Question

Two populations are described in each of the following cases. In which cases would it be appropriate to apply the small-sample t-test to investigate the difference between the population means?

a.Population 1: Normal distribution with variance ${{\mathbf{\text{σ}}}_{\mathbf{\text{1}}}}^{\mathbf{\text{2}}}$. Population 2: Skewed to the right with variance${{\mathbf{\text{σ}}}_{\mathbf{\text{2}}}}^{\mathbf{\text{2}}}\mathbf{=}{{\mathbf{\text{σ}}}_{\mathbf{\text{1}}}}^{\mathbf{\text{2}}}$.

b. Population 1: Normal distribution with variance ${{\mathbf{\text{σ}}}_{\mathbf{\text{1}}}}^{\mathbf{\text{2}}}$. Population 2: Normal distribution with variance ${{\mathbf{\text{σ}}}_{\mathbf{\text{2}}}}^{\mathbf{\text{2}}}\mathbf{\ne }{{\mathbf{\text{σ}}}_{\mathbf{\text{1}}}}^{\mathbf{\text{2}}}$.

c. Population 1: Skewed to the left with variance ${{\mathbf{\text{σ}}}_{\mathbf{\text{1}}}}^{\mathbf{\text{2}}}$. Population 2: Skewed to the left with variance${{\mathbf{\text{σ}}}_{\mathbf{\text{2}}}}^{\mathbf{\text{2}}}\mathbf{=}{{\mathbf{\text{σ}}}_{\mathbf{\text{1}}}}^{\mathbf{\text{2}}}$.

d. Population 1: Normal distribution with variance${{\mathbf{\text{σ}}}_{\mathbf{\text{1}}}}^{\mathbf{\text{2}}}$ . Population 2: Normal distribution with variance${{\mathbf{\text{σ}}}_{\mathbf{\text{2}}}}^{\mathbf{\text{2}}}\mathbf{=}{{\mathbf{\text{σ}}}_{\mathbf{\text{1}}}}^{\mathbf{\text{2}}}$ .

e. Population 1: Uniform distribution with variance${{\mathbf{\text{σ}}}_{\mathbf{\text{1}}}}^{\mathbf{\text{2}}}$ . Population 2: Uniform distribution with variance ${{\mathbf{\text{σ}}}_{\mathbf{\text{2}}}}^{\mathbf{\text{2}}}\mathbf{=}{{\mathbf{\text{σ}}}_{\mathbf{\text{1}}}}^{\mathbf{\text{2}}}$.

Short answer

A t-test is an inference statistic that is used to see if there is a substantial difference in the means of two categories that are connected in some way.

Step-by-step solution

Step-by-Step Solution Step 1: Definition of t-test.

The t-teststatistical examination was used to contrast the two groups' means. It assists us in determining whether or not there is a substantial disparity between the means of the two groups. When doing a t-test, basic assumptions include the measurement scale, accidental selection, normality of distribution of data, the sufficiency of sample size, as well as equality of variation in standard deviation.

The formula of the t-test is:

$\mathbf{\text{t=}}\frac{\overline{\mathbf{\text{x}}}\mathbf{-}\mathbf{\text{μ}}}{\frac{\mathbf{\text{σ}}}{\sqrt{\mathbf{\text{n}}}}}$

(a) State whether the t-test is appropriate when Population 1: Normal distribution with variance σ12 . Population 2: Skewed to the right with variance σ22=σ12 .

Population 2 is not normally distributed, so it will not be appropriate to apply the small-Sample t-test to investigate the difference between the population means.

(b) State whether the t-test is appropriate when Population 1: Normal distribution with variance σ12 . Population 2: Normal distribution with variance σ22≠σ12 .

The variances of Population1 and Population 2 are unequal, so it will not be appropriate to apply the small-Sample t-test to investigate the difference between the population means.

(c) State whether the t-test is appropriate when Population 1: Skewed to left with variance σ12 . Population 2: Skewed to left with variance σ22=σ12 .

The Populations are not normally distributed, so it will not be appropriate to apply the small-Sample t-test to investigate the difference between the population means.

(d) State whether the t-test is appropriate when Population 1: Normal distribution with variance σ12. Population 2: Normal distribution with variance σ22=σ12.

The Populations are normally distributed and the variances are also equal, so it will be appropriate to apply the small-Sample t-test to investigate the difference between the population means.

(e) State whether the t-test is appropriate when Population 1: Uniform distribution with variance σ12. Population 2: Uniform distribution with variance σ22=σ12.

The Populations are not normally distributed, so it will not be appropriate to apply the small-Sample t-test to investigate the difference between the population means.