Graphical Analysis In Exercises 13–16, a member is selected at...

Question

Graphical Analysis In Exercises 13–16, a member is selected at random from the population represented by the graph. Find the probability that the member selected at random is from the shaded region of the graph. Assume the variable x is normally distributed.

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Accepted Answer

Welcome back, everyone. The following graph shows the normal distribution of the heights in centimeters of a certain plant species. Find the probability that a randomly selected plant falls within the shaded region. Here we have our normal distribution of heights for a certain plant species, and for our answer choices. A says it's 0.0030. B 0.0371, C 0.0401, and D 0.756. Now what do we know here? Well, we're given a normal distribution that has a mean mu of 50 centimeters and a standard deviation sigma of 8 centimeters, and with that we are solving for, OK, we're solving for the probability that the height of our plant species X is between 64 and 72 centimeters. Now how can we figure that out? Well, since we're working with our normal distribution, it would first be a good idea to convert our heights to Z scores, and then we can be able to solve for the probability. So let's go ahead and do that. Recall, OK, that. A Z value or the Z score formula tells us that it will be equal to the value X minus the mean mu divided by the standard deviation sigma. In other words, it's telling us how many standard deviations or value is away from the mean. So, for our first value, 64 centimeters, then the first Z score will be equal to 64 minus 50 divided by 8. That's 14 divided by 8, which is 1.75. On the other hand, for a second value 72, its Z score will be equal to 72 minus 50 divided by 8, which is 22 divided by 8 or 2.75. So in other words, what we're seeing here. Is that This implies the probability that X is between 64 and 72. Will be equivalent to the probability that Z is between 1.75 and 2.75. Now, if we go to our diagram here. Or or distribution, if we think about it. That means here, if this is, let me make this here, 2.75, OK. And this is a Z score of 1.75, then this probability that it's going to be between those values would be equal to the probability it's less than 2.75 minus the probability it's less than 1.75, OK? So in other words, This is going to be equal to the probability it's. Sorry, Z is less than 2.75. Minus the probability z is less than 1.75, and we can find both of those probabilities with the standard normal distribution table. Now from our table then. OK. We, it tells us that the probability Z is less than 2.75 is equal or approximately equal to 0.9970. And the probability Z is less than 1.75, is approximately 0.9599. So this would imply then that the probability. That Z is between 1.75 and 2.75 is going to be equal to 0.9970. The probability Z is less than 2.75 minus 0.9599. The probability Z is less than 1.75, and that is Z equal to 0.0371. This tells us then That since it corresponds to our probability that X is between 64 and 72, the probability that a randomly selected plant has a height between 64 and 72 centimeters is approximately 0.0371. If we look back at our answer choices, that means B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Graphical Analysis In Exercises 13–16, a member is selected at random from the population represented by the graph. Find the probability that the member selected at random is from the shaded region of the graph. Assume the variable x is normally distributed.

Key Concepts

Normal Distribution

Probability

Z-scores

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