In Exercises 37–42, use the Standard Normal Table or technolog...

Question

In Exercises 37–42, use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area or percentile.

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Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says determine the Z score that corresponds to the cumulative area of 0.85 under the standard normal curve. And we're going 4 possible choices as our answers. For choice A, we have Z is approximately equal to 1.031. For choice B, Z is approximately equal to 1.037. For choice C, we have Z is approximately equal to 1.134, and for choice D, Z is approximately equal to 1.139. So, we need to determine the Z score that corresponds to a cumulative area of 0.85. And since our area here is actually greater than 0.5, we're gonna be using our table with our positive Z scores. Now, when we look up this cumulative area in our tables, we see that it falls between two different Z score values. So, the first value is going to be Z is equal to 1.03, and that has a cumulative area of 0.8. 485. And the other value is going to be Z is equal to 1.04, and this has a cumulative area of 0.8508. So, in order to find RZ score, we'll actually have to use interpolation. And recall for interpolation, we're going to take the difference, the ratio of our difference in Z values. So, on the left hand side of the equation, we'll have in the numerator, we'll have the quantity of Z. -1.03 in quantity, and that's going to be divided by the quantity of 1.04 minus 1.03, and this is going to be equal to the ratio in the difference of the areas. So, in the numerator, we'll have the quantity of 0.85, since that is the cumulative area for which we're looking for, and then we'll have minus the area for the 1.03. Which is going to be 0.8485. And that whole quantity is going to be divided by the difference in our areas for our two Z scores that we found at the table, which is going to be 0.8508. Minus 0.8485. So, we actually need to solve this equation for Z. So to do that, first thing we'll need to do is multiply both sides of the equation by the denominator on the left hand side. So we'll have Z minus 1.03 is equal to. And here we'll have our fractions, we'll have the quantity of 0.85 minus 0.8485, in quantity, divided by. The quantity of 0.8508 minus 0.8485 in quantity, and that fraction is going to be multiplied by our original denominator on the left hand side, which is going to be the quantity of 1.04 minus 1.03. In quantity. So now to software Z, we just need to add 1.03 to both sides of the equation, so we'll get Z. is equal to the quantity of 0.85 minus 0.8485. In quantity, divided by the quantity of 0.8508 minus 0.8485 in quantity. And then that gets multiplied by the quantity of 1.04 minus 1.03 in quantity, then we'll have plus. 1.03. And so evaluating this. This expression, we'll get that Z is going to be approximately equal to 1.037. So here we have rounded to three decimal places. So that 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 B. So I hope that this explanation has been useful, and I'll see you in the next video.

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In Exercises 37–42, use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area or percentile.

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Key Concepts

Z-Score

Standard Normal Distribution

Cumulative Area

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In Exercises 37–42, use the Standard Normal Table or technology to find the z-score that corresponds to the cumulative area or percentile. 0.1