Question

2.130 Suppose that a variable of a population has a reverse-$J$-shaped distribution and that two simple random samples are taken from the population.

a. Would you expect the distributions of the two samples to have roughly the same shape? If so, what shape?

b. Would you expect some variation in shape for the distributions of the two samples? Explain your answer.

Short answer

a. The expected shape of both samples is the same.

b. The expected shape of both samples will have variations.

Step-by-step solution

Part (a) Step 1: Concept Introduction

The distribution is called negative exponential when the reverse $J-$shaped curve plots as a

straight line with a negative slope on a semi-log scale.

Part (a) Step 2: Explanation

In this case, we take two simple random samples from a population whose variable has a reverse-$J$ distribution.

There is an assumption that a variable in a population has a reverse $J-$shape distribution, and two simple random samples from that population are selected.

A simple random sample is composed of values that are chosen primarily from the middle of the population.

Thus, we can expect an approximately reverse-$J-$shaped distribution for the two samples.

As such, the shape of the distribution of the two samples remains the same.

Part (b) Step 3: Concept Introduction

The distribution is called negative exponential when the reverse $J-$shaped curve plots as a straight line with a negative slope on a semi-log scale.

Part (b) Step 4: Explanation

In this case, we take two simple random samples from a population whose variable has a reverse-$J-$distribution.

There is an assumption that a variable in a population has a reverse $J-$shape distribution, and two simple random samples from that population are selected.

In a simple random sample, there is likely to be some variation between the two samples, even though they are taken from the same population, although it is not likely that both will have the same observations.

Accordingly, yes, there will be some variation in the shape of the two sample distributions.