Statistics 2 Dersi 4. Ünite Sorularla Öğrenelim

For two sided (two-tailed) test, express the null (H0) and alternative (H1) hypothesis.

H0: µ1 – µ2 = 0
H1: µ1 – µ2 ? 0

or

H0: µ1 = µ2 (There is no difference)
H1: µ1 ? µ2 (There is a difference)

For the right sided (upper tailed) test, express null (H0) and alternative (H1) hypothesis.

H0: µ1 – µ2 = 0
H1: µ1 – µ2 > 0

or

H0: µ1 = µ2
H1: µ1 > µ2

Express the rejection regions for the following cases:

If the alternative hypothesis is in the form of H1 : µ1 ? µ2 

If the alternative hypothesis is in the form of H1 : µ1 < µ2

If the alternative hypothesis is in the form of H1 : µ1 > µ2

If the alternative hypothesis is in the form of H1 : µ1 ? µ2 then rejection region is

z > z?/2 or z < – z?/2

If the alternative hypothesis is in the form of H1 : µ1 < µ2 then rejection region is

z < – z?

If the alternative hypothesis is in the form of H1 : µ1 > µ2 then rejection region is

z > z?

Explain the use of t test instead of z test in terms of sample size.

z test statistic can be used for difference between two means if sample sizes are greater than or equal to 30. In many real-life circumstances, it’s not possible or practical to attain the sample size of 30 or more due to the time, research budget and other relevant constraints. Also, usually the population standard deviations are unknown. Under these circumstances, t test is used to test the difference between means. At this time also, the assumption regarding the population distribution of the variables should be normal or approximately normal, and two samples are independently selected from these populations.

Covering the hypothesis test of the difference between two population proportions, and these parameters are depicted with p1 and p2, a sample proportion from each population is utilized to realize the z-test for the difference between two population proportions. The basic assumption of the test considered here is that there is no difference between the population proportions. Then, define the hypotheses for difference between two population proportions for two sided test and express null and alternative hypothesis.

For two sided (two tailed) test, the null hypothesis (H0) and the alternative hypothesis (H1) are in the following form,
H0: p1 = p2
H1: p1 ? p2

Covering the hypothesis test of the difference between two population proportions, and these parameters are depicted with p1 and p2, a sample proportion from each population is utilized to realize the z-test for the difference between two population proportions. The basic assumption of the test considered here is that there is no difference between the population proportions. Then, define the hypotheses for difference between two population proportions for right sided test and express null and alternative hypothesis.

For the right sided (upper tailed) test, the null hypothesis (H0) and the alternative hypothesis (H1) are in the following form,
H0: p1 = p2
H1: p1 > p2

Covering the hypothesis test of the difference between two population proportions, and these parameters are depicted with p1 and p2, a sample proportion from each population is utilized to realize the z-test for the difference between two population proportions. The basic assumption of the test considered here is that there is no difference between the population proportions. Then, define the hypotheses for difference between two population proportions for left sided test and express null and alternative hypothesis.

For the left sided (lower tailed) test, the null hypothesis (H0) and the alternative hypothesis (H1) are in the following form,
H0: p1 = p2
H1: p1 < p2

When there are more than two means define null and alternative hypothesis.

The null
hypothesis and alternative hypothesis for one-way analysis of variance test can be defined as follows,
H0: µ1 = µ2 = µ3 = ..... = µk (all population means under consideration are equal)
H1: At least one of population means is different µ1, µ2, µ3, ..... , µk.

Rejecting the null hypothesis, explain the practice to find the difference for more than two means.

rejecting the null hypothesis in an ANOVA test means that at least one of the means of the
population under consideration is different from the other population's means. To determine which of the population mean(s) are different we need additional statistical tests. F test is used if we consider more than two means of population.  By the help of the F test, all the population means are compared simultaneously. Otherwise, if you consider comparing two means at a time as we considered in the previous sections, the number of tests increases enormously as the number of populations and their means increase. For example, to compare three population's means at a time, we need three t-tests. In a similar manner, to compare five population's means at a time, we need 10 t-tests.

What are the requirements to utilize one-way ANOVA test ?

To utilize the one-way ANOVA test, the samples must be independent and randomly selected from a normal or approximately normal population. Also, each population variance under consideration must be equal.

Explain the one-way ANOVA test.

For the one-way ANOVA test, the test statistic is the ratio of two variances: namely the variance between samples and the variance within samples. These variances constitute the estimates of the population variance. The variance between samples are also called as between group variance, and the variance within samples are also called as within group variance. The variance between samples measures the differences associated with the treatment which is given to each sample and abbreviated as MSB . On the other hand, the variance within samples measures the differences associated with observations within the same sample abbreviated as MSW . The test statistic for a one-way ANOVA test is the ratio of two variances; namely the variance between samples (MSB) and variance within samples (MSW).

When comparing more than two means, explain the rejection of null hypothesis.

The F (one-way ANOVA) test to compare more than two population means is always a right tailed test. Therefore, H0 will be rejected if the test statistic F is greater than the critical value. From the F test statistic, it’s clear that the value of the test statistic is close to the 1 if MSB close MSW , means that there is little or no difference between the means. Then, the F test statistic value nearby to 1 suggests that we fail to reject the null hypothesis (H0).
If F test statistic value is greater than 1 we suggest that the null hypothesis should be rejected. This situation occurs when one population mean is significantly different than the other population's means, and this state is revealed if MSB is greater than MSW .