B.Sc.(H) Probability and Statistics
Practical problems using Microsoft Excel
Fitting of Binomial, Poisson, Negative Binomial, Normal Distributions.
1. (Example 8.21, p8.19): The following data due to Weldone shows the result of throwing 12
fair dice 4,096 times; a throw of 4, 5, or 6 being called success.
8
9 10 11 12
3
4
5
6
7
Success(X) 0 1 2
Frequency(f) 0 7 60 198 430 731 948 847 536 257 71 11 0
Fit a binomial distribution and find the expected frequencies.
2. (Ex8.22, p 8.21) Seven coins are tossed and number of heads noted. The experiment is
repeated 128 times and the following distribution is obtained:
No. Heads 0 1 2 3 4 5 6 7 Total
Frequencies 7 6 19 35 30 23 7 1 128
Fit a binomial distribution, (i) The coin is unbiased, (ii) The nature of the coin is not known,
(iii) Probability of a head for four coins is 0.5 and for the remaining three coins is 0.45.
3. (Ex. 8.55, p8.46)Fit a Poisson distribution to the following data:
Number of mistakes per page
Number of Pages
0
1
109 65
2 3 4 Total
22 3 1 200
4. (Ex. 8.57, p 8.47)Fit a Poisson distribution to the following data which gives the number of
doddens in a sample of clover seeds:
No. of doddens(X)
Observed frequency(f)
0
1
2
3 4
56 156 132 92 37
5 6 7 8
22 4 0 1
5. (Ex 8.63, p8.63)Given the hypothetical distribution:
0
1
2 3 4 5 Total
Number of cells(X)
213 128 37 18 3 1 400
Frequency(f)
Fit a negative binomial distribution and calculate the expected frequencies.
6. (Ex 8.7, p8.74)A set of 8 symmetrical coins was tossed 256 times and the frequencies of
throws observed were as follows:
0 1 2 3
4 5 6
7 8
Number of heads
Frequency of throws 2 6 24 63 64 50 36 10 1
Fit a binomial distribution and find mean and standard deviation of fitted distribution.
7. (Ex. 8.38, p 8.83)The following distribution relates to the number of accidents to 650 women
working on highly explosive shells during 5-week period. Show that a negative binomial
distribution rather than a geometric distribution, gives a very good fit to the data. How would
you explain this?
Number of accidents(X)
Frequency(f)
0
1
2 3 4 5 Total
450 132 41 22 3 2 650
Dharmendra Kumar, S.G.T.B. Khalsa College, University of Delhi-110007.
8. (p9.14)Obtain the equation of the normal curve that may be fitted to the following data:
60-65
65-70
70-75
Class
3
21
150
Frequency
Also, obtained the expected frequencies.
75-80
335
80-85
326
85-90
135
90-95
26
95-100
4
9. (p9.93)The following table gives baseball throws for a distance by 303 first year high school
girls:
Distance in
Number of
feet
Girls
15 -24
1
25-34
2
35-44
7
45-54
25
55-64
33
65-74
53
75-84
64
85-94
44
95-104
31
105-114
27
115-124
11
125-134
4
135-144
1
a. Fit a normal distribution and find the theoretical frequencies for the classes of the
above frequency distribution.
b. Find the expected number of girls throwing baseballs at a distance exceeding 105 feet
on the basis that the data fit a normal distribution.
10. (ex 15.15, p15.30) When the first proof of 392 pages of a book of 1200 pages were read,
the distribution of printing mistakes were found to be as follows:
1 2 3 4 5 6
Number of mistakes per page(X) 0
275 72 30 7 5 2 1
Number of Pages
Fit a Poisson distribution to the above data and test the goodness of fit.
Calculation of correlation coefficient, Rank Correlation, etc.
11. Calculate the correlation coefficient for the following heights(in inches) of fathers(X) and
their sons(Y):
X
Y
65
67
66
68
67
65
67
68
68
72
69
72
70
69
72
71
12. Ten competitors in a musical test were ranked by the three judges A, B, and C in the
following order:
Dharmendra Kumar, S.G.T.B. Khalsa College, University of Delhi-110007.
Ranks by A 1 6 5 10 3 2 4 9 7 8
Ranks by B 3 5 8 4 7 10 2 1 6 9
Ranks by C 6 4 9 8 1 2 3 10 5 7
Using rank correlation method, discuss which pair of judges has the nearest approach to
common likings in music
Fitting of polynomials and regression curves.
13. (ex 11.6, p11.14) For 10 randomly selected observations the following data were recorded:
Ovretime hrs.(X) 1 1 2 2 3 3 4 5 6 7
Additional unts(Y) 2 7 7 10 8 12 10 14 11 14
Determine the coefficients of regression and regression equation using the nonlinear
form:
14. (ex 11.7, p11.15)Fit an exponential curve of the form
to the following data:
1
2
3
4
5
6
7
8
X
1
1.2 1.8 2.5 3.6 4.7 6.6
9.1
Y
15. (11.9, p11.25)The job rating efficiency of an employee seems to be related to the number of
weeks of employment. For random sample of 10 employees, the following data were
observed:
55 50 20 55 75 80 90 30 75 70
Job Efficiency(X)
Weeks of Employment(Y) 2 4 1 3 5 9 12 2 7 5
Determine the coefficients of regression and regression equation using the non-linear form
16. The expected remaining life of an electronic part is belived to be related to the age of the part.
The ages of 10 of the these parts that were in use on a certain date were recorded in operating
hours. When each part burned out, the dlapsed time was recorded. The results were as follows
40 65 90 5 30 10 80 85 70 25
Age of Part(in hrs.)
Remaining Life(in hrs.) 30 20 10 80 40 65 15 15 20 50
Determine the coefficient of regression and regression curve using the non-linear form
.
Methods of estimation (MLE and method of Moments)
Selecting a simple random sample using random number tables.
17. Consider the following population of size 35.
No.
1
2
3
4
5
6
7
8
9
Region
South
East
West
MidWest
East
West
East
MidWest
West
Customer
Lucky's
QFC
QFC
WholeFoods
FredMyer's
QFC
Lucky's
TraderJoe's
Lucky’s
Used Coupon
Sales
no
$4,360
no
$1,340
no
$6,960
no
$14,375
no
$2,980
no
$7,560
yes
$4,460
no
$12,980
no
$4,950
Dharmendra Kumar, S.G.T.B. Khalsa College, University of Delhi-110007.
a.
b.
c.
d.
e.
f.
g.
10 West
GoodDeal
Yes
$3,625
11 West
CostLess
no
$520
12 South
GoodDeal
yes
$14,625
13 MidWest TraderJoe's no
$4,680
14 South
Lucky's
no
$1,400
15 East
FredMyer's no
$10,250
16 West
WholeFoods yes
$5,430
17 South
Lucky's
no
$2,460
18 South
GoodDeal
no
$3,188
19 West
Albertson's
yes
$2,750
20 South
CostLess
no
$1,180
21 West
Lucky's
no
$1,700
22 South
MegaFoods no
$6,600
23 West
WholeFoods no
$13,800
24 West
MegaFoods yes
$3,905
25 West
CostLess
no
$1,050
26 West
MarketWay yes
$520
27 West
MarketWay no
$1,000
28 MidWest FredMyer's no
$5,400
29 East
Safeway
yes
$605
30 West
Lucky's
no
$2,860
31 MidWest FredMyer's no
$5,920
32 East
Safeway
yes
$1,100
33 West
GoodDeal
no
$240
34 South
MarketWay no
$3,330
35 MidWest WholeFoods no
$2,563
Select a sample of size 10.
Find the point estimate of population mean for sales.
Find point estimate of population standard deviation for sales.
Find point estimate of population proportion for used coupons ‘Yes’.
Also, find population mean, standard deviation, and proportion for Sales, ‘Yes’.
Sampling error for the above criteria.
Find point estimate of population proportion for used coupons, ‘Yes’.
18.
Dharmendra Kumar, S.G.T.B. Khalsa College, University of Delhi-110007.