Question

Sampling and the Binomial Distribution. Refer to the discussion on the binomial approximation to the hypergeometric distribution that begins on page 240.

(a) If sampling is with replacement, explain why the trials are independent and the success probability remains the same from trial to trial-always the proportion of the population that has the specified attribute.

(b) If sampling is without replacement, explain why the trials are not independent and the success probability varies from trial to trial.

Short answer

Part (a) Because the sampled ones are added into the sample space before the following trail.

Part (b) Because the sampled ones are eliminated from the sample space for starting the next trail.

Step-by-step solution

Part (a) Step 1. Given information.

The statement given in the question is:

Refer to the discussion on the binomial approximation to the hypergeometric distribution that begins on page 240.

(a) If sampling is with replacement, explain why the trials are independent and the success probability remains the same from trial to trial-always the proportion of the population that has the specified attribute.

(b) If sampling is without replacement, explain why the trials are not independent and the success probability varies from trial to trial.

Part (a) Step 2. Explanation.

Because the sample ones are incorporated into the sample space before the following trail, the size of the sample space and the number of possible event outcomes do not vary from trail to trail in sampling with replacement.

As a result, each path is unique, and the chances of success will always be the same.

Part (b) Step 1. Explanation.

Because the sampled ones are eliminated from the sample space for the next trail, the size of the sample space and the number of possible event outcomes will vary from trail to trail when sampling without replacement is used.

As a result, each trail is always dependent on the prior track, resulting in changing success probabilities.