Question

When and way the following statistics methods are used, What type of test statistics can be used for hypothesis testing, explain in detail and provide Example for each.

• Population means, independent sample
• Population Means, Related Samples
• Population Variances, Two-Sample
• Multiple regression.

Answer 1

ANSWER:

1) We are asked to explain the following

a)

Population means, independent samples.

We use hypothesis test for population means, independent samples when we are given two samples which are independent of each other i.e., an observation in one sample group does not appear in the other group. We use t-test for independent samples if given sample size is small(less than 30) but if population variances are given, we can use z-test for two means also.

I will explain the steps involved in hypothesis testing by the way of an example.

Example: Two random samples of sizes 10 and 8 have the means 29 and 32 respectively. If the population variances are 7.52 and 6.84, test the significance difference between the sample means.

Given that

Null hypothesis

i.e., the two population means are equal or in other words the two samples have been drawn from the respective populations.

Alternate hypothesis

i.e., the two population means are not equal or in other words the two samples have not been drawn form the same population.

Test statistic under H₀

If population variances are not given we the test statistic becomes.

where

Since population standard deviations are given, we will use z-test for two means.

Inference

at 5% level of significance and for two-tailed test from standard normal tables.

=> we reject the null hypothesis

Hence we conclude that the two samples have not been drawn from populations respectively.

b)

We use hypothesis test for population means, related samples when we are given two samples where one sample appears in both the samples.

Example: A treatment was given to ten patients and readings were recorded before and after applying the treatment.

Patient Number	1	2	3	4	5	6	7	8	9	10
HB Percent before treatment	11.94	11.99	11.98	12.03	12.03	11.96	11.95	11.96	11.92	12.00
After treatment	12.00	11.99	11.95	12.07	12.03	11.98	12.03	12.02	12.01	11.99

Can we conclude that HB per cent was increased after treatment was given to the patients.

xi	yi	di=xi-yi	di2
11.94	12.00	-0.06	0.0036
11.99	11.99	0	0
11.98	11.95	0.03	0.0009
12.03	12.07	-0.04	0.0016
12.03	12.03	0	0
11.96	11.98	-0.02	0.0004
11.95	12.03	-0.08	0.0064
11.96	12.02	-0.06	0.0036
11.92	12.01	-0.09	0.0081
12.00	11.99	0.01	0.0001

Null hypothesis

i.e., HB percent remained the same before and after the treatment

Alternate hypothesis

i.e., HB percent was increased after the treatment was given.

Test statistic under H₀

Inference

at 5% level of significance and 9 d.f. for one-tailed test from t-tables.

=>We reject the null hypothesis

Hence we conclude that HB percent was increased after the treatment was given.

c)

We use hypothesis test for population variances, two samples when we are asked to test the equality of two variances.

We use F-test for the hypothesis.

Example: Two random samples of sizes 9 and 12 have the standard deviations 2.9 and 2.6 drawn form two normal populations. Test the significant difference between the sample variances.

Null hypothesis

i.e., there is no significant difference between the sample variances.

i.e., there is a significant difference between the sample variances.

Test statistic under H₀:

Since

Inference

at 5% significance level and at(8,11) d.f. from F-tables

=> We failt to reject the null hypothesis.

Hence we conclude there is a significant differences between the sample variances.

Note:- I couldn't find an example which would be easily explainable for hypothesis test for multiple regression. Hence I did not answer the multiple regression part.

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