Annual rainfall The annual rainfall in inches for San Francisc...

Question

Annual rainfall The annual rainfall in inches for San Francisco, California, is approximately a normal random variable with mean 20.11 in. and standard deviation 4.7 in. What is the probability that next year’s rainfall will exceed 17 in.?

Accepted Answer

Hello there. Today we are going to 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. Over the course of a year, the total snowfall in Denver is well modeled by a normal distribution with a mean mu is equal to 55.2 inches, and a standard deviation sigma is equal to 10.4 inches. What is the probability that next winter's snowfall will be greater than 65 inches? Round your answer to 3 decimal places. OK. So it appears for this particular prompt we're asked to take all the information that is provided to us and we're asked to determine what the probability will be that next winter's snowfall will be greater than 65 inches and we're asked to take our final answer and round it to 3 decimal places. So now that we know that we're trying to solve for this probability value, our first step that we need to take is we need to let our variable X be a random variable that represents this total snowfall in Denver over the course of a year or the winter season. So that will mean that the snowfall X will be normally distributed with a mean mu is equal to 55.2, and a, once again, a standard deviation sigma is equal to 10.4, and we want to solve for the probability P of X is greater than 65. Fantastic. So that will mean that the normal distribution, as we should recall, will be defined by the following probability density function, which states that fx is equal to 1 divided by sigma multiplied by the square root of 2 i multiplied by e to the power of 1/2 multiplied by parentheses of. X minus mu divided by sigma to the power of 2. So once again, based on our prior knowledge, we need to note that the normal distribution is symmetric about the mean mu, and the total area under the normal curve will be 1. So to help us visualize our normal distribution, let's do a sketch of the graph for our normal distribution. So as you can see, our curve for our normal distribution will be represented in blue, and it has the shape of an upside down parabola. And as we can see, the peak of our curve. Has a dotted line going down to the bottom, and that represents our mean, which the mean mu once again is equal to 55.2 inches. And as we can see, our curve starts to slope downwards as it's going towards 65. So that's our X axis, goes from 20 on the left to 90 on the right. So using our curve to help us like visualize what's happening for our normal distribution, we need to recall and note that half of the total area lies to the left and half to the right of the mean. Therefore, we could safely say and write that P X is less than 55.2 will be equal to P x is greater than 55.2, which will be equal to 1/2. And that will mean the following integral will represent the probability that X lies between 55.2, which is the mean, and 65. So that means we can write the integral evaluated from 55.2 to 65 of 1 divided by 10.4, multiplied by the square root of 2. I multiplied by e to the power of -1/2. Multiplied by parentheses of x minus 55.2 divided by 10.4 to the power of 2 dx, and we need to note that since the total area to the right of 55.2 is 1/2. And the area between 55.2 and 65 is the integral above, that will mean as a result, the area to the right of 65 will be written as p of x is greater than 60. 5 will be equal to 1/2 minus the integral evaluated from 55.2 to 65 of 1 divided by 10.4 multiplied by the square root of 2 pi. Multiplied by e to the power of 1/2 multiplied by parentheses of x minus 55.2 divided by 10.4. To the power of 2 dx, and when we plug this expression into our calculator and we round to 3 decimal places like the problem itself asks us, we will find that it's approximately equal to 0.173, and that's it. That is our final answer. Therefore, without a doubt, the probability that next winter's snowfall will be greater than 65 inches will be about 0.173. And that's it. So looking at the problem itself, we could safely say that our final answer once again, without a doubt, will be 0.173, and that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Annual rainfall The annual rainfall in inches for San Francisco, California, is approximately a normal random variable with mean 20.11 in. and standard deviation 4.7 in. What is the probability that next year’s rainfall will exceed 17 in.?

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