Finding a z-Score Given an Area In Exercises 23–30, find the i...

Question

Finding a z-Score Given an Area In Exercises 23–30, find the indicated z-score.

Find the positive z-score for which 12% of the distribution’s area lies between and z.

Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says determine the positive Z score, where the area between minus Z and Z is equal to 39% of the total distribution, and we're given 4 possible choices as our answers. For choice A, we have 1.15 for choice B, we have 0.15 for choice C, we have 1.51, and for choice D, we have 0.51. Now we have to determine the positive Z score where the area between YS Z and ZS equal to 39%. And so, the first thing we want to do is actually determine the area that is underneath the tails of our distribution. Now, since the total area of our distribution is equal to 1, we're gonna have 1 minus 0.39, where the 0.39 comes from converting the 39% into a decimal by dividing by 100. So since 39%, which is going to be 0.39% of our area. is in between our two Z values, what's going to be left outside in the tails is going to be 1 minus that value. So, when I do this calculation, we get a value of 0.61. So this is going to be the area underneath both of our tails in our distribution, but we're really only interested in the area underneath one of the tails. So since our uh normal distribution is symmetric, I can just divide this value by two. So the area under one of the tails is going to be 0.61. Divided by 2, which is going to be equal to 0.305. And so this is actually going to be the area underneath the tail to the right of our positive Z score. However, when we look up Z scores in our table, we actually need the area to the left of our Z score. So, again, since our total area is going to be equal to 1, we'll have 1 minus 0.305 in order to get the area to the left of our Z score. And when we calculate this, this is going to be equal to 0.695. And so now when we look up this area in our tables, we see that it corresponds to a Z value that is approximately equal to 0.51. And so this is the answer that we are looking for for this problem. And if we compare our answer here to our answer choices, we see that the correct answer choice corresponds to answer choice D. So, I hope that this explanation has been useful, and I'll see you in the next video.

Finding a z-Score Given an Area In Exercises 23–30, find the indicated z-score.

Find the positive z-score for which 12% of the distribution’s area lies between and z.

Key Concepts

Z-Score

Standard Normal Distribution

Area Under the Curve

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