Curving Test Scores A professor gives a test and the scores ar...

Question

Curving Test Scores A professor gives a test and the scores are normally distributed with a mean of 60 and a standard deviation of 12. She plans to curve the scores.

c. If the grades are curved so that grades of B are given to scores above the bottom 70% and below the top 10%, find the numerical limits for a grade of B.

Accepted Answer

Hello there. Today we're going to 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 company evaluates its employees using a performance test that is normally distributed with a mean score of 75 and a standard deviation of 8. As part of a performance-based incentive, the HR department plans to assign performance ratings. If the ratings of exceeds expectations are given to employees whose scores are above the bottom 67%, but below the top 14%. Find the numerical score range for an exceeds expectations rating. Awesome. So it appears for this particular problem we're asked to ultimately solve where we're trying to figure out that when for the condition where if the ratings of exceeds expectations are given to employees whose scores are above the bottom 67% but are also below the top 14%, whereas to ultimate. Find the numerical score range or an exceeds expectation rating. So we're trying to find the numerical score range for an exceeds expectation rating, given these conditions set to us by the prom itself. So with that said, let's read off our multiple choice answers to see what our final answer might be. A is 792 84, B is 76 to 84, C is 79 to 87, and finally D is 76 to 87. Awesome. So first off, in order to find the score range for the exceeds expectation rating. We will need to recall and use the properties of normal distribution. We need to find the Z scores that correspond to the bottom 67% and the top 14%, and then we must convert those Z scores to actual scores using the mean and standard deviation values. So, with that said, to find the Z score for the bottom 67%. We need to recall and note that the bottom 67%. Corresponds to a percentile of 67 or a cumulative probability of 0.67. So let's make a note of that in blue. So, 0.67, so that's how you write 67% as a decimal. So looking up 0.67 in the standard normal distribution table, which that should be in your textbook, we will find that Z subscript 0.67, which once again, this is our Z score, it's going to be equal to when we look at our table, 0.44. So now, at this point, we need to convert the Z score from the bottom 67% to an actual score value. So in order to do that, we need to recall and use the Z score formula, which states that Z is equal to X. Minus mu. All divided by sigma. Awesome. So we need to note that capital X is the actual score, mu is the mean, and sigma is the standard deviation. So let's make it, let's make a note here off to the side. Let's take a moment to note that mu, the mean is equal to 75. Let us also note that sigma is going to be equal to 8, as we're told by the problem itself. So now, at this stage, we need to recall and note the formula to help us convert a Z score to an actual score, and this equation states that X is equal to Z multiplied by sigma. Plus mu. So now we need to substitute in our known values, and when we do that, we will find that capital X is equal to 0.44 multiplied by 8 plus our value of mu, which is 75, which when we plug this into our calculator, we will find that it's equal to 78.52, which would mean that X is approximately equal to 79 when we round to the nearest whole number. So that is one of our final answers that we're trying to solve for because once again, we're trying to find the numerical score range. So this is one of our score values, so I'll underline it in green. So 79 is one of our numerical score range values. So now we need to switch gears and we need to find the Z score for the top 14%. So the top 14%, as we should recall, corresponds to a percentile of 86% because, let us note and we'll write this in blue, because 100% minus 14% is equal to 86%. And we need to note, or it will have a cumulative property of, when we write 86% as a decimal, 0.86. Fantastic. So when we look up a value of 0.86 in the standard normal distribution table, which once again should be located in your textbook, the Z score will be approximately equal to 1.08. And once again, this is our Z score for Z subscript 0.86. Fantastic. And then once again, that's when you look it up in your table in your textbook. So now our next step is we need to convert this Z score value for the top 14% to an actual actual score, so we need to take capital X is equal to 1.08 multiplied by 8. Once again, we're plugging in all of our known variables into our equation. So 0.8 multiplied by 8 plus 75, which will be equal to we plug that into our calculator. 83.64, which will mean that X, our value of capital X will be approximately equal to 84 when we round to the nearest whole number, and that is our second and final number in our numerical score range. So therefore, without a doubt, the score range for an exceeds expectation rating is from drumroll, 79 to 84, and that's it. That is our final answer. Hooray, we did it, and this corresponds to multiple choice answer letter A, which once again is 792 84. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Normal Distribution

Percentiles

Z-scores

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