret
4
question #4: [16 Points]
A study published in the American Journal of public Health was conducted to determine whether the use
of a seat belts in motor vehicles depends on ethnic status in San Diego country . A sample of 792
children treated for injuries sustained from motor vehicle accidents was obtained , and each child was
classified according to (1) Ethnic Status ( Hispanic or non Hispanic) and (2) Seat belt usage (worn or not
worn) during the accident . The number of children in each category is given in the table below
Seat belt usage
Ethnic Status
Total
Hispanic
Non_Hispanic
Seat belts worn
31
148
179
Seat belts not worn
283
330
613
Total
314
478
792
Using a chi-square test to see if there is a relationship between seat belts and ethnic status

(3 points)
State Ho:
Ha:

(6 points)

(3 points)

(2 points)

(2 points)
Compute the test Statistic:
At 5% level of significance , find the critical value
State your decision:
What is your final conclusion:
5
Question #5: [18 Points]
To test the effectiveness of a business school preparation course , 8 students took a general business
test before and after the course . The results are given below
Scores before Course 530 690 910 700 450 820 820 630
Scores after course
670 770 1000 710 550 870 770 610
At 5% level of significance , test whether the business school preparation course is effective in
improving exam. Scores ? Assume that the population of paired difference has a normal
distribution.

(2 points)
State Ho:
Ha:

(7 points)

(2 points)

(3 points)

(2 points)

(2 points)
Compute the test Statistic:
Find the critical value:
Construct a 90 % confidence interval for D
State your decision:
What is your final conclusion:
6
NOTE Solve only one question from the following two questions
Question #6: [18 Points]
Apolitical analyst was curious if younger adults were becoming more conservative . He decided to
see if the mean age of registered Republicans was lower than that of registered Democrats . He
selected an SRS of 128 registered Republicans from a list of registered Republicans and
determined the mean age to be X 1 = 39 years , with a standard deviation S 1 = 8 years. He also
selected an independent SRS of 200 registered Democrats from a list of registered Democrats and
determined the mean age to be X 2 = 40 years , with a standard deviation S 2 = 10 years. Let 1 and
 2 represent the mean ages of the populations of all registered Republicans and Democrats ,
respectively . Suppose that the distribution of age in the population of registered Republicans and
of registered Democrats have the same Standard deviation
 (2 points)State Ho:
Ha:

(8 points)

(2 points)

(2 points)

(2 points)

(2 points)
Compute the test Statistic:
Compute the “P-value” or the rejection region:
Compute the critical value by using a level of significance   0.05
State your decision:
What is your final conclusion
7
Question #7: [18 Points]
A corporation randomly selects 150 salespeople and finds that 66% who have never taken a self –
improvement course would like such a course . The firm did a similar study 10 years ago in which
60% of a random sample of 160 salespeople wanted a self – improvement course . The groups are
assumed to be independent random samples . Let 1 and  2 represent the true proportion of
workers who would like to attend a self – improvement course in the recent study and the past study
, respectively. At 0.05 level of significance , test whether a great proportion of workers would
currently like to attend a self – improvement course than in the past.
 (2 points)State Ho:
Ha:

(8 points)

(2 points)

(2 points)

(2 points)

(2 points)
Compute the test Statistic:
Compute the “P-value” or the rejection region:
Compute the critical value by using a level of significance   0.05
State your decision:
What is your final conclusion
8
Formulas:
n
n
xi
x 
, D 
i 1
n
n
x
r
i 1
i
yˆ  b0  b1 x , b1  r

2
2 
2
2
  x i  nx    y i  ny 
 i 1
  i 1

X Y
b1 
i
i 1
i
n X Y
2
i
 nX
n
X
i 1
2
Sy

Sx
b0  y  b1 x
1 n
Di  D 2

n  1 i 1
SD 
2
SD
n
  f 0  f e 2 

  


fe
all cells


df   r 1c 1
n
S D2 
2

n
xz
D  D  t 
n
n
n
1 n
2
x i  x 
n  1 i 1
S
i 1
y i  nx y
n
t 
 Di
x  
z
x  
, df  n  1
s n
t

xt
n
D
SD
n
Di  X1i  X 2i ,
,
p̂  z
D
i 1
 nD 2
2
i
n 1
s
, df  n  1
n
ˆ  p)
ˆ
p(1
n
df  n  1
Z 
x  x
x   ,  x 
x
x1
 x 2   t sp

x
1 x 2
Sp
1 1

n1 n 2
1 x 2
 1  2
p   ,  p 
n
1
1

n1 n 2
df  n1  n 2  2
x
x1  x 2
t 
sp
Sp 
1
1

n1 n 2
 1   
 n1  1 S12   n 2  1 S22
n1  n 2  2
df  n1  n 2  2
z
p
 1  

, p
n
x
n
 p1  p2   z
p1 (1  p1 ) p2 (1  p2 )

n1
n2
 Z 
n 
 , e is the margin of error
 e 
2
z
 p1  p2    1  2 
1 1 
p(1  p)   
 n1 n 2 
n
p1 
p
x1
x
and p 2  2
n1
n2
2
 Z
n    p 1  p 
e
x1  x 2
n1  n 2
P 
9
n1 p1  n 2 p 2
n1  n 2
10
11
12