Chapter 6: Q2 (page 263)

Sampling with Replacement The Orangetown Medical Research Center randomly selects 100 births in the United States each day, and the proportion of boys is recorded for each sample.
a. Do you think the births are randomly selected with replacement or without replacement?
b. Give two reasons why statistical methods tend to be based on the assumption that sampling is conducted with replacement, instead of without replacement.

Short Answer

Expert verified

a. The births will be selected without replacement because, in general, the researcher does not want to repeat the same units while extracting samples.

Moreover, since the sample size of 100 is less than 5% of the population of births every day in the United States, it can be said that the sample is selected with replacement.

b. The two reasonswhy statistical procedures assume that sampling is conducted with replacement instead of without replacement are given below:

If the sample size is small, the two methods of sampling can be considered the same.
Since sampling without replacement involves independent events whose outcomes do not depend on the outcomes of the previous events, the calculations for such samples are easier and understandable.

Step by step solution

Given information

It is given that 100 births are selected randomly out of all the births in the United States every day.

The proportion of boys is recorded for every sample.

Sampling with replacement vs. sampling without replacement

The births will be selected using the sampling method, named as without replacement because, in general, the researcher does not want to repeat the same units while extracting samples.

However, since the number of births every day in the United States is large compared to the sample size of the births selected each day, it can be safely assumed that the selection is in accordance with the 5% rule.

The 5% rule states that if the sample size is no more than 5% of the population size, the sample can be assumed to be selected with replacement.

Here, 100 is certainly less than 5% of the population of all births each day. Therefore, the sample can be assumed to be selected with replacement.

Advantages of sampling with replacement

The two major reasons why all statistical procedures involve the use of sampling with replacement technique are mentioned below:

There is no difference whether the sample is selected with replacement of units or without replacement of units when the size of the sample is comparatively small as that of the population.

Sampling with replacement technique results in independent events. That is, outcomes of current events are unaffected by outcomes of previous events. Moreover, independent events are easier to interpret and involve simple calculations.