1 Testing of Hypothesis Two Sample test Dr. T. T. Kachwala

Objective of Hypothesis Testing for differences between Means: 2 In many decision making situations, we need to determine whether the parameters of the two populations are alike or different. For example: i)Do female employees earn less than male employees for the same work in a company? ii)Do students of one division score more marks than other division? In the above examples, we are concerned with the parameters of two populations. We are not interested in the actual values of the population means as we are in the relation between the values of the two population means, i.e. how these population means differ. The objective of two sample test is to assess whether or not there is a significant difference between the two population means

Hypothesis Testing for difference between Means Procedure (Large Sample Size – z test): 3 In testing of difference between two means, we must choose whether to use a one-tailed hypothesis test or a two-tailed hypothesis test. If the test concerns whether two means are equal or are not equal, use a two tailed test. However, if the test concerns whether one mean is significantly higher or significantly lower than the other, a one-tailed test is more appropriate. H 0 :  1 =  2 H 1 :  1   2 H 0 :  1 =  2 H 1 :  1   2 or H 0 :  1 =  2 H 1 :  1 >  2

Hypothesis Testing for difference between Means Procedure (Large Sample Size – z test): 4 Step (i) H 0 :  1 =  2 H 1 :  1   2 (assuming Two Tail Test) Step (ii)Select α  0.05, select Normal distribution, select two tailed; z critical   Step (iii)

Hypothesis Testing for difference between Means (Large Sample Size – z test): 5 x z stat Accept H 0 x z stat Reject H 0 Accept H 1 If ‘z’ is in acceptance area, Accept H 0 :  1 =  2 and we conclude that there is no significant difference in the population mean. If ‘z’ is in the rejection area, Reject H 0 :  1 =  2 Accept H 1 :  1   2 and we conclude that the population means differ significantly. Step (iv) & (v): Decision Rule & Conclusions

Hypothesis Testing for difference between Means Small Sample Size – t test: (assuming variations of two population are equal) 6 When sample sizes are small, there are two changes in our procedure for testing the differences between means: (i)Estimated standard error of the difference between two Sample mean ii) Use of t  distribution. The following is the procedure for t test Step (i) H 0 :  1 =  2 H 1 :  1   2 (assuming Two Tailed Test) Step (ii) α = 0.05 ; t distribution, 2 tailed, = n 1 +n t critical = (from table) Note: Degree of freedom for two samples: = n 1 – 1 + n 2 – 1 = n 1 + n 2 - 2

Hypothesis Testing for difference between Means Small Sample Size – t test: (assuming variations of two population are equal) 7 Step (iii)Calculation of t statistic Where ‘S’ is the pooled estimate of standard deviation  (weighted average), N is the combined sample size

Hypothesis Testing for difference between Means Small Sample Size – t test: (assuming variations of two population are equal) 8 x t stat Accept H 0 x t stat Reject H 0 Accept H 1 Step (iv) & (v): Decision Rule & Conclusion If ‘t’ is in acceptance area Accept H 0 :      Conclusion: There is no significant difference in the population means. If ‘t’ is in rejection area Reject H 0 :      Accept H 1 :      Conclusion: There is a significant difference in the population means

Paired difference Test (Dependent Samples) - Objective 9 1.In the earlier examples, our samples were chosen independently of each other; for example students from two different colleges, light bulbs of two different manufacturers, height of sailors and soldiers, sample of chicken with high protein diet and another sample with low protein diet etc. 2.Sometimes however it makes sense to take samples that are not independent of each other. Often the use of such dependent (or paired) samples enable us to perform a more precise analysis because they will allow us to control for extraneous factors. 3.With dependent samples, we follow the same basic procedure that we have followed when testing hypothesis about a single mean.

Paired difference Test (Dependent Samples) 10 Step (ii) t distribution (critical value depends on Degree of Freedom, level of significance  and single tailed or two tailed test). H 0 :  =  H 0 {  H 0 =  d = Mean of difference value of the population} H 1 :    H 0 Step (i) Procedure t critical = (from table)

11 Step (iii) Paired difference Test (Dependent Samples) Calculation of test statistics {  H 0 =  d }

Paired difference Test (Dependent Samples) 12 Step (iv)& (v) Decision Rule & Conclusion: There are two possibilities as indicated in diagram below: x t stat Accept H 0 x t stat Reject H 0 Accept H 1 There is no significant difference between Sample Mean & There is a significant difference between Sample Mean &

13 Hypothesis Testing for difference between Proportions (Large Sample Size – z test) - Objective: In many decision making situations, we need to determine whether the proportions of the two populations are alike or different. For example: i)Is the proportion defective of the first factory different from the proportion defective of the second factory? ii)Is the proportion of wheat consumers of one town different from the proportion of wheat consumers of the second town? In the above examples, we are concerned with the proportion of two populations. We are not interested in the actual values of the proportion as we are in the relation between the values of the two proportions, i.e. how these proportions differ. The objective of two sample test is to assess whether or not there is a significant difference between the two population proportions

Hypothesis Testing for difference between Proportions (Large Sample Size – z test) - Procedure: 14 In testing of difference between two proportions, we must choose whether to use a one-tailed hypothesis test or a two-tailed hypothesis test. If the test concerns whether two proportions are equal or are not equal, use a two tailed test. However, if the test concerns whether one Proportion is significantly higher or significantly lower than the other, a one-tailed test is more appropriate. H 0 : p 1 = p 2 H 1 : p 1  p 2 H 0 : p 1 = p 2 H 1 : p 1  p 2 or H 0 : p 1 = p 2 H 1 : p 1 > p 2

Hypothesis Testing for difference between Proportions (Large Sample Size – z test) - Procedure: 15 Summary Procedure Step (i) H 0 : p 1 = p 2 H 1 : p 1  p 2 (assuming Two Tailed Test) Step (ii) α = 0.05, Normal Distribution, 2 tailed  z critical =  = Standard Error for the difference in proportion of success

16 Hypothesis Testing for difference between Proportions (Large Sample Size – z test): is the best estimate of the overall proportion of success in the population (combined proportion of success or weighted proportion of success)

17 Hypothesis Testing for difference between Proportions (Large Sample Size – z test): Step (iv)& (v) Decision Rule & Conclusion: There are two possibilities as indicated in diagram below: If z statistics lies in acceptance area Accept H 0 : p 1 = p 2 If z statistics lies in rejection area Reject H 0 : p 1 = p 2 Accept H 1 : p 1  p 2 Conclusion: There is a significant difference in the population proportions Conclusion: There is no significant difference in the population proportions

18 Thanks and Good Luck Dr. T. T. Kachwala