Answer :
The second [8, 2, 2, 2, 8], and fourth samples [8, 8, 8, 8, 8] are valid bootstrap samples, while the first and third ones are not. Therefore, the answer is [False, True, False, True].
Bootstrapping is a resampling technique which serves to estimate the sampling distribution of an estimator. Bootstrap samples are formed by sampling with replacement from an original sample. Therefore, the size of a bootstrap sample must always be the same as the original sample and it should only contain numbers from the original sample.
Starting with our original sample [9, 3, 12, 2, 8], we can now examine each potential bootstrap sample to assess its validity: For the sample [9, 3, 12, 23, 12, 2, 8, 9], it has more elements than the original sample, disqualifying it as a valid bootstrap sample. Moreover, it contains the number 23, which does not exist in our original sample.
Examining the sample [8, 2, 2, 2, 8], we find it has the same number of elements as the original sample and only contains numbers that exist in our original sample, thus it is a valid bootstrap sample. The third sample, [9, 3, 12, 4, 8], contains the number 4, which does not exist in our original sample. Because of this, it cannot be a valid bootstrap sample.
Our last sample [8, 8, 8, 8, 8], despite consisting of just one unique number, is indeed a valid bootstrap sample. It has the same number of elements as the original sample, and its numbers are all representative of a value that is present in our original sample.
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