Help on Prob Dist Question? Have I done it correctly?

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madanalyst

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this question is from essential econometrics 3rd edition by Damodar N Gujarati Chp 4 (question 4.12)
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Question
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The grade point average in an econometrics test was normally distributed with a mean of 75. In a sample of 10 percent of students it was found that the grade point average was greater than 80. Can you tell what the stand deviation of the grade point average was?
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My solution
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z = (x-u)/sd
sd = (x-u)/z
as 10% of the population is > 80 so the corresponding z value is 1.285.
putting the values in the above sd eq
sd = 3.89
 
come on guys ….. i will appreciate your help …
Joey anyone plz???
 
I don’t even understand the question.
“The grade point average in an econometrics test was normally distributed with a mean of 75.” <- Got that
“In a sample of 10 percent of students it was found that the grade point average was greater than 80.” So the sample mean of 10 students is normal with mean 75 and std dev = sigma/sqrt(10), so I can give you a probability of drawing such a sample for any sigma
P(drawing sample) = P(Z > (80 - 75)/(sigma/sqrt(10)))
But you drew the sample, which might be a probability 1/100000000000 event so the numbers don’t help us here.
 
The key is that you are told that the distribution is normal. That means that the CDF should be equal to 0.9 at a value of 80%, and if you use =normsinv(.9) in excel, that works out to be 1.2816. Thus, mean +/- 1.2816 standard deviations should leave 10% in the right tail. To you, that means 75 + 1.28156(stdev)= 80, so your stdev = 5/1.28156 = 3.9 which is how you computed it.
 
Except it didn’t say that the highest 10% scored greater than 80. It said a sample of 10% scored greater than 80. Big difference.
Edit: should be “averaged greater than 80”
 
“Can you tell what the standard deviation was?”
No, not with that info. I saw this post and figured that maybe a of the more math specialist types could find an answer. But I don’t see how you would do it.
Now, if you took a whole bunch of samples, and you know that 10% of those samples scored above 80, you might get somewhere, if you know the sample size. But that wouldn’t depend on the population being normal.
 
I am confused with the same logic that you said above but if question would have specified that the top 10% scored >80 it would haven been very easy to solve it. I have taken that assumption.
I think that is the assumption you have to take. I can see no other solution. If somebody have the solution manual of the book, can he/she give me the correct answer?
 
I think the fact that it asks “Can you tell what the stand deviation of the grade point average was?” is indicative of the fact that the answer is “No,” for reasons JDV stated in his post.
If it were possible to answer, the question would have asked “What is the standard deviation?” or “Can you tell what the stand deviation of the grade point average was, and if so, what is it?”
It’s a conceptual question, not a math question. They just didn’t format it in the CFA Yes/Yes Yes/No No/Yes No/No. ;-)
 
bchadwick your solution makes sense here … It was a tricky one
 
Hold on now, I have been thinking of this. If you have a normal distribution with a mean of 75, it is quite possible, based on the value of the standard deviation, that you can take a random sample of 10% of the students and have a mean above 80. It should be possible to figure the bounds on a standard deviation from this.
Let’s say that I have 100 students, and I use matlab (or excel) to generate random normal variables with mean 75, stdev 5: I just did this and got the following samples (where I have sorted the results in descending order). It’s quite possible to take 10 random samples and get a mean of 80. The mean of the top ten, for example, is 83ish. So I would approach it by finding the minimum standard deviation that would give you a mean of the top 10% = 80. Yes, I know it’s a random sample and you wouldn’t expect to pick the top ten as your sample, but this would be a minimum bound on the standard deviation. For example a standard deviation of 3 would not give you this result, and just iterating a bit says that 3.5 is pretty close.
Datapoint Grades
1 86.54643743 Mean 74.60918345
2 85.68154211 Stdev 4.784512826
3 84.31700307
4 83.85050446
5 82.6050662
6 81.39042073
7 81.38226237
8 81.34890924
9 81.18277826
10 81.13723995
11 81.06722247
12 80.36343134
13 80.32059574
14 80.27824194
15 79.95057436
16 79.72099863
17 79.56570409
18 79.44586309
19 78.40219292
20 78.34077517
21 78.17637067
22 78.1017507
23 77.9346928
24 77.75592356
25 77.6231934
26 77.49912834
27 77.42748854
28 77.40067911
29 77.31024006
30 76.98287659
31 76.89611811
32 76.62772992
33 76.58267907
34 76.40440244
35 76.3083123
36 76.30404199
37 76.2154735
38 76.0944956
39 75.86832834
40 75.51712223
41 75.42995297
42 75.35686432
43 75.27970339
44 75.0486708
45 75.03762243
46 74.99591485
47 74.97497463
48 74.94357219
49 74.94106338
50 74.93411664
51 74.80780618
52 74.60839402
53 74.43388005
54 74.41940753
55 74.33432775
56 74.08959066
57 73.77306852
58 73.75281858
59 73.74396313
60 73.71191442
61 73.67993323
62 73.62666507
63 73.6189107
64 73.39497654
65 73.26408405
66 72.99166633
67 72.53456041
68 72.38720349
69 72.29395335
70 72.26091927
71 72.09867999
72 71.99293937
73 71.84360172
74 71.77660542
75 71.518976
76 71.48222869
77 71.47849136
78 71.43957274
79 71.08553478
80 70.96175435
81 70.29313903
82 69.90931392
83 69.89429157
84 69.8551245
85 69.50079773
86 69.46465053
87 69.40480212
88 69.12719859
89 68.84181733
90 68.71704946
91 68.64749898
92 68.33234635
93 67.95235755
94 67.41230434
95 66.68196774
96 66.67994562
97 64.97718339
98 64.39786656
99 63.37394436
100 63.17705076
 
This is a classic example of people making far too many assumptions about normal distributions, and coming up with methods out of thin air to generate some fact about a data set. The question says that a sample of 10% of the students was taken. What if the population was 10? Then the sample has one value, which is greater than 80, and would also be the mean of that sample. Does the knowledge that one out of 10 tests was greater than 80 tell you what the std. dev. for the population is? NO!!
 
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