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.

b. Find the mean of the sampling distribution.

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. Find the mean of the sampling distribution of the proportion of blue butterflies. Option A is 2/5. Option B says 3/5, Option C says 9/25, and option D says 12/25. All right, so for this question, the first thing you want to do is define the possible outcomes for each draw, right? So, here we want to see the probability of drawing a blue butterfly or piece of B versus the probability of drawing an orange butterfly or piece of. So here, because we have 3 blue butterflies out of a total of 5, piece of B. is equal to 3/5, whereas piece of O is equal to 2/5. So from here, We have to list all possible samples of size 2 with replacement. This gives us 4 different possibilities, because we can have 2 blue butterflies. One blue and one orange. One orange and one blue. And 2 orange butterflies. And so for each sample combination, we now have to find pea hat, which corresponds to the proportion of blue butterflies in each sample. So When you're choosing two blue butterflies, for example, Phat is equal to 1, because in this case, all of the butterflies in your sample are blue. By contrast, P is equal to 0. In the case where you have two orange butterflies. So for both blue orange and orange blue, Phat is going to be equal to 1/2. Because in this case, only 1 out of the 2 butterflies in that in those samples are blue. And now we can move on to finding the probability of each sample. So, to accommodate the space needed for the probability, I'm going to go ahead and move the possible samples and the corresponding values for Phat towards the bottom of the screen here. So here, for the sample in which we have two blue butterflies, the probability. Is going to be equal to. The product of the probability for each butterfly. So here, for example, because both butterflies are blue, and the probability of drawing a blue butterfly is 3/5, the probability for that sample would simply be 3/5 multiplied by 3/5, to yield 9/25. So our next two samples have 1 blue butterfly. And one orange butterfly. Which means that for both of these, you would multiply 3/5s by 2/5. And both samples in this case would yield the same probability of 6/25. And the probability where both butterflies drawn are orange is equal to 2/5 multiplied by 2/5, which equals 4/25. So before finding the mean of the sampling distribution, we should first combine the probabilities of samples with the same sample proportion. In this case, as mentioned before, we have two samples where one butterfly is blue and the other is orange. Which means that we can combine. Both sample proportions, they yield 12/25. And now we can find the mean. Of the sampling distribution. So here, that's going to be the sum of. All calculations of Phat. Multiplied by the corresponding probability of that Phat value. So for a Phat equal to 1, for example, that would be equal to 1. Multiplied by the probability, which is 9/25. Next, you would add that to the probability of a peahat of 1/2. It should be 1/2 Multiplied by 12/25, which is a probability. And then you have Phat equal to 0, multiplied by its probability of 4/25. And this ultimately yields 3/5 as your final answer. Making the correct choice option B. 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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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.

b. Find the mean of the sampling distribution.

Key Concepts

Sampling Distribution

Mean

Random Sampling with Replacement

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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.b. Find the mean of the sampling distribution.