This ppt have problems and solutions for the test of proportion of success and test difference of proportion of success

1. Test for the proportion of success &
difference between proportion of
success
(Problems and Solutions)
Mrs.K.Sudha Rameshwari
Assistant Professor,
Department of Biochemistry,
V.V.Vanniaperumal college for women, Virudhunagar,
Tamilnadu, India

2. Test for the proportion of success
If the success in each sample is provided in
proportions , we follow this method.
The standard error of proportion of success is
expressed in the following formula:

3. Problem 1: A wholesaler in bananas claims that only
3% supplied by him are defective. A random sample
of 700 banana contained 45 defective bananas. Test
the claim of the wholesaler
Solution:
The cultivator claims only 3% of bananas are
defective, hence , 95% confidence limits are
given by
Defective bananas =p=0.03
Therefore , non defective bananas = 1-p
=1-0.03
=0.97
n=700

4. The number of defective banana is expected to lie
between 13 and 29. Since the actual number of
defective is 45, the cultivator’s claim that only 3%
defective cannot be accepted.

5. Problem 2. A wholesaler in apples claims that only 4% of the apples
supplied by him are defective. A random sample of 600 apples contained
36 defective apples. Test the claim of the wholesaler.

6. Problem 3: 500 apples are taken at random from a large basket
and 50 are found to be bad. Estimate the proportion of bad
apples in the basket and assign limits within which the
percentage most probably lies.

7. Problem 4: Out of 20000 customer’s ledger accounts, a sample of 600 was
taken to test the accuracy of posting and balancing and 45 mistakes were
found. Assign limits which the number of mistakes can be expected at 95%
level of confidence.
Hence at 5% level of significance it is expected that the
number of mistakes would vary between 5.3 to 9.7Percent

8. Problem 5 : A factory is producing 50000 pairs of shoes daily. From a sample
of 500 pairs, 2% were found to be of sub-standard quality. Estimate the
number of pairs that can be reasonably expected to be spoiled in the daily
production and assign limits at 95% level of confidence.

9. Test for difference of proportions

10. Problem1: Before an increase in dosage of antibiotics of fish reared in a
research station, 1000 out of 1200 were in good condition. After an increase
in dosage if antibiotics , 1300 fish were in good condition in a sample of
1400 fish. Do you think that there has been any significant increase in good
condition of the fish after the increase in dosage?
Since the difference is more than 2.58SE at 1% level of significance , we
could not accept the hypothesis and thus , we conclude that the
increase in dosages of antibiotics resulted in increase in health
condition

11. Problem 2:In a village ‘A’ out of a random of 2000 persons, 200 were found to be
vegetarians while in another village ‘B’ out of 3000 persons 360 were found to be
vegetarians. Do you find a significant difference in the food habits of the two villages?
Solution: Let us take hypothesis that there is no significant difference in the
food habits of the two villagers.
Applying the test of the difference of proportions:
x1+x2 200+360
P=---------- = -----------------
n1 +n2 2000+3000
560
= ------- =0.112
5000
Therefore, q=1-p
=1-0.112
=0.888
P1=200/2000
=0.1
P2=360/3000
=0.12

12. Difference 0.12-0.10 0.02
-------------- = ------------------- = ---------- = 2.19
S.E 0.0091 0.0091
Since the calculated value 2.19 is higher than table value 1.96 at 5%
significance level, we cannot accept the hypothesis that there is
no significant difference in the food habits between the two
villagers
On the other hand, if we take 1% significance level, where the table
value is 2.58 which is greater than calculated value 2.19 we can
accept the hypothesis. Thus , at 1% significance level, we can
accept the hypothesis. But at 5% level of significance, we cannot
accept the hypothesis.

13. Problem 3:Before an increase in excise duty on tea 400 people out of a
sample of 500 persons were found to be tea drinkers in a sample of 600
people. Do you think that there has been a significant decrease in the
consumption of tea after the increase in the excise duty?
Since the difference is more than 2.98
S.E(1% level of significance) , reject the
hypothesis, we concluded that there is a
significant decrease in the consumption
of tea after increase in the excise duty
Since the difference is more than 2.98 S.E(1% level of
significance) , reject the hypothesis, we concluded that
there is a significant decrease in the consumption of tea
after increase in the excise duty

14. Problem 4: A machine produced 16 defective articles in a batch
of 500. After the machine is overhauled it produced 3 defective
in a batch of 100. Has the machine improved?

15. Difference p1-p2 0.032-0.03
------------- = -------- = --------------- = 0.105
S.E 0.019 0.019
Since the difference is less than 1.96SE (at 5%
level ), hypothesis is accepted i.e we conclude
that machine has not improved significantly

16. Problem 5: A machine puts out 10 defective units in a sample
of 200units. After the machine is overhauled it puts out 4
defective units in a sample of 100 units. Has the machine been
improved?

17. Problem 6: A machine produced 20 defective articles in
a batch of 400. After overhauling it produced 10
defectives in a batch of 300. Has the machine
improved?

18. 7. In a random sample of 1000 persons from
town A, 400 are found to be consumers of
wheat. In a sample of 800 from town B, 400
are found to be consumers of wheat. Do these
data reveal a significant difference between
town A and town B, so far as the proportion of
wheat consumers is concerned?
8. In a simple random sample of 600 men taken
from a big city 400 are found to be smokers. In
another simple random sample of 900 men
taken from another city 450 are smokers. Do
the data indicate that there is a significant
difference in the habit of smoking in the two
cities?

19. 9. In a sample of 300unit of a manufactured product,
65units were found to be defective and in another
sample of 200units, there were 35 defectives. Is
there significant difference in the proportion of
defectives in the samples at the 5%level of
significance?
10. In a sample of 1000 men from a certain city, 450
men are found to be smokers. In a sample of 800
from another city 400 are found to be smokers. Do
the data indicate that the two cities are significantly
different with respect to prevalence of smoking habit
among men?
Solution follows

20. Problem 7: In a random sample of 1000 persons from town A, 400 are found
to be consumers of wheat. In a sample of 800 from town B, 400 are found to
be consumers of wheat. Do these data reveal a significant difference between
town A and town B, so far as the proportion of wheat consumers is
concerned?

21. Problem 8: In a simple random sample of 600 men taken from a
big city 400 are found to be smokers. In another simple random
sample of 900 men taken from another city 450 are smokers. Do
the data indicate that there is a significant difference in the habit
of smoking in the two cities?
450

23. Problem 9: In a sample of 300unit of a manufactured product, 65units were
found to be defective and in another sample of 200units, there were 35
defectives. Is there significant difference in the proportion of defectives in the
samples at the 5%level of significance?
Solution: let us take the hypothesis that there is no significant
difference in the proportion of defectives in the two
samples

24. I
Problem 10: In a sample of 1000 men from a certain city, 450 men are found
to be smokers. In a sample of 800 from another city 400 are found to be
smokers. Do the data indicate that the two cities are significantly different
with respect to prevalence of smoking habit among men?

25. Comparing the proportion of a sample with the
proportion given by the population in such a case the
following formula is applicable:

26. Problem 1: There are 1000 students in a college. Out of 20,000 in the whole
university, in a study 200 were found smokers in the college and the 1000 in
the whole university. Is there a significant difference between the proportion
of smokers in the college and university?

27. Since the difference is more than 2.58SE at 1% level of
significance, we could not accept the hypothesis and thus
conclude that there is a significant difference in proportion of
smokers in the college and university
(20,000)

28. References
• Statistical methods –Dr.S.P.Gupta
• Statistical methods for biologists
(Biostatistics)- S.Palanichamy & M.Manoharan