Hybridization A hybridization experiment begins with four peas...

Question

Hybridization A hybridization experiment begins with four peas having yellow pods and one pea having a green pod. Two of the peas are randomly selected with replacement from this population.

a. After identifying the 25 different possible samples, find the proportion of peas with yellow pods in each of them, then construct a table to des

Accepted Answer

All right. Hello, everyone. So this question says, in a biology study, a population consists of 3 blue butterflies and 2 orange butterflies. Two butterflies are randomly chosen with replacement. What is the sampling distribution of the proportion of blue butterflies in the samples? All right, so first and foremost, what are the possible outcomes for drawing either a blue butterfly or an orange one? Well, The probability or capital B of a blue butterfly. Is equal to 3 out of 5 in this case. And the probability of an orange butterfly is equal to 2. Over 5. So now The next step is to list all possible samples of size 2 with replacement. This gives you 4 distinct possibilities. One where both butterflies drawn are blue. 2 Where one butterfly is blue and the other is orange. And then another where both butterflies drawn are orange. With each sample size, what you do now is find the proportion of blue butterflies in each sample. Now the proportion of blue butterflies. Is Pihat So for the first sample, Phat is equal to 1, because both butterflies in the sample happen to be blue. When both butterflies are orange, pea hat is equal to 0 instead. So now, for the remaining two sample sizes, he hat is equal to 1/2. Because only one of the two butterflies happens to be blue. So now for each value of Phat, or rather for each sample, You would then go ahead and find the probability. Now the probability is going to be equal to. The product Of the probability of drawing each individual butterfly. So starting, for example, with the first sample where both butterflies are blue. That would be equal to 3/5 multiplied by 3/5. Which equals 9/25? For the middle two samples where one butterfly is blue and the other is orange. Pea hat or the probability rather. Is actually going to be equal. Because in both cases, we have one blue butterfly. And one orange butterfly. So 3/5 multiplied by 2/5. Is equal to 6/25. And for the final sample where both butterflies are orange, that's 2/5 multiplied by 2/5. And this equals 4/25. So now You would combine probabilities for samples that have the same sample proportion. As mentioned before, there are two samples that both have a probability equal to 6/25. So adding those together gives you 12/25. And from here The sampling distribution is as follows. So for the final answer, right, the sample proportions. Are as follows. We have 1 with a probability of 9/25, 1/2 with a probability of 12/25, and 0 with a probability of 4/25. And there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

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Key Concepts

Hybridization

Sampling with Replacement

Proportion

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