Normal distribution, mean=10, st. deviation = 4...question

[yancey]

A normal distribution has a mean of 10 and a standard deviation of 4. Which of the following statements is most accurate?
A) 50% of all the observations will fall between 6 and 14.
B) The probability of finding an observation below 2 is 5%.
C) The probability of finding an observation at 22 or above is 1%.
D) 81.5% of all the observations will fall between 6 and 18.
Your answer: A was incorrect. The correct answer was D) 81.5% of all the observations will fall between 6 and 18.
68 percent of all observations will fall in the interval plus or minus one standard deviation from the mean (6 to 14). 95 percent will fall in the interval plus or minus two standard deviations from the mean (2 to 18), so 2.5 percent will fall below 2. 99 percent will fall in the interval plus or minus three standard deviations from the mean (-2 to 22), so 0.5 percent will fall above 22.
My first question is: which LOS is this?
Am I correct?
68% = 1 std = 6-14
95% = 2-18
97.5% = -2 - 22
I don’t understand why B is not correct.
Anything helps, I’m confused on this subject. thanks.

 

[maratikus]

B is incorrect because the probability of finding an observation below 2 is one half of probability of being outside two standard deviations.
(1-.95)/2 = 2.5%

 

[njblain]

You need to know some basic figures from the normal distribution. In this case you only need (1) 68% between +/-1 SD (i.e. 34% from 0 to 1), and (2) 95% between +/- 2 SD (i.e. 47.5% from 0 to 2).
Now 6 to 18 is -1 through +2, i.e. -1 to 0 plus 0 to 2 = 34% + 47.5% = 81.5%.

 

[Konstantin]

yancey
+-1,96xsigma is 95%
so 2-18 is a little more thant 95%
so below 2 is (100-95%)/2= little more than 2% (symmetrical distribution)
So B is incorrect
So correct is D:
6-18 is 1sigma to the left and sigma to the right from mean (10). So
to the left 1sigma = 68%/2 = 34%
to the right 2 (rounded 1,96) sigma = 95%/2 = 47,5%
So 34%+47,5% = 81,5%

 

[yancey]

thanks for all your help!