Suppose the scores earned on Professor McArthur’s third statis...

Question

Suppose the scores earned on Professor McArthur’s third statistics exam are normally distributed with mean 64 and standard deviation 8. Professor McArthur wants to curve the exam scores as follows: The top 6% get an A, the next 14% get a B, the middle 60% get a C, the bottom 6% fail, and the rest earn a D. Any student who can determine these cut-offs earns five bonus points. Determine the cut-offs for Professor McArthur.

Accepted Answer

Hey everybody, glad to have you back. Let's take a look at this next problem. A statistics test in Professor Gomez's class is normally distributed with a mean of 58 and a standard deviation of 7. Grades are assigned as follows. Top 5% get an A, next 15% get a B, Middle 60% get a C. Bottom 5% get an F, and the rest are in a D. What are the score cut-offs for each rate? So, let's think about what this distribution of grades looks like to think about how we calculate this. So, it's a normal distribution, we'll draw that curve. It's a little better there. And the mean, right there in the middle, is going to be 58. So I'm going to write that 58 up top just to not be quite so confusing. And then we have the probabilities representing, you know, each grade. So it says top 5% get an A. So, let's draw a line. The line will represent the Z value that corresponds to an A. So, we'll put Z. Sub A And then we know that the Area to the right of that Z sub A under that curve. Is 5%, the top 5%. On the test received an A. Then the next 15% get a B. So, 5% plus 15% represents 20%. So, now, not the B grade, but Z sub B, that line. Represents the Z value corresponding to a probability. That represents Or I should say, excuse me, a probability of 20%. Now, again, 20% is the whole area under the curve representing the B and A score, but the line that delineates the 20% represents that Z value corresponding to the B grade. A little bit confusing. This is the middle 60% get a C, so. We have middle 60%, we already know the top 20% get A's and B's, then add 60%, 80%. So, the bottom 20% get D's and Fs, and that makes sense. Bottom 5% get an F. So, we'll draw a line for Z sub F. And then highlight in yellow here, this yellow area. To the left of that Z F. That area represents 5%. And then it says the rest are a D since. The lower 20% earned Ds and F's, that means that, which makes sense symmetrically to the B, the next 15%, so corresponding to An area under the to the left of the Z sub D. Of 20%. And we'll draw red stripes here for that area. Now recall what a Z value signifies. A z value. Oops, let's write that in blue a Z value of 1. Represents. The point at which you're one standard deviation above the mean. So to convert from these Z values to a score. We would say that our score of X will be equal to the mean score. And then we add The Z value we calculate multiplied by. This standard deviation. Since the values are essentially units of standard deviation. So let's go ahead and calculate those Z-values. Now recall that. On a Z table, the probability that corresponds is a probability left of our Z value. So, for Z sub A, Since we're looking for the shaded blue area to the right of that Z sub A value, we need to look up the probability. equal to the area to the left of the curve, which will be 0.95, 1 minus 0.05. And if you look that up on a Z table, that's going to give you a Z value of approximately 1.65. Now, Z sub B, the probability again, we need to look at the probability to the left. Of that ZB value. Since the area to the right, my blue striped area is 20%, I need to look at where the probability is 0.80. And when I looked that up. That gives me approximately. Oops, and excuse me, I, I have made my made my Z value equal to my P value, but that was very confusing, so. The semicolon there, so I would say the ZA corresponding to that probability is approximately equal to. 1.645. And the sub B is approximately equal to. 0.841. Now, when we get to Z sub D and Zub F, This is pretty straightforward. We don't have to do any more calculations because recall, we're looking at standard deviations away from the mean. So Z sub D is just going to be a negative Z value. And it's going to be, you know, in equidistant from the mean, as Z sub B. So Z sub D, our Z value is going to be approximately equal to negative 0.841. And Z F Will be the negative of the Z value for A. Again, equidistant from the mean just on the other side. So the Z value for F will be approximately negative 1.645. So last of all, we need to convert these to scores, given that formula. So we have X sub A. So the score to get A will equal the mean of 58. We remind ourselves that mu equals 58 and standard deviation sigma equals 7 from our problem. So, for X sub A, we have that equals 58 plus Z sub A is 1.645, and we multiply that by 7. And that's going to give us a value for X sub A of. 69. 0.5. Rounded and approximate value. And then X B will equal 58 plus our Z value of 0.841, multiplied by 7. Which will give us approximately 63 9 X D. C equal to 58 plus, and then our Z value of negative, 0.841, multiplied by 7. Will give us approximately 52 0.1 and we double-check, that's below the mean, as we would expect, to make sure we haven't lost a negative sign there or anything.X F is equal to 58 plus our Z value of -1.645, multiplied by 7. It's going to give us approximately 46.5. And then of course, we know that a C score would be in between a B and D score. So our To get a grade of an A. You need to get above a 69.5 on the test. A bee Is going to be a score of 63.9. 269.5 A C will be, when you start at the lower bound of D from 52.1 to 63.9. A D will be And again referring to the graph, you need to look from the lower point of F, so 46.5 through 52.1. My 5 looks a little weird there. To 52.1. And the f will be below. 46.5. So, there we go. We were given the um the the information that the grades are normally distributed with a mean of 58, the standard deviation of 7. Given the assignment of grades, the top 5% get an A, next 15% get a B, middle 60% get a C, bottom 5% get an F, rest or a D, what are the score cutoffs for each grade? And that is that an A is above 69.5, B from 63.9 to 69.5, C from 52.1 to 63.9, D from 46.5 to 52.1, and F below 46.5. See you in the next video.

Key Concepts

Normal Distribution

Z-Scores and Standardization

Percentiles and Cut-Off Scores

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