Question

3. Regarding the pooled and nonpooled $f$-procedures,

a. what is the difference in assumptions between the two procedures?

b. how important is the assumption of independent simple random samples for these procedures?

c. how important is the normality assumption for these procedures?

Short answer

a) The difference in assumptions is given.

b) For both procedures, random samples must be simple and independent.

c) For both pooled and non-pooled t procedures, the normal assumption is more important for small samples than for large samples.

The central limit theorem should be used with large sample size.

Step-by-step solution

Part (a) Step 1: Given Information

To state the difference of assumptions between pooled and nonpooled $t$ procedure.

Part(a) Step 2: Explanation

The following are the assumptions for the pooled$t-$test:

1. A simple random sample was drawn from two populations.

2. The chosen samples are independent.

3. The population is roughly normal.

4. The standard deviations of the populations are equal.

The following are the assumptions for the non-pooled $t-$test:

1. A simple random sample was drawn from two populations.

2. The chosen samples are independent.

3. The population is approximately normal.

4. It is not necessary for the standard deviations of populations to be equal.

Part(b) Step 1: Given Information

To explain the importance of independent simple random samples for the assumption of pooled and nonpooled $t$ procedure.

Part(b) Step 2: Explanation

The following are the assumptions for the pooled$t$-test:

1. A simple random sample was drawn from two populations

2. The chosen samples are independent.

3. The population is roughly normal.

4. The standard deviations of the populations are the same.

The following are the assumptions for a non-pooled $t$-test:

1. A simple random sample was drawn from two populations

2. The chosen samples are independent.

3. The population is roughly normal.

4. Population standard deviations do not have to be equal.

As a result, the random samples must be simple and independent for both procedures.

Part(c) Step 1: Given Information

To explain the importance of normal approximation for the assumption of pooled and nonpooled$t$ procedure.

Part(c) Step 2: Explanation

The following are the assumptions for the pooled $t$-test:

1. A simple random sample was drawn from two populations

2. The chosen samples are independent.

3. The population is roughly normal.

4. The standard deviations of the populations are the same.

The following are the assumptions for a non-pooled $t$-test:

1. A simple random sample was drawn from two populations

2. The samples chosen are distinct.

3. The population is roughly normal.

4. Population standard deviations do not have to be equal.

For both pooled and non-pooled $t$ procedures, the normal assumption is more important for small samples than for large samples.

The central limit theorem should be used with large sample size.