Question 360290: If the results on a nationally administered university entry exam are normally distributed with a mean of 45 points and a standard deviation of 4 points, determine the following:
(a) Describe the graph of this distribution (if you can do so, produce an electronic sketch of the graph to the right, otherwise adequately describe the distribution graph through its shape and horizontal scale values.)
(b)Find the z-score for a single exam that had 35 points. Then find the z-score for one with 45 points.
(c)If x represents a possible point-score on the exam, find P(x < 41).
(d)Find P(40 < x < 50) and give an interpretation of this value.
(e)Suppose a certain university requires one to score in the top 40% of all such scores to be admitted. What is the minimum number of points one must score on this exam for admission?
Answer by neatmath(302)   (Show Source):
You can put this solution on YOUR website! Okay, so since it is normally distributed, that means it has a smooth bell-shaped curve, and it would be drawn as any other normal curve.
a.) Specifically, since the mean is 45, the midpoint value of the curve would be 45, or where the mound-shape is highest. Then all you need to do is use the standard deviation to make several intervals on this graph.
The first two points you would draw a vertical line on are 41 and 49, representing 1 SD away from the mean. The second two points with a vertical ine would be 37 and 53, representing two SD away from the mean. The last two points with a vertical line would be 33 and 57, representing 3 SD away from the mean.
b.)The z-score is defined as (x-xbar)/s where xbar is the mean, and s is the standard deviation.
Thus, for a score of 35, the z-score would be 
For a score of 45, the z-score would be 
c.) Find the p(x<41)
Since we can see that 41 is 1 standard deviation away from the mean, we can then conclude that the probability that an exam score will be less than 41 will be 16%. This is found by adding up all the probabilities below 41, and that would be 2.5% + 13.5%
d.) Find P(40 < x < 50)
We must first find the corresponding z-scores for each value.
For 40, the z-score is 
For 50, the z-score is 
So now we are looking for the area under the normal curve corresponding to P(-1.25 < z < 1.25)
We know for a fact that this probability should be greater than 68% because both 40 and 50 are more than 1 standard deviation away from the mean.
P(-1.25 < z < 1.25) is approximately equal to 78.87% which you can obtain from a z-score table or z-score calculator.
e.) Now you need to know which exam score corresponds with the top 40% of all exam scores.
In order to do this, you have to recognize which corresponding z-score we are looking for. Basically, I need the z-score that corresponds to 60% below the z-score, and 40% above the z-score.
With the table I am looking at, that means I need the right-tailed z-score that corresponds to 10% greater than the mean, because then I will have 60% to the left of the z-score, and 40% to the right of the z-score, which is exactly what we are looking for.
Consulting the z-score tables or calculator, I find that a z-score of 0.25 will put me right around the mark I am looking for. Now, I just need to convert the z-score of 0.25 to a corresponding exam score and we are all set. So,
So this means if I want to have an exam score in the top 40% of all the exam scores so that I can gain admission to the school, I need to score a 46 or better!!!
I hope this helps!
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