In a standardized IQ test, scores are normally distributed, wi...

Question

In a standardized IQ test, scores are normally distributed, with a mean score of 100 and a standardized deviation of 15. Use this information in Exercises 3–10. (Adapted from 123test)

What is the highest score that would still place a person in the bottom 10% of the scores?

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 university entrance exam has scores that are normally distributed with a mean of 70 and a standard deviation of 8. What is the highest score that would still place an applicant in the bottom 15% of all scores? Awesome. So it appears for this particular prom, we're ultimately trying to determine what is the highest score that would still place an applicant in the bottom 15% of all the scores. So with that in mind, 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 72, B is 68, C is 54, and D is 61. Awesome. So first off, in order to find the highest score that places an applicant in the bottom 15%, We need to perform two steps. First, we need to find the Z score that corresponds to the bottom 15% of the standard normal distribution, and our second step is we need to convert that Z score back to the original exam score using the given mean and standard deviation. So with that in mind, let us do the first step. So once again, we're trying to find the Z score for the bottom 15%. So we're trying to look for the Z score where the probability P, so P of Z is less than or equal to lowercase z. So the probability of capital Z is less than or equal to lowercase z. And to make sure we know that it's a capital Z, I'll add little bars to it. So once again, the probability of capital Z is less than or equal to lowercase z is all equal to 0.15. So, using a Z table or R calculator, which you should have a Z table in your textbook, you will find that Z lowercase z, is approximately equal to -1.04. Awesome. So now our second step is we need to convert the Z score to an exam score. So with that in mind, we need to recall that Z lowercase z, is equal to x minus m, all divided by sigma, and when we rearrange this expression to isolate and solve for X, we will find that X is equal to Z multiplied by sigma plus mu. Fantastic. So what do all these variables mean? Well, obviously Z is our Z score. X is our value that we're ultimately trying to solve for because we're asked to determine the highest score that would still place an applicant in the bottom 15% of all scores, and new is the mean. And sigma is the standard deviation. So with that in mind, we now need to plug in all of our known variables to solve for X. So X is equal to -1.04 multiplied by 8. Plus, our mean, which we're told is 70. So when we plug in that expression into our calculator, we will find that it's equal to 61.68 when we round to two decimal places. And we need to recall that since the score must be an integer value, and it needs to ensure that the applicant is still in the bottom 15%, we need to round down. And when we round down, we will determine that it will be our final answer will be approximately equal to 61. So that means our final answer, when we round down, has to be equal to 61, and this corresponds to multiple choice answer letter D, which once again is 61. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

In a standardized IQ test, scores are normally distributed, with a mean score of 100 and a standardized deviation of 15. Use this information in Exercises 3–10. (Adapted from 123test)
What is the highest score that would still place a person in the bottom 10% of the scores?

Key Concepts

Normal Distribution

Percentiles

Z-scores

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