If a test has a mean of 500 and standard deviation of 100(assuming a normal distribution), with N = 1000, about how many individuals score between 300 and 700

Question

If a test has a mean of 500 and standard deviation of 100(assuming a normal distribution), with N = 1000, about how many individuals score between 300 and 700

If a test has a mean of 500 and standard deviation of 100(assuming a normal distribution), with N = 1000, about how many individuals score between 300 and 700

Answer 1

Hello Jenn,

First, get the z-score for 300

z = x - mean/standard deviation, correct? 

Therefore: 

z = 300 - 500/100 

z = -200/100

z = -2.00 

Now, get the z-score for 700, okay

z = x - mean/standard deviation 

z = 700 - 500/100

z = 200/100 

z = 2.00 

Therefore, your two z-scores are -2.00 and +2.00  You should notice by now that this means 2 standard deviations above and below the mean, correct? All a z-score is doing is translating the data or observation. 

Now, let's go a bit further, shall we? 

Find the probabilities: That is, 

Find P (Z < 2) You will note that this is the first z-score

P(Z<2) 

P(z<2) = .9773 (I rounded up due to value being .97725


Find P (Z < -2) You will note that this is the second z-score 
 
P (Z<-2)
P(z<-2) = .0228 (I rounded up due to value being .02275


Subtract the two above values. 

.9773
.0228
_______

.9545


Now, multiply by 1000 and you will arrive at the precise number of students who scored between 300 and 700 

.9545 x 1000 = 954.5 students scored between 300 and 700

I would round this number up to 955 students Jenn. 


Please let me know if my work makes sense to you, or if you need further explanation by follow-up question here or to  my Inbox. 

Have a great weekend. 

Jenn, I want to add something here to the answer I provided you with. I have seen arguments by professors and teachers alike on BOTH sides of the argument as to whether one should or should NOT round up. Therefore, I am going to provide you again with the two original probability values I obtained for my z-scores. They were as follows: .97725 .02275 You still will subtract the two values and then multiply by 1000 (N). You can round up as your professor or instructor tells you. I would not want you to get a wrong answer due to my rounding up in a manner your professor or teacher does not agree with. As I said both sides have a valid argument, and so you will have one professor say round up with the value given in the chart, and another who says DO NOT round until you are done working the problem. Either way you come up with the same answer. You have everything you need however, for your answer. Please do follow up so I KNOW you have this. Thanks again!