This PPT deals with the problems and solutions for sampling of large variables and relate, compare the observations with the exception of the population sample ie testing the difference between means of two samples, standard error of the difference between two standard deviations.

1. Test of significance of large samples
(PROBLEMS AND SOLUTIONS)
MRS.K.SUDHA RAMESHWARI
ASSISTANT PROFESSOR,DEPARTMENT OF BIOCHEMISTRY
V.V.VANNIAPERUMAL COLLEGE FOR WOMEN
VIRUDHUNAGAR
TAMILNADU,INDIA

2. Test of significance of large samples
• We deal with the problems of sampling of variables
such as weight, height etc., which may take any value.
• The problems relating the sampling of variables are
studied to compare the observations with the
expectation ,
 to estimate from sample,
 some characteristics of the parent population etc;
• A sample is to be recorded as large only if its size
exceeds 30.
• The test of significance used for dealing with problems
relating to large samples are different from the ones
used for small samples.

3. Problem 1: A feeding experiment conducted on 100 experimental animals
showed an average increase in weight of 5kgs and the standard deviation of
1kg. Test the hypothesis that the expected increase in 4kg. against the
alternative that it is more at the 0.05 level of significance.
Difference 5-4 1
-------------- = ---------- = ------ = 10
S.E 0.1 0.1
Since the calculated value is more than 1.96 at 5% level of significance , the
hypothesis is rejected. Therefore , we may conclude that the average increase in
weight is 4Kg is not correct

4. Problem 2: A sample of 100 sugarcanes is taken from a field. The mean height
is 164 inches and the standard deviation 6inches. Can it be reasonably
regarded that the sugarcane mean height is 166 inches.

5. Problems
3. A sample of 100 tyres is taken from a lot. The mean life
of tyres is found to be 39350 kms with a standard
deviation of 3260. Could the sample come from a
population with mean life of 40,000kms? Establish
99% confidence limits within which the mean life of
tyres is expected to lie.
4. An auto company decided to introduce a new six
cylinder car whose mean petrol consumption is
claimed to be lower that of the existing auto engine. It
was found that the mean petrol consumption for the
50 cars was 10km per liter with a standard deviation of
3.5 km per litre. Test for the company at 5% level of
significance, whether the claim the new car petrol
consumption is 9.5 km per litre on the average is
acceptable.

6. Problems
5. The mean lifetime of 100 fluorescent light bulbs produced by a company is
computed to be 1570 hours with a standard deviation of 120 hours. If µ is the
mean lifetime of all the bulbs produced by the company, test the hypothesis
µ=1600 hours against the alternative hypothesis µ≠1600 hours using a level of
significance of (i)0.05 (ii)0.01
6. An educator claims that the average I.Q of American college students is at
most 110 and that in a study made to test this claim 150 American college
students, selected a random had an average I.Q of 111.2 with a standard
deviation of 7.2. Use a level of significance of 0.01 to test the claim of the
educator.
7. A sample of 100 households in a village was taken and the average income tax was
found to be Rs.628 per month with a standard deviation of Rs.60 per month. Find
the standard error of mean and determine 99% confidence limits within which the
incomes of all the people in this village are expected to lie.
8.A sample of 400 male students is found to have a mean height of 171.38cm. Can it
be reasonably regarded as a sample from a large population with mean height
171.17cm. And standard deviation 3.30cm?
9.The mean breaking strength of the cables supplied by a manufacturer is 1800 with a
standard deviation 100. By a new technique in the manufacturing process it is
claimed that the breaking strength of the cables has increased . In order to test
this claim a sample of 50 cables is tested. It is found that the mean breaking
strength is 1850. Can we support the claim at a 0.01 level of significance.
Solutions follows

7. Problem 3: A sample of 100 tyres is taken from a lot. The mean life of tyres is
found to be 39350 kms with a standard deviation of 3260. Could the sample
come from a population with mean life of 40,000kms? Establish 99%
confidence limits within which the mean life of tyres is expected to lie

8. Problem 4:An auto company decided to introduce a new six cylinder car whose mean petrol
consumption is claimed to be lower that of the existing auto engine. It was found that the mean
petrol consumption for the 50 cars was 10km per liter with a standard deviation of 3.5 km per
litre. Test for the company at 5% level of significance, whether the claim the new car petrol
consumption is 9.5 km per litre on the average is acceptable.

9. Problem 5: The mean lifetime of 100 fluorescent light bulbs produced by a company is
computed to be 1570 hours with a standard deviation of 120 hours. If µ is the mean
lifetime of all the bulbs produced by the company, test the hypothesis µ=1600 hours
against the alternative hypothesis µ≠1600 hours using a level of significance of (i)0.05
(ii)0.01
Solution: let us take hypothesis that there is no
significant difference between the sample mean and
hypothetical population mean i.e., µ=1600
Since the difference is more than1.96SE (at 5% level of
significance), the null hypothesis is rejected. Hence µ≠1600 .
However , at 1% level of significance the null hypothesis is
accepted since the difference is less than 2.58SE.

10. Problem 6: An educator claims that the average I.Q of American college
students is at most 110 and that in a study made to test this claim 150
American college students, selected a random had an average I.Q of 111.2
with a standard deviation of 7.2. Use a level of significance of 0.01 to test the
claim of the educator.
Solution: Let us take the hypothesis that there is no significant
difference in the claim of the educator and the sample results.
Since the difference is less than 2.58 SE (1% level of significance ), the
hypothesis is accepted. Hence, the claim of the educator is valid.

11. Problem 7: A sample of 100 households in a village was taken
and the average income tax was found to be Rs.628 per month
with a standard deviation of Rs.60 per month. Find the standard
error of mean and determine 99% confidence limits within which
the incomes of all the people in this village are expected to lie.
99% confidence limits
mean±2.58SE
=628±2.58(5)
=628±12.9
=615.1 to 640.9
Hence the limits within which the incomes of all the
people in this village are expected to be are Rs.615.1
to Rs.649.9

12. Problem 8:A sample of 400 male students is found to have
a mean height of 171.38cm. Can it be reasonably
regarded as a sample from a large population with mean
height 171.17cm. and standard deviation 3.30cm?
Solution: let us take the hypothesis that there is no
significant difference in the sample mean and the
population mean.
Since the difference is less than 1.96 SE (5% level of
significance), the hypothesis is accepted. Hence, there
is no significant difference in the sample mean and
population mean

13. Problem 9: The mean breaking strength of the cables supplied by a manufacturer is
1800 with a standard deviation 100. By a new technique in the manufacturing process
it is claimed that the breaking strength of the cables has increased . In order to test
this claim a sample of 50 cables is tested. It is found that the mean breaking strength
is 1850. Can we support the claim at a 0.01 level of significance.

14. Testing the difference between means of two samples
• When two independent random samples are
drawn from same population, then S.E of the
difference between sample
• When two random samples are drawn from
different population , then the S.E of the
difference between the mean is given by the
following formula:

15. Problem 1: 150 wheat earheads of C306 variety gave an average 45
grains/earheads with a standard deviation of 3 and 100 earheads of kalyan
variety gave an average of75 grains/ earheads with a standard deviation of 5.
Do you conclude that kalyan variety has more grains /earheads at 0.05%level
of significance

16. Difference 75-45 30
------------- = -------- = --------------- = 53.57
S.E 0.56 0.56
Since the calculated value 53.57 is greater
than 1.96 at 5% level of significance, the
hypothesis is rejected. Therefore, we may
conclude that the Kalyan has more
grains/earheads than C 306 variety

17. Problem 2: The number of accidents per day was studied for 144
days in a town A and 100 days in town B and the following
information was obtained:
Is the difference between mean accidents of the two towns
statistically significant?
Town A Town B
Mean no. of accidents 4.5 5.4
Standard deviation 1.2 1.5

18. Problems
3. The mean population of a random sample of 400 villages in Jaipur district was found to
be 400 with a standard deviation of 12. The mean population of a random sample of
400 villages in Meerut district was found to be 395 with a standard deviation of 15. Is
the difference between the two district was found to be 395 with standard deviation of
15. Is the difference between two districts means statistically significant?
4. Two randomly selected groups of 50 employee each of a very large firm are taught an
assembly operation by two different methods and then tested for performance if the
first group average 140 points with a standard deviation of 10 points while the second
group points with a standard deviation of 8 points, test at 0.05 level whether the
difference between their mean scores is significant.
5. An examination was given to two classes consisting of 40 and 50 students respectively. In
the first class the mean mark was 74 with a standard deviation of 8, while in the second
class the mean mark was 78 with a standard deviation of 7. Is there a significant
difference between the performances of the two classes at a level of significance of
0.05?
6. You are working as a purchase manager for a company. The following information has
been supplied to you by two manufactures of electric bulbs.
Company A Company B
Mean life(in hours) 1300 1248
Standard deviations (in hours) 82 93
Sample size 100 100
Which brand of bulbs are you going to purchase if you desire to take a risk of 5%?

19. PROBLEM 3 :The mean population of a random sample of 400 villages in Jaipur district was found
to be 400 with a standard deviation of 12. The mean population of a random sample of 400
villages in Meerut district was found to be 395 with a standard deviation of 15. Is the difference
between the two district was found to be 395 with standard deviation of 15. Is the difference
between two districts means statistically significant?
Solution : Let us take the hypothesis that the difference between the
mean population of the two villages is not statistically significant.
Since the difference is more than 2.58 (1% level of significance) the
hypothesis is rejected. Hence the difference between the mean
population of the two villages is statistically significant

20. Problem 4 : Two randomly selected groups of 50 employee each of a very large firm are taught an
assembly operation by two different methods and then tested for performance if the first group
average 140 points with a standard deviation of 10 points while the second group points with a
standard deviation of 8 points, test at 0.05 level whether the difference between their mean
scores is significant.

21. Problem 5 : An examination was given to two classes consisting of 40 and 50
students respectively. In the first class the mean mark was 74 with a standard
deviation of 8, while in the second class the mean mark was 78 with a
standard deviation of 7. Is there a significant difference between the
performances of the two classes at a level of significance of 0.05?
Solution: let us take the hypothesis that there is no significant difference in
the mean marks of the two classes.
Difference 78-74
----------------------- = ----------------- =2.49
SE 1.606
Since the difference is more than 1.96 SE.(5% level of significant, the
hypothesis is rejected. Hence, there is a significant difference in the
performance of the two classes at 5% level.

22. Problem 6: You are working as a purchase manager for a company. The following
information has been supplied to you by two manufactures of electric bulbs.
Company A Company B
Mean life(in hours) 1300 1248
Standard deviations (in hours) 82 93
Sample size 100 100
Which brand of bulbs are you going to purchase if you desire to take a risk of 5%?

23. Problem 7: Intelligence test on two groups of boys and girls gave the following
results:
Is there a significant difference in the mean scores obtained by boys and girls
Solution :Let us take hypothesis that there is no significant difference
in the mean scores obtained by boys and girls.
Since the difference is more than 2.58 SE (1% level of significance) ,
the hypothesis is rejected. We conclude that there is a significant
differences in the mean scores obtained by boys and girls.
Mean S.D N
Girls 75 15 150
Boys 70 20 250

24. Problem 8 : A man buys 50 electric bulbs of ‘philips’ and 50 electric bulbs of ‘HMT’. He
finds that ‘Philips” bulbs an average life of 1500 hours with a standard deviation of 60
hours and ‘HMT’ bulbs gave an average life of 1512 hours with a standard deviation of
80 hours. Is there a significant difference in the mean life of the two make of bulbs?
arisen due to fluctuation of sampling. Hence the
difference in the mean life of the two makes is not
significant

25. Problem 9: A simple sample of the height of 6400 Englishmen has a mean of 67.85
inches and a standard deviation of 2.56 inches while a simple sample of heights of
1600 austraians has a mean of 68.55 inches and standard deviation of 2.52 inches . Do
the data indicate that the Austrians are on the average taller than the Englishmen?
• Solution: let us take hypothesis that there is
no significant difference in the mean height
of Englishmen and Austrians

26. Problem 10: In a survey of buying habits, 400 women shoppers are chosen at random
in super market A located in a certain section of Mumbai city. Their average monthly
food expenditure is Rs.250 with a standard deviation of Rs.40. For 400 women
shoppers chosen at random in super market B in another section of the city, the
average monthly food expenditure is Rs.220 with a standard deviation of Rs.55. Test at
1% level of significance whether the average food expenditure of the two populations
of shoppers from which the samples were obtained are equal.

27. Problem11 :Two samples of 100 electric bulbs has a means 1500 and 1550,
standard deviation 50 and 60. Can it be concluded that two brands differ
significantly at 1% level of significance in equality.
• Solution: let us take hypothesis that there is no difference in the
mean life of two makes of bulbs.
Since the difference is more than 2.58 SE(1% level of
significance), the hypothesis is rejected. Hence there is a
signiicant difference in the man life of the two brands of
bulbs.

29. Support the hypothesis. Thus, we can conclude that there is an no significant difference in
standard deviation between paddy and wheat.

30. Problems
1. In a sample of 1000 the mean is 17.5 and the
s.d.2.5. in another sample of 800 the mean is
18 and s.d.2.7. Assuming that the samples are
independent discuss whether the two samples
can have come from a population which have
the same s.d.

31. • Solution: let us take the hypothesis that there is
no significant difference in the standard deviation
of the two samples
• Since the differe3nt is more than 1.96 SE at 5%
level of significance the hypothetical is rejected.
Hence the two samples have not come from a
population which has the same standard
deviation..

32. Reference
• Statistiscal methods for biologists (Biostatistiscs)-
S.Palanisamy & M.Manoharan
• Statistical methods –SP.Gupta