Which Is More Likely? Assume that the fertility rates in Exerc...

Question

Which Is More Likely? Assume that the fertility rates in Exercise 32 are normally distributed. Are you more likely to randomly select a state with a fertility rate of less than 65 or to randomly select a sample of 15 states in which the mean of the state fertility rates is less than 65? Explain.

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. The weights of apples in an orchard are normally distributed. Is it more probable to randomly pick an apple weighing less than 120 g, or to randomly select a group of 8 apples whose average weight is less than 120 g? Awesome. So it appears for this particular problem, we're trying to determine whether or not it is considered more probable to randomly pick an apple that weighs less than 100 g, or if it is more probable to randomly select a group of apples whose average weight is less than 120 g. So with that in mind, now that we know what we're ultimately trying to solve for, we need to recall and use our key concept for this particular problem, which is the central limit theorem, CLT for short. So let's make a note of that, CLT. Once again, our central limit theorem. So, as you should recall for this particular prompt, individual weights of apples are normally distributed. The average of the sample of apples, in this case we're told there's 8 apples, is also normally distributed. But the standard deviation of the sample mean is smaller than that of a single apple, and this is crucial to note. So now let's consider our definitions. So let us state that mu will be the population mean weight of the apples, and let us state that sigma is the population standard deviation. Therefore, we can go ahead and state that for the individual apple weight, which we'll call this value X, we could say that X is roughly Capital N multiplied by sigma. Fantastic. We could also say that for the sample mean of 8 apples, which we'll call this value. X bar subscript 8, so X bar subscript 8, because there's 8 apples, is roughly N multiplied by mu sigma divided by the square root of 8. Fantastic. And as a note that both of our values for X, or rather it's for X is in parentheses, and our value for X barred subscript A is also in parentheses for parentheses mu com some value. So, We need to note that X bar subscript 8 has less variability than X because sigma divided by the square root of 8 is less than sigma. Fantastic. So what does this particular piece of information or what does all this information mean to us? So what does this mean for our probabilities when we consider these pieces of information? So, once again, we need to note that we are comparing, so I'll make a little line here. We are comparing the probability P, so P of X is less than 120, so this is the probability that one apple weighs less than 120 g, and we also have the probability, the probability of X bar subscript 8 is less than 120. Fantastic. So considering these two pieces of information, once again, P X is less than 120, the probability that one apple weighs less than 120 g, and P X bar subscript 8 is less than 120, which is the probability that the average weight of 8 apples is less than 120 g. So therefore, as a result, as we should recall, Since the distribution of sample means is narrower, the probability that the average that the average will fall or will fall far from the mean, like below 120 g is lower than for a single value. So therefore, in conclusion, we could safely write that P of X is less than 120. is greater than PXar subscript 8. is less than 120. So therefore, without a doubt, As a result, it is more probable to pick one apple weighing less than 120 g. So that means our final answer, without a doubt has to be once again more probable to pick an apple weighing less than 120 g. And that's it. We've officially solved for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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