In Exercises 27–32, the random variable x is normally distribu...

Question

In Exercises 27–32, the random variable x is normally distributed with mean mu=74 and standard deviation sigma=8. Find the indicated probability.

P(72 < x < 82)

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 random variable X follows a normal distribution with a mean mu equals 40 and a standard deviation sigma equals 4. Find the probability that X falls between 36 and 46. Awesome. So it appears for this particular problem, based on all the information that is provided to us by the problem itself, we're asked to determine what the probability that X will fall between 36 and 46. So now that we know that we're ultimately trying to figure out what this probability value is, that's our final answer 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 0.1587. B is 0.7745. C is 0.9332 and D is 0.5187. Awesome. So first off, in order to solve this problem, we need to standardize the values 36 and 46 to Z scores by recalling and using the Z score equation, which states that the Z score Z is equal to X minus mu divided by sigma. So let us quickly note that our value for X1 will be equal to 36 and our value for X2 will be equal to 46. So when we plug in our different values to solve for our two different Z scores, we will find that Z1 is equal to 36 minus 40 divided by 4. is equal to -1, and we will find that Z2 is equal to 46 minus 40 divided by 4, which will be equal to 1.5. Fantastic. So now we need to find the probabilities corresponding to these specific Z scores by using a standard normal distribution table, and you should have a standard normal distribution table in your textbook. So we will find that the probability P of Z is less than -1 will be approximately equal to 0.15. 87, and the probability P of Z is less than 1.5, will be approximately equal to 0.9332. And once again we get both of these values from looking at a standard normal distribution table. So now our next step, as we should recall, is we need to subtract the lower probability from the higher probability in order for us to determine the probability that x will fall between 36 and 46, so we need to find, or rather we need to note that the Probability P of 36 is less than X and X is less than 46 will be equal to the probability P of Z is less than 1.5 minus the probability P of Z is less than -1. And when we plug in our known variables, we will find that this is equal to 0.9332 minus 0.1587. So when we plug that into our calculator, we will find that this will be equal to when we round to four decimal places like all of our multiple choice answers. It will be equal to 0.7745. And that is our final answer. Hooray, we did it. So therefore, without a doubt, the probability that X falls between 36 and 46 will be 0.7745, and this corresponds to multiple choice answer letter B, which once again is 0.7745. 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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In Exercises 27–32, the random variable x is normally distributed with mean mu=74 and standard deviation sigma=8. Find the indicated probability.

P(72 < x < 82)

Key Concepts

Normal Distribution

Z-scores

Probability Calculation

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In Exercises 27–32, the random variable x is normally distributed with mean mu=74 and standard deviation sigma=8. Find the indicated probability.P(72 < x < 82)