1.

The average IQ of working women in Bangladesh is suspected to be more than 110, on average. A
random sample of 64 of such women yielded an average IQ of 115.5 and a standard deviation of 20. Can
you conclude that the average score of the women in the average score of the women in the population is
really more than 110? Test this at 5% level of significance.

2.
Ten persons were appointed as probationary officers in an office. Their performance was noted by taking
a test and the marks were recorded out of 100. After training for a 6 month period, another test was
conducted. The marks obtained by the officers before training and after training were as follows-

Employees A B C D E F G H I J
Before training 80 76 92 60 70 56 74 56 70 56
After training 84 70 96 80 70 52 84 72 72 50

Were the employees benefited by the training?

3.
You are working as a purchase manager for a company. The following information has been supplied to
you by two manufacturing of electric bulbs:
Company A Company B
Mean life (in hours) 1300 1288
S.D. (in hours) 82 93
Sample size 100 100

Is there any difference between the qualities of the electric bulbs of two brands?
4.
Twelve secretaries, already working for different periods, at an office were asked to take a special three
day intensive course to improve their keyboard skill. At the beginning and again at the end of the course,
they were given a particular two page letter to type and the improved flawless numbers of words typed
are recorded. The recorded data are shown in the following table.
Secretary A B C D E F G H I J K L
Number of years of experience 2 6 3 8 10 5 10 11 12 9 8 10
Improvement (words per minute) 9 11 8 12 14 9 14 13 14 10 9 10

i) Compute the product moment correlation coefficient and test its significant against a positive
alternative.
ii) Fit a regression line of improvement on experience and test the significant of regression coefficient.
5.
Two sources of raw materials are under consideration by a company. Both sources seem to have
similar characteristics but the company is not sure about their respective uniformity. A sample of
15 lots from source A yields a variance of 225 and a sample of 11 lots from source B yield a
variance of 200. Is it likely that the variance of source A is significantly greater than the
variance of source B at 1% level of significance
6.
The managing director of a firm claims that his firm produces 110 items on average daily. A random
sample of 15 days gives the following data set:
 110, 118, 130, 140, 142, 146, 112, 100, 95, 98, 96, 122, 123, 124, 130

It is known that the number of items produced by the firm follows normal distribution with variance
300. Can we conclude at 1% level of significance that the average daily production of items of that
firm is 110 items?
7.
Suppose your company wants to improve sales. Past sales data indicate that the average sale was $100 per
transaction. After training your sales force, recent sales data (taken from a sample of 25 salesmen)
indicates an average sale of $130, with a standard deviation of $15.
a) Set up an appropriate null and alternative hypothesis for this problem.
b) Is it one tailed or two tailed test?

Did the training work? Test your hypothesis at a 5% alpha level.

8.
An employer in a garment factory argues that it is justified to pay more than female workers on the
ground that they are more efficient. Can the employer’s claim be entertained at the 5% level of
significance if the weekly output of a sample of 15 workers produced a mean of 350 men’s wears with the
standard deviation of 18 and the weekly outputs of 10 female workers produced a mean of 345 such
wears with a standard deviation of 21?
9.
The average time required to perform a certain industrial task is known to be 12.5 minutes. Suppose 10
are hired and trained. During a testing their times for completion of the same task are as follows:
9.3, 12.1, 15.7, 10.3, 12.2, 14.8, 15.1, 13.2, 15.9, 14.5
a) Define One tailed and Two tailed test.
b) Test the hypothesis that this mean is not different from the given average at 10% level of significance.
10.
The managing director of a firm claims that his firm produces 110 items on average daily. It is known that
the number of items produced by the firm follows normal distribution with variance 300. A random
sample of 15 days gives the following data set:
110, 118, 130, 140, 142, 146, 112, 100, 95, 98, 96, 122, 123, 124, 130
a) Set up an appropriate null and alternative hypothesis for this problem.
b) Which test you have to apply for testing the null hypothesis?
c) Is it one tailed or two tailed test?
d) Can we conclude at 5% level of significance that the average daily production of items of
that firm is 110 items?
11.
It is wanted to investigate if male and female typists earn comparable wages, the sample data for daily
wages of male and female typist earn comparable wages. The sample data for daily wages of male and
female provide with the following information. Table of sample mean and variance of male and female
typists-
Sample size Male Female
60 60
Mean Tk.158.50 Tk.141.60
SD (Population) Tk.18.20 Tk.20.60
a) Define Statistical hypothesis and Errors in decision making.
Test whether the mean wages of male typist is more than the female typist at 5% level of significance.
12.
A process that produces bottles of shampoo, when operating correctly, produces bottles whose contents
weigh, on average, 20 ounce. A random sample of nine bottles from a single production run yielded the
following weights-
21.4, 19.7, 19.7, 20.6, 20.8, 20.1, 19.7, 20.9, 20.3
Assuming that the population distribution is normal, test the hypothesis that the process is operating
correctly at 5% level of significance.
13.
Suppose a machine produces 12% faulty items. A manufacturer of the same types of machine claims that
there machine is better that this machine. In order to test the manufacturers claim a random sample of 300
items were checked and 30 items were found to be faulty. On the basis of the information, comment on
the claim of manufacturer.
14.
Sample of two different types of bulbs were tested for length of life and the following data were
obtained:
Type I Type II
Sample size 8 7
Sample mean 1234hrs 1136hrs
Sample S.D. 36hrs 40hrs
Is the difference in mean life of two types of bulbs significant?
15.
Suppose a consumer group suspects that the proportion of households that have three cell phones
is not known to be 30%. A cell phone company has reason to believe that the proportion is 30%.
Before they start a big advertising campaign, they want to conduct a hypothesis test. Their marketing
people survey 150 households with the result that 43 of the households have three cell phones. On the
basis of the information, comment on the belief of cell phone company.
16.
You are given the following data about the life of two brands of bulbs:

Mean life Standard deviation Sample size

Brand A 2000 hours 250 hours 12
Brand B 2230 hours 300 hours 15
Do you think there is a significant difference in the quality of the two brands of bulbs?
17.
A professor taught two sections of an introductory marketing course using different styles. In the first
section approach was extremely formal and rigid, while in the second section an independent, more
relaxed and informal attitude was adopted. At the end of the course, a common final examination was
administered. In the first section the 72 students obtained a mean score of 71.03 and the sample standard
deviation was 22.91. In the second section there were 64 students, with mean score 80.92 and standard
deviation 23.11. Assume that these two groups of students can be regarded as independent random
samples from the population of all students who might be exposed to those teaching methods. Test at 5%
level of significance i) whether the performance of these two methods are the same, ii) Whether second
method is better than first one.
18.
The management at Private Bank claims that the mean waiting time for all customers at its branches is
less than that at the Public Bank, which is its main competitor. A business consulting firm took a sample
of 200 customers from the Private Bank and found that they waited an average of 4.5 minutes before
being served. Another sample of 300 customers taken from the Public Bank showed these customers
waited an average of 4.75 minutes being served. Assume that standard deviations for the two populations
are 1.2 and 1.5 minutes, respectively.
a) Define Null and Alternative hypothesis. Set up an appropriate null and alternative hypothesis for this
problem.
b) Is it one tailed test or Two tailed test?
c) Test at the 2.5% significance level whether the claim of the management of the Private Bank is true.
19.
A buyer of computer bought 100 computers each of two famous brands. Upon testing these he found that
brand A had a mean life of 1500 hours with a S.D. of 50 hours where as brand B had a mean life of 1530
hours with S.D. of 60 hours. Can it be concluded at 5% level of significance, that the two brands differs
significant quality of the computers?
20.
The average useful life of a random sample of 33 similar calculator batteries made
on a production line is found to be 99.5 hours continuous use. The sample variance
is 18.49 hours2. Test the null hypothesis that the population mean lifetime is 100
hours against the alternative that it is less. Use the 5% level of significance.

21.
Manager of a factory I claims that the average wage of its workers is higher than that of factory II. A firm
conducted a survey on daily wages of workers of two factories to see if the claim of manager of factory I
is justified. The results are given in the following table:
Factory I Factory II
Sample size 16 11
Sample mean wage Tk. 290 Tk. 250
Sample standard deviation 15 50
a) Write the applications of t-test.
b) Test at 1% level of significance whether the manager claim is true.
22.
Ten oil tins are taken at random from an automatic filling machine. The mean weight of the tins is 15.8kg
and standard deviation is 0.50kg. Does the sample mean differ significantly from the intend weight of
16kg?
23.
 The tensile strength of fabric is required to be at least 50 kg/cm2 . From past experience it is known that
the standard deviation of tensile strength is 2.5 kg/cm2 . From a random sample of 9 specimens, it is
found that the mean tensile strength is 48 kg/cm2 . (i) State the appropriate hypotheses for this
experiment and test the hypotheses # = +What is your conclusion? (ii) What is your decision based on
the p-value? SOLUTION: (i) The hypotheses to be tested are H0: 50 kg/cm2 H1: < 50 kg/cm2 Since the
standard deviation is known, the test statistic is 0 0 _ 48 50 = = 2.5/ 9 2 = = 2.41 0.83 x X Z     Reject
H0, if Z0 < – Z0.05 From standard normal table, the critical value for Z0.05 = 1.65. Hence, we reject the
null hypothesis and conclude that the tensile strength is less than 50 kg/cm2 . (ii) p-value approach:
From standard normal table, the tail area under the curve beyond –2.41 is 0.008. So, the p-value for the
test is 0.008. ,-
-
 )
24.
A manufacturer of florescent tubes claims that his tubes have life time of 1950 burning hours. A
random sample of 100 tubes is taken from a day’s output and tested for burning life. It is found
to have a mean burning lifetime of 1900 hours with a standard deviation of 150 hours. Can the
claim of the manufacturer be accepted at 5% level of significance?
25.
10 plots of same area are chosen at random and the yield of certain paddy variety are recorded in kg., they
are 63,63,66,67,68,69,70,70,71 and 71.in the light of the above data can you suggest that the population
mean production of that paddy variety is 66kg. for same area
26.
A manufacturer of electronic equipment has developed a circuit to feed current to a
particular component in a computer display screen. While the new design is
cheaper to manufacture, it can only be adopted for mass production if it passes the
same average current to the component. In tests involving the two circuits, the
following results are obtained.
Test Number 1 2 3 4 5 6 7 8 9 10 11 12
Circuit 1 - Current (mA) 80.1 82. 84.1 82.6 85.3 81.3 83.2 81.7 82.2 81.
3 4
Circuit 2 - Current (mA) 80.7 81. 84.6 81.7 86.3 84.3 83.7 84.7 82.8 84. 85.2 84.9
3 4

On the assumption that the populations from which the samples are drawn have
equal variances, should the manufacturer replace the old circuit design by the
new one? Use the 5% level of significance.

27.
The results of a state mathematics test for random samples of students taught by two different teachers at
the same department are shown below. Can you conclude that there is a difference in the mean
mathematics test scores for the students of the two teachers? Use α = 0.01. Assume the populations are
normally distributed and the population variances are not equal.
Teacher 1 Teacher 2
x 1 =473 x 2 =459
s1= 39.7 s2= 24.5
n1= 8 n2= 18

28.
500 units from factory A are inspected and 12 are found to be defective, 800 units from factory B are
inspected and 12 are found to be defective. Can it be conclude at 1% level of significance that production
of factory B is better than Factory A?
29.
The mean life time of a sample of 100 light bulbs produced by a company found to be 1580 hours with
standard deviation of 50 hours. Test the hypothesis that the mean lifetime of the tubes produced by the
company is 1600 hours.
30.
A professor wants to know if her introductory statistics class has a good grasp of basic math. Six students
are chosen at random from the class and given a math proficiency test. The professor wants the class to be
able to score above 70 on the test. The six students get scores of 62, 92, 75, 68, 83, and 95. Can the
professor have 90 percent confidence that the mean score for the class on the test would be above 70?
31.
Suppose we are interested in a population of 20 industrial units of the same size, all of which are
experiencing excessive labour turnover problems. The past records show that the mean of the
distribution of annual turnover is 320 employees, with a standard deviation of 75 employees. A
sample of 5 of these industrial units is taken at random which gives a mean of annual turnover as 300
employees. Is the sample mean consistent with the population mean? Test at 5% level.
32.
The mean of a certain production process is known to be 50 with a standard deviation of 2.5. The
production manager may welcome any change is mean value towards higher side but would like to
safeguard against decreasing values of mean. He takes a sample of 12 items that gives a mean value
of 48.5. What inference should the manager take for the production process on the basis of sample
results? Use 5 per cent level of significance for the purpose.

33.
The specimen of copper wires drawn form a large lot have the following breaking strength (in kg.
weight):
578, 572, 570, 568, 572, 578, 570, 572, 596, 544
Test (using Student’s t-statistic)whether the mean breaking strength of the lot may be taken to be
578 kg. weight (Test at 5 per cent level of significance). Verify the inference so drawn by using
Sandler’s A-statistic as well.

34.
Raju Restaurant near the railway station at Falna has been having average sales of 500 tea cups
per day. Because of the development of bus stand nearby, it expects to increase its sales. During
the first 12 days after the start of the bus stand, the daily sales were as under:
550, 570, 490, 615, 505, 580, 570, 460, 600, 580, 530, 526
On the basis of this sample information, can one conclude that Raju Restaurant’s sales have
increased? Use 5 per cent level of significance.

35.
36.

Sample of sales in similar shops in two towns are taken for a new product with the following results:

Is there any evidence of difference in sales in the two towns? Use 5 per cent level of significance
for testing this difference between the means of two samples.

37.
Memory capacity of 9 students was tested before and after training. State at 5 per cent level of
significance whether the training was effective from the following scores:
Use paired t-test as well as A-test for your answer.
38.
The sales data of an item in six shops before and after a special promotional campaign are:
Shops A B C D E F
Before the promotional campaign 53 28 31 48 50 42
After the campaign 58 29 30 55 56 45

Can the campaign be judged to be a success? Test at 5 per cent level of significance. Use paired
t-test as well as A-test.
39.
A tobacco company claims that there is no relationship between smoking and lung ailments. To
investigate the claims a random sample of 300 males in the age group 40 -50 are given medical
test. The observed results are shown below:
Found lung ailment No lung ailment Total
 Smokers 75 105 180
Non - Smokers 25 95 120
Total 100 200 300
On the basis of the information, can it be concluded that Smoking and lung ailments are independent?
40.
A random sample of 200 employees was drawn from a business enterprise to see if there is any
relationship between educational attainment and job performance. The data on this appear below-
Educational Attainment
Job performance Below primary College University Total
Excellent 10 40 10 60
Good 30 30 20 80
Fair 10 30 20 60
Total 50 100 50 200
Do you find any relationship between educational attainment and job performance of these employees?
Test this at 5% level of significance.
41.
Four machines A, B, C, and D are used to manufacture certain machine parts which are classified as 1 st
grade, 2nd grade and 3rd grade. The quality control engineer wants to test whether the quality of the
product and machines are independent. Data collected is as follows:
Machines
Grade A B C D Total
1st 620 750 400 430 2200
2nd 130 200 140 130 600
3rd 50 50 60 40 200
Total 800 1000 600 700 3000
On the basis of the information can you conclude at 5% level of significance that the quality of the
product and machines are independent?
42.
A survey on a number of consumers regarding the quality of a product in rural and urban areas
products the following results-
Opinion Rural Urban
High satisfactory 40 150
Satisfactory 55 130
Not satisfactory 125 90
On the basis of the information, can it be concluded at 5% level of significance that the opinion
is independent of the area?

43.
A company bought a total of 500 color television sets. Three different brands were purchased, and their
repair records were kept for each set’s of operation. The data is given below:

Number of repairs
Brand 0 1 2 or more Total
A 143 70 37 250
 B 90 67 43 200
C 17 13 20 50
Total 250 150 100 500

Is there any relationship between Brand and number of repairs?

44.
Use the following data, test at the α = 0.01 significance level whether a person’s ability in mathematics is
independent of his or her interest in statistics:
Low Ability Average Ability High Ability
Low Interest 63 42 15
Average Interest 58 61 31
High Interest 14 47 29

45.
A fashion house is interested to see whether there is any association between the preferences of colour
and gender of customers. She collected the information regarding the preference of colour from a
randomly selected sample of 200 customers. The result is summarized in following table:
Colour Male Female
Red 10 40
Green 70 30
White 30 20
Comment on the association between the preference of colours and gender of customers at 1% level of
significance
46.
The following table gives the number of good and bad parts produced by each of three shifts in a
factory:
Good Bad
Day 900 130
Evening 700 170
Night 400 200
Is there any association between the shift and the quality of parts produced?
47.
A survey is conducted of 175 young adults whose parents are classified either as wealthy, middle class or
poor to determine their highest level of schooling (graduated from university, graduated from high school
or neither). The results are summarized on the Following table. Based on the data collected is the person’s
level of schooling independent of their parents’ wealth?
Person’s level of schooling
Parents’ wealth University High school None
Wealthy 20 15 10
Middle class 40 25 20
Poor 8 14 23

48.
The following table gives the number of good and bad parts produced by each of three shifts in a
factory:
Good Bad
 Day 900 130
Evening 700 170
Night 400 200

Is there any association between the shift and the quality of parts produced?
49.

Is age independent of the desire to choose different types of sharees? A random sample of 610 women
was surveyed. Each woman was asked their interest in selecting different types of sharees and their age.
The data that resulted from the survey is summarized in the following table:

Types of Sharees Age

18-24 25-34 35-49 50-64 Total

Silk sharees 60 54 46 71 231

Georgette sharees 40 44 53 57 194

Designer sharees 80 72 21 12 185

Total 180 170 120 140 610

Is there evidence to conclude, at the 0.05 level, that the desire to choose sharees depends on age?
50.
An automobile manufacturing firm is bringing out a new model. In order to map out its advertising
campaign, it wants to determine whether the model appeal depends on age group or not. The firm takes a
random sample from persons attending a preview of the new model and obtained the results summarized
below-

Person who Age groups

Under 20 20-40 40-50 50 & over Total
Liked the car 146 78 48 28 300
Disliked the car 54 52 32 62 200
Total 200 130 80 90 500

Test whether the model appeal and age groups are independent.

51.
Suppose there is a city of 1 million residents with four neighborhoods: A, B, C, and D. A random sample
of 650 residents of the city is taken and their occupation is recorded as "blue collar", "white collar", or
"no collar". Test whether each person's neighborhood of residence is independent of the person's
occupational classification. The data are tabulated as:
A B C D Total
White collar 90 60 104 95 349
Blue collar 30 50 51 20 151
No collar 30 40 45 35 150
Total 150 150 200 150 650

52.
A study is undertaken on 500 items of goods of different qualities in order to verify the dependence of
customers’ preference of goods on the quality. The results are summarized as below:
Quality of goods Preferred Not-preferred
High 95 55
Medium 80 100
Low 125 45

Test whether customers’ preference of goods depends on the quality.

53.
A study was conducted to see whether the uses of internet in using computer depend on age. A random
sample of 395 people was surveyed. Each person was asked their interest in using computer and their age.
The data that resulted from the survey is summarized in the following table:

Interest in using Age

Computer

18-24 25-34 35-49 50-64 Total

Yes 60 54 46 41 201

No 40 44 53 57 194

Total 100 98 99 98 395

Is there evidence to conclude, at the 0.05 level, that the desire to use a computer depends on age?

54.
You may be interested in finding out whether or not certain sets of data are independent. Suppose you
collect data on the favorite color of T-shirt for men and women. You may want to find out whether color
and gender are independent or not. The results are given on the table:
Black White Red Blue Totals
Male 48 12 33 57 150
 Female 35 46 42 27 150
Totals 83 58 75 84 300

Perform a appropriate test, at 1% significance level, to determine whether color and gender are
independent or not.

55.
Electronic devices are made on three production lines. Records are kept of faults found on de-
vices made on each line. Faults are classified as “electronics”, “power supply” or “mechanical”.
The data are as follows.

Production Line

1 2 3

Electronic 13 33 15

Power supply 7 4 11

Mechanical 18 10 14

Test the hypothesis that there is no association between production line and type of fault. Use the
5% level of significance.
Answers: Expected frequency
Wear No wear Total
Standard 9 11 20
New compound 9 11 20
Total 18 22 40

Test statistic= 3.636

Critical value: χ2(5%) = 3.841
The result is not significant at the 5% level. There is insufficient evidence to conclude that there
is a difference between the wear rates.
56.
Washing machines are made on three production lines in a factory. A record is kept of faults
reported, during the guarantee period, in machines produced by each of the three lines. The faults
are classified into three types A, B and C. The results are given in the table below.

Fault type

Production line A B C Row Totals

1 40 28 34 102

2 27 39 32 98
 3 45 26 29 100

Column Totals 112 93 95 300

Use a χ2-test at the 5% level of significance to determine whether fault type is related to the
production line on which the machine was produced.

Answer
The hypotheses are:
H0 : fault type is independent of production line,
H1 : fault type is not independent of production line
The expected frequencies are calculated are follows:

102 × 102 × 102 ×

E11 = = 38.08, E12 = = 31.62, E13 = = 32.30
11230 9330 9530
98 0× 980× 980× E21 =
= 36.59,E22 = = 30.38,E23 = = 31.03
112
30 9330 9530
0×
100 0 ×
100 0 ×
100
= 37.30, E32 = = 31.00, E33 = =E31 =
31.70
11230 9330 9530
The test statistic
0 0 0
is

Σ 33Σ (Oijij− E ) 2
W=
i=1 j=1 Eij
(40 − (28 − (34 − (27 − (39 −
= + + + +
38.08)2
38.0 31.62)2
31.6 32.30)2
32.3 36.59)2
36.5 30.38)2
30.3
(32 8
− (45 2− (26 0
− (29 9− 8
+ + + +
31.03)2
31.0 37.30)2
37.3 31.00)2
3 31.70)2
31.
3 0 1 7
= 0.097 + 0.414 + 0.089 + 2.512 + 2.446 + 0.030 + 1.590 + 0.806 + 0.230 =
8.214
and the number of degrees of freedom is (r − 1) × (c − 1) = (3 − 1) × (3 − 1) = 4 so
critical
that thevalue from tables is 0.05,
χ2= 9.49.
4
Since 8.214 < 9.49 we do not have sufficient evidence to reject the null hypothesis and so
we should
conclude that there is no evidence that the distribution of fault types differs between
production lines.