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.

P85

Accepted Answer

Hi, everyone. Let's take a look at this practice problem. This problem says define the Z score that corresponds to the 70th percentile P 70 of the standard normal distribution. And here we're getting 4 possible choices as our answers. For choice A, we have Z is approximately equal to 0.423. For choice B, we have Z as approximately equal to 0.428. For choice C, we have Z is approximately equal to 0.524, and for choice D, we have Z is approximately equal to 0.528. So here we're asked to find the C score that corresponds to the 70th percentile P70. As you recall that the 70th percentile is value below which 70% of the observations may be found. So, when we convert that 70% into a decimal by divided by 100, We see that that's going to be equal to 0.70. So, using the table for standard normal distribution, we're going to look for a Z value such that the area to the left of Z is 0.70. And when we look through our table, we see that 0.70 falls in between two Z values. The first value is going to be Z equal to 0.52. And this corresponds to a cumulative area of 0.6985. And the other Z value, it's going to be Z is equal to 0.53. And this corresponds to a cumulative area of 0.7019. And so, in order to find our Z value, we're actually going to find it using interpolation. So, recall our formula for interpolation. On the left hand side, we're gonna have the ratio of the differences between our Z values. So, in the numerator, we're gonna have Z. Minus 0.52. And Z here is the Z value that we're looking for. And so we're going to divide that quantity. By the quantity of the difference of the Z values that we found at the table. So we'll have 0.53 minus 0.52. And then this is going to be equal to, and on the right hand side, we're gonna have the ratio of the differences of the areas. So, in the numerator, we're going to have the quantity of 0.70, since that is the area that we're actually looking for for the Z value minus the area for the Z value of 0.52, which is going to be 0.6985, and that quantity is divided by the quantity of the difference of the areas of the Z values that we found in the table, that it's going to be 0.7019. Minus 0.6985. So now what we need to do is to solve this equation for Z. So, the first thing to do is to multiply both sides of the equation by the denominator on the left hand side. So doing so, we'll have Z minus 0.52. is equal to The quantity of 0.70 minus 0.6985 in quantity divided by the quantity of 0.7019 minus 0.6985 in quantity. And then that fraction is going to be multiplied by the quantity of 0.53 minus 0.52 in quantity. So now to solve for Z, we just need to add 0.52 to both sides. In doing so, we'll get that Z is equal to the quantity of 0.70 minus 0. 6985 in quantity, divided by the quantity of 0.7019 minus 0.6985 in quantity, multiplied by the quantity of 0.53 minus 0.52 in quantity, plus 0.52. And so when we evaluate this expression, this is going to be approximately equal to. 0.524. So here we've just rounded to 3 decimal places. And so, this is the answer that we're looking for for this problem. And when we compare our answer here to our answer choices, we see that the correct answer choice corresponds to answer choice C. So, I hope that this explanation has been useful, and I'll see you in the next video.

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

P85

Key Concepts

Z-Score

Standard Normal Distribution

Cumulative Area

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