SOLUTION: Let X be a random variable normally distributed with parameters μ= 70, σ= 10. Estimate P(65<X<85). (Give the probability in the form of a decimal number (example: 0.123)! T

Question 1174294: Let X be a random variable normally distributed with parameters μ= 70, σ= 10. Estimate P(65 (Give the probability in the form of a decimal number (example: 0.123)!

This is the question that I have and I have done it this way.
P8(65 Z(85) = (85-70)/10 = 1.5
P(X<85) = 0.066807
Z(65) = (65-70)/10 = -0.5
However, I am not quite sure how I get P, I used an online calculator, and in the results it should be 0.624
> pnorm(85,70,10)-pnorm(65,70,10)=0.6246553
Can someone explain this to me? Thank you.
Found 3 solutions by Boreal, Theo, ewatrrr:
Answer by Boreal(15235)   (Show Source): You can put this solution on YOUR website!
I assume you want the probability between 65 and 85, but 85 didn't appear in the above question. The probability of 65 is 0, because each unique point is dimensionless and therefore has no probability.
The probability you want is between z of -0.5 and 1.5, and that probability is 0.6247 rounded.
You have the probability of z > 85, which is 0.0668; you want the probability of z between the intervals above.
You can do the probability of z < 1.5 and subtract from that the probability z < -0.5, but that is more work.
You can also do it with the calculator 2nd VARS 2normalcdf(65,85,70,10)ENTER and get the same answer.
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
your mean is 70 and your standard deviation is 10.
you want to know the probability of getting a score between 65 and 85 if i understand you correctly.

using my ti-85 plus calculator, i got the same probability that you did as .6246553.

if you use the normal distribution tables, you would do the following.

first you want to find the z-score for 65 and 85 with a mean of 70 and a standard deviation of 10

the z-score formula is:
z = (x - m) / s
for x = 65, this formula becomes z = (65 - 70) / 10 = 1.5
for x = 85, this formula becomes z = (85 - 70) / 10 = -.5

you go to the z-score table to find the area to the left of 1.5 and the area to the left of -.5.
you will get area to the left of z-score of -.5 = .30854 and area to the left of z-score of 1.5 = .93319.
subtract the smaller area from the larger area to get .62465.

if you round to 3 decimal, the calculator gets you .625 and the table gets you .625.

if you truncate to 3 decimal places, you will get .624.

your calculator results of .6246553 will be equal to .625 when you round to 3 decimal places.

outside of the rounding discrepancy, the results i gave you pretty much agree with what your calculator gave you.

if i rounded .93319 to 3 decimal places, it would be equal to .933.
if i rounded .30854 to 3 decimal places, it would be equal to .309.
subtract .309 from .933 and you will get .624.

if you used the stattrek online calculator, that calculator rounds the probability to 3 decimal places.
that would explain the discrepancy.

the table i used can be found at https://www.math.arizona.edu/~rsims/ma464/standardnormaltable.pdf

let me know if this answers your question and whether this is enough for you to understand how to do it.

theo

Answer by ewatrrr(24785)   (Show Source): You can put this solution on YOUR website!

Hi μ= 70, σ= 10. IF You mean between P( 65 ≥ x ≤ 85) One can directly find that: normcdf(65,85,70,10) = .624 ............ IF You are trying to estimate P (x=65), normcdf(64.5,65.5,70,10) = .0352 recommend smaller boundaries: P (x ≤ 65.5) is .3264 P (x ≤ 64.5) is .2912 P(x= 65) = (.3264 - .2912) = .0352 ---------------- Or Wish You the Best in your Studies.

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