# 3447-Statistics & Probability_VUsolutions.com

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```							                        ALLAMA IQBAL OPEN UNIVERSITY
(Department of Mathematics and Statistics)

[

WARNING
1.   PLAGIARISM OR HIRING OF GHOST WRITER(S) FOR SOLVING THE
ASSIGNMENT(S) WILL DEBAR THE STUDENT FROM AWARD OF
DEGREE/CERTIFICATE, IF FOUND AT ANY STAGE.
2.   SUBMITTING ASSIGNMENTS BORROWED OR STOLEN FROM
OTHER(S) AS ONE’S OWN WILL BE PENALIZED AS DEFINED IN
“AIOU PLAGIARISM POLICY”.

Course: Statistics & Probability (3447)                                    Semester: Autumn, 2011
Level: B.S.                                                                      Total Marks: 100
Pass Marks: 40
ASSIGNMENT No. 1
(Unit 1-3)
Note: Attempt all questions. All questions carry equal marks.

Q1. A manufacturer of outboard motors receives a shipment of shearpins to be used in the
assembly of its motors. A random sample of ten pins is selected and tested to determine the
amount of pressure required to cause the pin to break. When tested the required pressures to
the nearest pound are 19, 23, 27, 19, 23, 28, 27, 28, 29, and 27.
a)    Calculate the measures of central tendency and interpret your results.           (10)
b)    Calculate the measures of variation and point out the most suitable measure by giving
a valid reasoning.                                                               (10)

Q2. The following data is a sample of the accounts receivable of a small merchandising firm.
37      42       44      47      46      50       48      52      90
54      56       55      56      58      59       60      62      92
60      61       62      63      67      64       64      68
67      65       66      68      69      66       70      72
73      75       74      72      71      76       81      80
79      80       78      82      83      85       86      88

Using a class interval of 5, i.e. 35 – 39,
a)   Make a frequency distribution table and construct its frequency polygon.       (10)
b)   Make a cumulative frequency distribution and construct a cumulative percentage
Ogive.                                                                         (10)

Q3. a)        A bag contains 4 white balls, 6 black balls, 3 red balls, and 8 green balls. If one ball
is drawn from the bag, find the probability that it will be either white or green. (10)
b)   What is the probability of making a 7 in one throw of a pair of dice?               (10)

1
Q4. a)     How many distinct permutations can be made from the letters of the word
“INFINITY”?                                                                    (10)
b)   Three cards are drawn in succession, without replacement, from an ordinary deck of
playing cards. Find the probability that the event A1  A2  A3 occurs, where A1 is
the event that the first card is a red ace is, A2 is the event that the second card is a 10
or a jack, and A3 is the event that the third card is greater than 3 but less than 7. (10)

Q5.   a)   A restaurant chef prepares a tossed salad containing, on average, 5 vegetables. Find
the probability that the salad contains more than 5 vegetables on a given day and on 3
of the next 4 days.                                                               (10)
b)   Given a normal distribution with   30 and   6 , find the two values of x that
contain the middle 75% of the normal-curve area.                                  (10)

ASSIGNMENT No. 2
(Unit 4-7)
Note: Attempt all questions. All questions carry equal marks.

Q1. a)     A consumer report shows that in testing 8 tires of brand A the mean life of the tires
was 20,000 miles with a standard deviation of 2,500 miles. Twelve tires of brand B
were tested under similar conditions with a mean life of 23,000 miles and a standard
deviation of 2,800 miles. If a 0.05 level of significance is used, does the data present
sufficient evidence to indicate a difference in the average life of the two brands of
tires?                                                                              (10)
b)   A population consists of 5 members. The marital status of each member is given
below:
Member               1         2           3           4              5
Marital Status      M          S          M            S             M
Where M and S stands for Married and Single respectively.
Draw all possible samples of size 3 without replacement from this population.
Construct the sampling distribution for the sample proportion of married persons and
verify the properties:                                                              (10)
PQ  N  n 
p = P ,               p =
2
       
n  N 1 

2
Q2. a)   The amount of caffeine (milligrams) in randomly sampled cups of coffee was as
follows:
112.8, 86.4, 45.9, 110.36, 100.3, 93.3, 101.9, 115.7, 92.57, 117.3, 105.6, 81.6.
It is claimed that the average contents of caffeine of a cup is more than 95.8 grams.
Test the hypothesis whether the claim is accurate or not. Also interpret your results.(10)
b)   Let X 1 , X 2 ,..., X n be a random sample from the population with a mean  and a
1 n
variance  2 . Show that the sample variance S 2          ( X i  X ) 2 is the baised
n i 1
estimate of  2 .                                                                   (10)

Q3. a)   Given the following population distribution:
X             1               2                    3                    4
f(x)           1/7             3/7                  2/7                  1/7
Find the sampling distribution of the mean, if a sample of three numbers is taken
without replacement. Compare the mean and variance of the sampling distribution
with the population mean and variance?                                               (10)
b)   A retailer places an order for 400 automobile tyres with a supplier who claims that no
more than 12 percent of his output is ever returned unsatisfactory. In time 35 out of
400 tyres were unsatisfactory. Should the retailer continue to trust his supplier’s word
as to the rate of return? Use 1% level of significance.                              (10)

Q4. a)   A certain brand of cigarettes is advertised by the manufacturer as having a mean
nicotine content of 15 milligrams per cigarette. A sample of 200 cigarettes is tested
by an independent research laboratory and found to have an average of 16.2
milligrams of nicotine content and a standard deviation of 3.6. Using a 0.01 level of
significance, can we conclude based on this sample that the actual mean nicotine
content of this brand of cigarettes is greater than 15 milligrams?                   (10)
b)   A recent report claims that college non-graduates get married at an earlier age than
college graduates. To support the claim random samples of size 100 were selected
from each group, and the mean age at the time of marriage was recorded. The mean
and standard deviation of the college non-graduates were 22.5 years and 1.4 years
respectively, while the mean and standard deviation of the college graduates were 23
years and 1.8 years. Test the claims of the report at the .05 level of significance. (10)

Q5. a)   A quality control inspector at the Cocoa Fizz soft drink company has taken twenty-
five samples with four observations each of the volume of bottles filled. The data and
the computed means are shown in the table. If the standard deviation of the bottling

3
operation is 0.14 ounces, use this information to develop control limits of three
standard deviations for the bottling operation.                              (10)
Observations
Sample                   (bottle volume in ounces)       Average         Range
X1           X2           X3            X4
1      15.85          16.02       15.83        15.93        15.91         0.19
2      16.12          16.00       15.85        16.01        15.99         0.27
3        16           15.91       15.94        15.83        15.92         0.17
4      16.20          15.85       15.74        15.93        15.93         0.46
5      15.74          15.86       16.21        16.10        15.98         0.47
6      15.94          16.01       16.14        16.03        16.03         0.20
7      15.75          16.21       16.01        15.86        15.96         0.46
8      15.82          15.94       16.02        15.94        15.93         0.20
9      16.04          15.98       15.83        15.98        15.96         0.21
10      15.64          15.86       15.94        15.89        15.83         0.30
11      16.11          16.00       16.01        15.82        15.99         0.29
12      15.72          15.85       16.12        16.15        15.96         0.43
13      15.85          15.76       15.74        15.98        15.83         0.24
14      15.73          15.84       15.96        16.10        15.91         0.37
15      16.20          16.01       16.10        15.89        16.05         0.31
16      16.12          16.08       15.83        15.94        15.99         0.29
17      16.01          15.93       15.81        15.68        15.86         0.33
18      15.78          16.04       16.11        16.12        16.01         0.34
19      15.84          15.92       16.05        16.12        15.98         0.28
20      15.92          16.09       16.12        15.93        16.02         0.20
21      16.11          16.02       16.00        15.88        16.00         0.23
22      15.98          15.82       15.89        15.89        15.90         0.16
23      16.05          15.73       15.73        15.93        15.86         0.32
24      16.01          16.01       15.89        15.86        15.94         0.15
25      16.08          15.78       15.92        15.98        15.94         0.30
b)   A quality control inspector at Cocoa Fizz is using the data from part (a) to develop
control limits. If the average range R for the twenty-five samples is .29 ounces and
the average mean X of the observations is 15.95 ounces, develop three-sigma control
limits for the bottling operation.                                              (10)

4

```
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