I am supposed to solve this task:
Provide the following probabilities for X ~ N(35, 9) (without using Excel or a calculator):
a) P(32 ≤ X ≤ 38)
b) P(29 ≤ X ≤ 41)
c) P(26 ≤ X ≤ 44)
d) P(35 − k ∙ 3 ≤ X ≤ 35 + k ∙ 3)
Since we worked all the time with Excel and calculator I was not able to solve it. I looked in the solutions and this is what is given without any other explanation:
sigma = 3
2*sigma = 6
3*sigma = 9
a) P(35-sigma<=X<=35+sigma) = 68%
b) P(35-2*sigma<=X<=35+2*sigma) = 95,4%
c) P(35-3*sigma<=X<=35+3*sigma) = 99,7%
d) k = 2
P(35-k*sigma<=X<=35+k*sigma) = 0,9545
I tried to understand which formula was used to arrive at these results but I can't figure it out. Also, what is the meaning of k in this context and why is sigma multiplied for 2 and 3?
Thank you very much in advance for your help!
One thing to typically try in such problems is to transform your random variable $X$ into standard form:
$ Y = (X - \mu)/\sigma$, where $\mu$ is the mean of $X$ and $\sigma$ is the std dev of $X$.
In your problem, $\mu=35, \sigma=3$, so $Z = (X-35)/3$, or, $X = 3Z + 35$
Let's try the first one: $P(32 \leq X \leq 38) = P(32 \leq 3Z+35 \leq 38) = P(-3 \leq 3Z \leq 3) = P(-1 \leq Z \leq 1) \approx 68\%$.
This is well known for a standard normal RV: https://en.wikipedia.org/wiki/68–95–99.7_rule
You can solve the other problems similarly.
This seems like an illustrative exercise and uses "well-known" rules. In general, you need a calculator or at the very least a normal table for Z-values to solve these problems.