Test of significance of large samples

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 .

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

29. 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 .

30. 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..

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