Life Spans of Fruit Flies The life spans of a species of fruit...

Question

Life Spans of Fruit Flies The life spans of a species of fruit fly have a bell-shaped distribution, with a mean of 33 days and a standard deviation of 4 days.

b. The life spans of three randomly selected fruit flies are 29 days, 41 days, and 25 days. Using the Empirical Rule, find the percentile that corresponds to each life span.

Accepted Answer

Welcome back, everyone. In this problem, the heights of a population of sunflowers are normally distributed with a mean of 150 centimeters and a standard deviation of 12 centimeters. If a randomly selected sunflower is 138 centimeters tall, what percentile does this height correspond to according to the empirical rule? A says it's the 0.1 5th percentile, B 2.5th percentile, C 16th percentile, and D 84th percentile. Now, before we figure out the percentile that the height corresponds to, it would be a good idea for us to find the Z score for this given value, this given observed value in our normal distribution. Now if we make a note here. We know that our observed value X is 138 centimeters. Our mean mu is 150 centimeters, and our standard deviation sigma is 12 centimeters. And recall that the Z square is going to be equal to the observed value minus the mean divided by the standard deviation. So in this case it would be 138 minus 150 divided by 12. That's -12 divided by 12, which equals -1. So how will this Z score of -1 help us to figure out what percentile the height corresponds to? Well, let's think about what we know in the empirical rule, OK? What do we know about the empirical rule? Let me make sure I spell that properly. OK. Now to help us best understand it, let's just do a quick sketch of a normal distribution. Now, according to the empirical rule, what it really tells us, it basically says that on our normal distribution, OK. Then 68% of values fall within one standard deviation of the mean. In other words, between -1 sigma or between negative sigma and positive sigma, there are, well, 68% of our values will be in this region. Next, it tells us that 95% of our values are between 2 sigma and negative 2 sigma within two standard deviations of the mean and about 99.7%. Or, or 99.7% of the data, that is, uh goes from, or it falls within three standard deviations. That is, it goes from -3 sigma to 3 sigma. OK? So that's why the empirical rule is also known as the 6895 99.7 rule, OK? So now, if we think about it on our normal distribution from end to end, it will go from the 0th percentile, let me put that in blue here, to go from the 0th percentile all the way to the 100th percentile, OK? So if we take our Z score for the observed value and put it on our normal distribution, then we should be able to tell what percentile our value will be. Now remember that our value had a Z score of -1. OK. So on our on our distribution. That means our value would fall at this point. OK? Let me highlight that in blue. And no, remember that 68% of the data is between negative sigma and sigma. So, if 68% of the data is between here, OK? And if we take one half of our, one half of our, uh, Or or a normal distribution right then the rest of the data. OK, goes from that point to the 0th percentile, and if all of this is 50% or half of the graph, OK. And we know that 68% falls here, so that means this part must be 34%. Then the remaining or the remaining percentage, the difference between 50 and 34, which would be 16%, would tell us that 16% of the values lie below 11 standard deviations of the data. So here That means that the 16th percentile is the value at which 16% of the data lies at or below it, because the nth percentile is the value below which 1% of the data falls, including the value itself. Therefore, that means the height of 138 centimeters is approximately at the 16th percentile. If we look back at our answer choices, that tells us that C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Normal Distribution

Empirical Rule

Percentiles

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