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Expert: - 4/10/2009


Question
A test bank in the public domain consists of 500 questions. A random selection of 60 questions is used for certification, with a minimum passing score of 70%. We have the same test bank and use a random selection of 60 questions. What is the minimum score on a prep test to ensure that 90% of the time a person scores at least 70% on the "real" test?

Answer

Test Results
500   number of questions               
60   number on a test               
0.7   percent needed               
1.28   standard deviation needed over 70%               
The data below looks a lot better in the image attached.
It's most likely all scrunched together here.                  
number known   number not known   expected number right   percent score   variance   standard deviation   epected minimum
450   50   54   0.900   0.013   0.116   0.752
449   51   53.88   0.898   0.014   0.116   0.749
448   52   53.76   0.896   0.014   0.117   0.746
447   53   53.64   0.894   0.014   0.117   0.744
446   54   53.52   0.892   0.014   0.118   0.741
445   55   53.4   0.890   0.014   0.118   0.739
444   56   53.28   0.888   0.014   0.119   0.736
443   57   53.16   0.886   0.014   0.119   0.733
442   58   53.04   0.884   0.014   0.120   0.731
441   59   52.92   0.882   0.014   0.120   0.728
440   60   52.8   0.880   0.015   0.121   0.726
439   61   52.68   0.878   0.015   0.121   0.723
438   62   52.56   0.876   0.015   0.121   0.721
437   63   52.44   0.874   0.015   0.122   0.718
436   64   52.32   0.872   0.015   0.122   0.716
435   65   52.2   0.870   0.015   0.123   0.713
434   66   52.08   0.868   0.015   0.123   0.711
433   67   51.96   0.866   0.015   0.123   0.708
432   68   51.84   0.864   0.015   0.124   0.706
431   69   51.72   0.862   0.015   0.124   0.703
430   70   51.6   0.860   0.016   0.125   0.701
429   71   51.48   0.858   0.016   0.125   0.698

I'm not sure if the columns come out perfect,
but the way to do it is know it all hinges on the percent known.

Let n be the number of questions known.

The number right is then 60*n/500 = 3n/25.

The percent score (on the average) is the percent known * 500 = A.

The variance is n(500-n)/500.

The standard deviation is the squareroot of variance and will be sd.

The value in the expected minimum must be A-1.28*sd,
since we want the chance to be greater than 90%.

The table above starts off with 90% of the questions being known,
and drops down by one question at a time.  The expected minimum
is determined for each of the number of scores known.

From this data, you can see that the person needs to know at least 430 question out of the 500 (86%) to have a 90% chance of getting at least 70% right out of 60 questions chosen.  

Scott A Wilson

Expertise

I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can even tell you it takes me over 2,000 steps to go a mile, but is that relevant?

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Experience in the area; I have tutored people in the above areas of mathematics for almost two years in AllExperts.com. I have tutored people here and there in mathematics since before I received a BS degree almost 25 years ago. In just two more years, I received an MS degree as well, but more on that later. I tutored at OSU in the math center for all six years I was there. Most students offering assistance were juniors, seniors, or graduate students. I was allowed to tutor as a freshman. I tutored at Mathnasium for well over a year. I worked at The Boeing Company for over 5 years. I received an MS degreee in Mathematics from Oregon State Univeristy. The classes I took were over 100 hours of upper division credits in mathematical courses such as calculus, statistics, probabilty, linear algrebra, powers, linear regression, matrices, and more. I graduated with honors in both my BS and MS degrees. Past/Present Clients: College Students at Oregon State University, various math people since college, over 7,500 people on the PC from the US and rest the world.

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I earned high honors in both my BS degree and MS degree from Oregon State. I was in near the top in most of my classes. In several classes in mathematics, I was first. In a class of over 100 students, I was always one of the first ones to complete the test. I graduated with well over 50 credits in upper division mathematics.

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