Question # A population of values has a normal distribution with \mu=221.6 and \sigma=44.1. You intend to draw a random sample of size n=42. Find the probability

Random variables
ANSWERED A population of values has a normal distribution with $$\displaystyle\mu={221.6}$$ and $$\displaystyle\sigma={44.1}$$. You intend to draw a random sample of size $${n}={42}$$.
Find the probability that a single randomly selected value is between 221.6 and 229.1.
$$\displaystyle{P}{\left({221.6}{<}{X}{<}{229.1}\right)}=$$? 2021-02-22

Step 1
Given information-
Population mean,$$\displaystyle\mu={221.6}$$
Population standard deviation, $$\displaystyle\sigma={44.1}$$
Sample size, $$\displaystyle{n}={42}$$
Let, X be the randomly selected value is approximately normally distributed.
$$\displaystyle{X}\sim{N}{\left({221.6},{44.1}\right)}$$
Step 2
The probability that a single randomly selected value is between 221.6 and 229.1 is $$\displaystyle{P}{\left({221.6}{<}{X}{<}{229.1}\right)}=$$
$$\displaystyle{P}{\left({221.6}{<}{X}{<}{229.1}\right)}={P}{\left({\frac{{{221.6}-{221.6}}}{{\frac{{44.1}}{\sqrt{{{1}}}}}}}{<}{\frac{{{X}-\mu}}{{\frac{\sigma}{\sqrt{{{n}}}}}}}{<}{\frac{{{229.1}-{221.6}}}{{\frac{{44.1}}{\sqrt{{{1}}}}}}}\right)}$$
$$\displaystyle{P}{\left({221.6}{<}{X}{<}{229.1}\right)}={P}{\left({0}{<}{Z}{<}{0.1701}\right)}$$
$$\displaystyle{P}{\left({221.6}{<}{X}{<}{229.1}\right)}={P}{\left({Z}{<}{0.170}\right)}-{P}{\left({Z}{<}{0}\right)}$$
$$\displaystyle{P}{\left({221.6}{<}{X}{<}{229.1}\right)}={0.5675}-{0.5}={0.0675}$$
(From excel using formula = NORM.S. DIST (0.170,TRUE))
(From excel using formula = NORM.S. DIST (0,TRUE))
Hence, the probability that a single randomly selected value is between 221.6 and 229.1 is 0.0675.