Download 1) - Academic Information System (KFUPM AISYS)

1
KING FAHD UNIVERSITY OF PETROLEUM & MINERALS
DEPARTMENT OF MATHEMATICAL SCIENCES
DHAHRAN, SAUDI ARABIA
STAT 213: STAT213 STATISTICS METHODS FOR ACTUARIES
Final Exam, Term 112
Time: 7.00- 9.30 p.m., Saturday May 19, 2012
Instructors: Prof. Hassen Muttlak
Student Surname:
ID#
You are allowed to use electronic calculators and other reasonable writing accessories that
help write the exam. Try to define events, formulate problem and solve.
Do not keep your mobile with you during the exam, turn off your mobile and leave it aside.
Question No
1
Full Marks
15
2
8
3
9
4
15
5
16
6
16
7
8
8
13
Total
100
Marks Obtained
Note: You may assume  = 0.05 for testing and 95% for confidence interval estimation
if not otherwise stated.
2
Q1. [4+4+3+4=15] A study of the ability of individuals to walk in a straight line reported that
accompanying data on cadence (strides per seconds) for a sample of n =14 randomly selected
healthy men:
81
93
95
93
86
92
93
100
78
106
103
96
85
92
a. Calculate and interpret a 95% confidence interval for a population mean cadence.
b. Test the hypothesis the average cadence for healthy men is 96 cadences using 1%
level of significance.
c. Determine the sample size to estimate the population mean to be within 5 with 99%
confidence level.
d. Test the hypothesis that the population variance is 70, using 5% level of significance.
3
Q2. [4+4=8] The motor vehicles office records indicate that of all vehicles undergoing
emissions testing during the previous year, 60% passed on the first try in the countrywide. A
random sample of 150 vehicles tested in a particular city during the current year yields 80
that passed on the first test.
a. Does this suggest that the true proportion for this city during the current year smaller
from the previous countrywide proportion? Report your p-value.
b. Determine the sample size to estimate proportion of all vehicles to within 4% with
90% confidence level.
4
Q3. [ 5+4=9 ] Tensile strength tests were carried out on two different grades of wire rod
resulting in the accompanying data:
Grade Sample
Size
Sample Mean
Sample St. Dev.
2
(kg / mm )
AISI 1064
m  100
x  110
s1  19
AISI 1078
n  110
y  128
s2  30
a. Does the data provide compelling evidence for concluding that true average strength
for the 1078 grade exceeds that for the 1064 grade by more than 10 kg / mm2 ? Test the
appropriate hypotheses using the P -value approach with 1% level of significance.
And make your final recommendation.
b. Estimate the difference between true average strengths for the two grades in a way
that provides information about precision and reliability.
5
Q4. [4+4+3+4=15] A study includes the accompanying data on compression strength (lb) for
a sample of 12-oz aluminum cans filled with strawberry drink and another sample filled with
cola.
Beverage
Sample Mean
Sample St. Dev.
Sample Size
Strawberry drink
500
21
25
Cola
535
16
17
You may assume equal variances i.e.  1   2 .
Estimate the true difference between the two population means using 95% confidence
interval. Would you agree with the claim that there is no difference the two
populations mean?
Does the data suggest that the extra carbonation of cola results in a higher average
compression strength? Using 1% level of significance. What is your final
recommendation?
What assumptions are necessary for your analysis in the previous parts?
Test our assumption of equal variances, do you agree with this assumption? Explain.
2
a.
b.
c.
d.
2
6
Q5.[5+3+4+4=16] Consider the following data set were x = rainfall volume (m 3 ) and y =
runoff volume (m 3 ) for a particular location were given.
x
5
12
14
17
23
30
40
47
y
4
10
13
15
15
25
27
46
You may use the following results:  x  188,  y  155,  x 5892 ,
2
 xy  4914 and s yx  4.1844
 y 2  4205 ,
a. Fit the least square regression line and use it to predict the average runoff volume
when rainfall volume is 50.
b. What proportion of the observed variation in runoff volume can be attributed to the
simple linear regression relationship between runoff and rainfall?
c. Test the hypothesis that there is a positive linear relation between the rainfall and
runoff volume using 5% level of significance.
d. Construct a 95% confidence interval estimate for the average runoff volume when
rainfall volume is 50 and interpret your finding.
7
Q6. [4x4=16] Answer the following questions
a. Let A1 and A2 be mutually exclusive and exhaustive events, with P ( A1 ) = 0.10 and
P ( A2 ) = 0.25. Let B by any event such that P ( B / A1 ) = 0.50 and P ( B / A2 ) = 0.80, find
P(B).
b. For a sample of size 5, if x1  x  5, x2  x  9, x3  x  7,and x4  x  2, find the
sample standard deviation.
c. Each of 12 refrigerators of a certain type has been returned to a distributor because of
the presence of a high-pitched oscillating noise when the refrigerator is running.
Suppose that 5 of these 12 have defective compressors and the other 7 have less
serious problems. A sample of size 5 refrigerators is randomly selected, what is the
probability that exactly two refrigerators are having less serious problems?
d. The probability density function of a continuous random variable X is
f ( x)  5e 5 x ; x  0 , what is the probability that X is more than 1?
8
Q7. (3+5=8) A loan officer in a bank wished to determine if the marital status of loan
applicants was independent of the approval loans. The following table presents the results of
his survey:
Approved
Rejected
Total
Single
213
189
402
Married
374
231
605
Divorced
358
252
610
Total
945
672
1617
a. State the relevant hypotheses.
b. Conduct the appropriate test and state your final conclusion. Use 5% level of
significance.
9
Q8: Answer the following question by choosing the right answer
1. Which of the following statements are correct?
a. Constructing a histogram for continuous data (measurements) entails subdividing the
measurement axis into a suitable number of class intervals or classes, such that each
observation is contained in exactly one class.
b. The reaction time to a particular stimulus is an example of a discrete variable.
c. Constructing a histogram for discrete data is generally not different from constructing
a histogram for continuous data.
d. The total area of all rectangles in a density histogram is 100.
2. The major difference between the binomial and hypergeometric distributions is that with
the hypergeometric distribution
a.
b.
c.
d.
e.
the probability of success must exceed .5
the trials are independent of each other
the probability of success is not the same from trial to trial
the random variable of interest is continuous
None of the above statements are true
3. Which of the following statements is (are) true about a Poisson probability distribution
with parameter  ?
a.
b.
c.
d.
e.
The mean of the distribution is 
The variance of the distribution is 
The parameter  must be greater than zero
All of the above statements are true
None of the above statements are true
4. The cumulative distribution function F(x) for a continuous random variable X is defined
for every number x by which of the following inequalities?
a.
b.
c.
d.
F ( x )  P( X  x)
F ( x )  1  P( X  x )
F ( x )  P( X  x)
F ( x)  P( X  x )  P ( X  x )
5. Which of the following statements about the percentiles for the standard normal
distribution are correct?
a.
b.
c.
d.
e.
The 90th percentile is approximately –1.28
The 10th percentile is approximately 1.28
The 75th percentile is approximately .67
The15th percentile is approximately 1.04
None of the above answers is correct
10
6. Let X1 , X 2 ,
, X n be a random sample from a distribution with mean  and variance  2 .
Then if n is sufficiently large, the mean X has approximately a normal distribution with
mean  x   and variance  x   / n . This result is well known as
2
a.
b.
c.
d.
e.
2
Approximation theorem
Central limit theorem
Estimation theorem
Mean value theorem
None of the above
7. Which of the following statements are true?
a. A point estimate of a population parameter  is a single number that can be regarded
as a sensible value of  .
b. A point estimate of a population parameter  is obtained by selecting a suitable
statistic and computing its value from the given sample data. The selected statistic is
called the point estimator of  .
c. The sample mean X is a point estimator of the population mean  .
d. The sample variance S 2 is a point estimator of the population variance  2 .
e. All of the above statements are true.
8. Which of the following statements are not true?
a. A correct interpretation of a 100(1   )% confidence interval for the mean  relies on the
long-run frequency interpretation of probability.
b. It is correct to write a statement such as P[ lies in the interval (70,80)]=.95
c. The interval x  1.645   / n is a 90% confidence interval for the mean  .
d. None of the above statements are true.
9. Which of the following statements are true?
a.
b.
c.
d.
Many statisticians recommend pooled t procedures over the two-sample t procedures.
The significance level for the pooled t test is exact.
The significance level for the two-sample t test is only approximate.
All of the above statements are true.
11
10. Which of the following statements are true?
a. The null hypothesis, denoted by H o , is the claim that is initially assumed to be true
(the “prior belief” claim).
b. The alternative hypothesis, denoted by H a , is the assertion that is contradictory to the
null hypothesis H o .
c. The null hypothesis H o will be rejected in favor of the alternative hypothesis only if
sample evidence suggests that H o is false.
d. If sample evidence does not strongly contradict the null hypothesis H o , we will
continue to believe in the truth of H o .
e. All of the above statements are true.
11. If  denotes the parameter of interest, and the simplified null hypothesis has the form
Ho :   o , where o is a specified number called the “null value” of the parameter,
then the alternative hypothesis will be
a. H a :   o (so the implicit null hypothesis is    o )
b. H a :   o (so the implicit null hypothesis is    o )
c. H a :    o
d. The alternative hypothesis will look like any of the above three assertions.
e. The alternative hypothesis must be the assertion specified in (C).
12. Which of the following statements are not generally true?
a. A type I error is usually more serious than a type II error.
b. A type II error is usually more serious than a type I error.
c. A test with significance level  is one for which the type I error probability is
controlled at the specified level.
d. When an experiment and a sample size are fixed, then decreasing the size of the
rejection region to obtain a smaller value of  (probability of type I error) results in a
larger value of

(probability of type II error) for any particular parameter value
consistent with the alternative hypothesis H a .
e. None of the above statements are true.
13. Which of the following statements are not true?
a. In regression analysis, the independent variable is also referred to as the predictor
or explanatory variable.
b. In regression analysis, the dependent variable is also referred to as the response
variable.
c. A first step in a regression analysis involving two variables is to construct a scatter
plot.
d. The simple linear regression model is Y  0  1 x   , where the quantity  is a
random variable, assumed to be normally distributed with E( )  0 and V ( )  1.
e. All of the above statements are true.