Question

Replacement

Why does the bootstrap method require sampling with replacement? What would happen if we used the methods of this section but sampled without replacement?

Short answer

If the sampling is done with replacement, it can be repeatedly picked in subsequent draws.

If sampling is done without replacement, the confidence interval for a bootstrapped sample is not going to work because all the samples are going to have the same summary statistics.

Step-by-step solution

Describe the concept of the bootstrap sample

A bootstrap sample is a sample obtained from another randomly selected sample.

Thus, a resampling method where large numbers of smaller samples of the same size are repeatedly drawn with replacement from a single original sample is bootstrapping.

State the reason of bootstrap method requires sampling with replacement

The bootstrap samples are the same size as the original data set. If sampling is done with replacement, then a large number of smaller samples of the same size are repeatedly drawn to ensure that an appropriate confidence interval can be constructed.

Explain the bootstrap sampling method under without replacement

If sampling is done without replacement, it would end up with the same data over and over.

Therefore, the means or standard deviation or proportions, or variance would be the same for the original sample and bootstrap sample.

Hence, the confidence interval for the bootstrapped sample is not going to work because all of those samples are going to have the same summary statistics.