# Need major help in this statistics problem

#### sabbycat

##### New Member
When looking at my text book. To answer the A, B, and C. at the bottom, we had to refer to our previous question in our book to help us with the problem at hand. I went and got the solution for the previous problem. Marked 23, and now need to use that to help with number 24. Hope this is understandable # 23

Inverse Normal Distribution: SAT and ACT Scores:. What is the probability that a randomly selected high school senior's score on the mathematics part of the SAT will be?

(a) more that 675? P(x > 675) = P(z > 1.75) = 0.0401

(b) less than 450? P(x < 450) = P(z < 0.50) = 0.3085

(c) between 450 and 675? P(450 ≤ x ≤ 675) = P(-0.50 ≤ z ≤ 1.75) = 0.6514

What is the probability that a randomly selected high school senior's score on the mathematics part of the ACT will be

# 24

(d) more than 28? P(x > 28) = P(z > 1.67) = 0.0475

(e) more than 12? P(x > 12) = P(z > -1.00) = 0.8413

(f) between 12 and 28? P(12 ≤ x ≤ 28) = P(-1.00 ≤ z ≤ 1.67) = 0.7938

When answering the questions below refer to the SAT and Act information from above.

a) Suppose that an engineering school honors program will accept only high school seniors with mathematics SAT or ACT score in the top 10%. What is minimum SAT score in mathematics for this program? What is the minimum ACT scores in mathematic for this program?

b) Suppose that an engineering school will accept only high school seniors with mathematics SAT or ACT score in the top 20%. What is the minimum SAT score in mathematics for this program? What is the minimum ACT score in mathematics for this program?

c) Suppose that an engineering school will accept only high school seniors with mathematics SAT or ACT score in the top 60%. What is the minimum SAT score in mathematics for this program? What is the minimum ACT score in mathematics for this program?

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#### sabbycat

##### New Member
Just to let ya know, I have no clue on even how to start on this.

#### jamesmartinn

##### Member
For these kinds of questions, you must find the Z-score that separates the upper x% from the lower y%

So for the first one, you want the z-score that separates the upper 10% from the lower 90% of the distribution. You need to consult your normal distribution table and look under the column (area in the tail, since its 'upper 10') till you find the value closest to 10% or .10, which is z = +1.28

Now, to find the score that separates the upper 10%, you use the formula:

X = mean + zscore * standard deviation

X = mean of SAT + 1.28 * std-dev of SAT

And that should give you the score. Follow this outline with the rest of the remaining questions and you should be okay.

#### sabbycat

##### New Member
Ok thought that I should add this, forgot to put it in my post. For that first part where I already did it. The mean score for the SAT was 500, with a standard Deviation of 100. Now for the ACT the mean was 18 with a standard deviation of 6

now you said to use this formula.

x =mean + zscore *standard deviation
x = mean of SAT + 1.28 (which you said was the z-score) * standard deviation.

now according to what i added above, it would be

x = 500 (mean of SAT) + 1.28 (z-score) * 100 (standard deviation) which would give me = 50128 (not sure if this is right)

so then when I do the ACT one, i would do it this way?

x = 18 (mean of ACT) + 1.28 (z-score) * 6 (standard deviation) which would give me = 115.68 ??

are these the minimum ACT and SAT scores in the program?

#### jamesmartinn

##### Member
You have the right idea, but I think your computations are a bit off.

for the SAT:

X = mean(SAT) + z score * stddev(SAT)
= 500 + 1.28*100
= 500 + 128
= 628

for the ACT:

X = mean(ACT) + z score *stddev(ACT)
= 18 + 1.28*6
= 25.68

These are the minimum scores for part A. Follow the same general outline for parts B and C, but of course using the appropriate z scores. Note the formula I suggested to you is the regular formula for finding the Z score, just rearranged so that we can solve for X.

#### sabbycat

##### New Member
You have the right idea, but I think your computations are a bit off.

for the SAT:

X = mean(SAT) + z score * stddev(SAT)
= 500 + 1.28*100
= 500 + 128
= 628

for the ACT:

X = mean(ACT) + z score *stddev(ACT)
= 18 + 1.28*6
= 25.68

These are the minimum scores for part A. Follow the same general outline for parts B and C, but of course using the appropriate z scores. Note the formula I suggested to you is the regular formula for finding the Z score, just rearranged so that we can solve for X.
doh hold on ,i see where i messed up

#### sabbycat

##### New Member
oh one more thing. How exactly did you figure out that the z-score was 1.28 for the 10% one? Is there a formula for which i can use to figure out say the next one, of 20%

#### jamesmartinn

##### Member
There's no formula; well there is but it involves knowledge of integral calculus.

You should have a normal distribution table in the appendix of your textbook, if not there are many available online. What I did was find the closet value of .1000 that was located in the "tail of the distribution" and then used its corresponding Z-Score.

Do the same with 20%. Look up the closest value to .2000 in the "tail" column of the distribution and use its corresponding z-score for your calculations.

#### jamesmartinn

##### Member
What you need to do is look up the closest value of .1000 in the body of the table. This value is .1003, and if you read across to the left for the Z-score, you see its 1.2 to the left, and 0.08 at the top. This is 1.28.

The normal distribution is symmetrical, and the Z-values here are negative. This isn't an issue, however. Since you are looking for the UPPER 10%, you look for the closest value to .1000, which is .1003, so this corresponds to +1.28. If you were looking for the LOWER 10%, you would still look for the closest value of .1000, which would still be .1003, but your z value would equal -1.28.

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#### sabbycat

##### New Member
Oh ok, lol I see now. I also looked at the one in the back of my book and your correct, mine also says -1.2 (which since you said that I am looking for the upper scores, it would just be 1.2.. So when i do my calulations, should i just use the 1.2???

#### sabbycat

##### New Member
ok will be back, going to do all the calculations and see if i am right, hope i can do it #### sabbycat

##### New Member
Ok how is this:

for the SAT: for 10%

X = mean(SAT) + z score * stddev(SAT)
= 500 + 1.28*100
= 500 + 128
= 628

for the ACT:

X = mean(ACT) + z score *stddev(ACT)
= 18 + 1.28*6
= 25.68

For 20%

X = mean(SAT) + z score * stddev(SAT)
= 500 + 0.84*100
= 500 + 84
= 584

for the ACT:

X = mean(ACT) + z score *stddev(ACT)
= 18 + 0.84*6
= 23.04

For 60% (the two closest i find is .5987 and .6026 (not sure which one I would use. But now I know that i look on the left and it would be 0.2 and depending on which number that i gave you, would determine if i add .05 or .06 to it.

X = mean(SAT) + z score * stddev(SAT)
= 500 + 0.5*100
= 500 + 50
= 550

for the ACT:

X = mean(ACT) + z score *stddev(ACT)
= 18 + 0.5*6
= 21

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#### jamesmartinn

##### Member
Sorry, I made a slight edit to my last post, please read it and I believe you'll need to re-calculate the scores.

#### sabbycat

##### New Member
Yea i re edited mine, i see what you mean. As for the 60% i wasnt sure which number i should go with that was closest to .6000

by the way, I am so grateful for your help, means a whole lot #### jamesmartinn

##### Member
np, and the calculations look good. As for the 60%, go with the .5987; its closer to .6000 than .6026.

Best of luck

#### sabbycat

##### New Member
Thats what I thought, but just wanted to make sure. Thank you for all your help. lol i am sure i will be back, before my statistics class ends

#### jamesmartinn

##### Member
Not a problem, I'll be glad to help as will other members of the board, as long as you show some of the work, which you've been doing! gj