Question

We want to estimate the proportion of nuts that are peanuts inside a bag of mixed nuts. We get a jar of the nuts and assume that the nuts in that container represent a random sample of all the mixed nuts sold by that company. We open the jar and count 100 nuts of which 52 are peanuts.

1.1 - (20 points) How could we physically use the jar of nuts to construct one bootsrap sample? What would we record to find the bootstrap statistic?
1.2 - (20 points) If we create a bootstrap distribution by collecting many bootstrap statistics, describe the center and anticipated shape of the distribution.

Answer 1

1.1 This can be done by chosing with replacement. We take the 100 nuts jar, pick one nut at random, record whether the nut is peanut, and put the nut back again. This process is to be done a total of 100 times and the bootstrap statistic would be the proportion of peanuts obtained in the experiment.

1.2 Essentially, this is a question of sampling. If we do the experiment many times the mean would be expected to be centred at 0.52. Also, since the process is repeated many times, we can expect the distribution to be normal. That is a bell shaped curve.