The random variable x is normally distributed with the given p...

Question

The random variable x is normally distributed with the given parameters. Find each probability.

c. μ = 5.5, σ ≈ 0.08, P(5.36 < x < 5.64)

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. A variable Y is normally distributed with a mean of 20 and a standard deviation of 2. What is the probability that 18 is less than Y and Y is less than 22? Awesome. So it appears for this particular prompt, using the information that is provided to us by the prom itself, we're asked to determine what is the probability that 18 is less than Y and Y is less than 22. Awesome. So now that we know what we're ultimately trying to solve for, let's read off our multiple choice answers to see what our final answer might be. A is 0.3174. B is 0.6826, C is 0.2686, and finally, D is 0.6286. Awesome. So our first step to solve this particular problem is we need to convert to a Zsco value. So in order to convert to a Zco value, we need to recall and use the Z2 equation, which states that Z to Z score is equal to Y minus mu, which mu represents the mean divided by sigma where sigma denotes the standard deviation. So, if we're told in this case that Y is equal to 18 and we plug in all of our other known variables, we will find that Z is equal to 18 minus 20. We're told that our mean mu is equal to 20, and we're told that our standard deviation sigma is equal to 2, so we need to take that Z is equal to 18 minus 20 divided by 2, which will give us an answer of -1. Awesome. So, using our second value for Y, since we know that Y is equal to 18, and we're also told that Y. is equal to 22. So plugging in a value of 22 now into our equation for the Z score, we will find that Z is equal to 22 minus 20 divided by 2, which is going to be simply equal to 1, whereas when Y is equal to 18, we get a Z value of -1. So, what we want to know, once again, is we're trying to determine the probability P of -1 is less than Z, specifically, capital Z. Awesome, and we need to note that -1 is less than Z and Z is less than 1. So once again, -1 is less than Z and Z Z is less than 1. So now, With that in mind, our second step now is we need to recall and use our Z table, which once again a Z table should be located in your textbook, and we will determine that the probability P of capital Z. is less than 1 is equal to 0.8413, and the probability P of capital Z is less than -1 will be equal to 0.1587. So that will mean ultimately that the probability P of -1. is less than capital Z. Which is less than 1, will be equal to 0.8413 minus 0.15. 87, which when we perform that subtraction, you will find that it's equal to 0.6. 826. And that's it. That is our final answer. Hooray, we did it. So the probability of -1 is less than Z and Z is less than 1 is equal to 0.6826, which means that our final answer without a doubt has to be multiple choice answer letter. B, which is 0.6826. And that's it. We've officially solved this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

The random variable x is normally distributed with the given parameters. Find each probability.

c. μ = 5.5, σ ≈ 0.08, P(5.36 < x < 5.64)

Key Concepts

Normal Distribution

Standard Normal Distribution

Probability Calculation

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