Question

# A population of values has a normal distribution with \mu=198.8 and \sigma=69.2. You intend to draw a random sample of size n=147. Find the probability that a single randomly selected value is between 184 and 205.1. P(184 < X < 205.1) =? Write your answers as numbers accurate to 4 decimal places.

Random variables
A population of values has a normal distribution with $$\displaystyle\mu={198.8}$$ and $$\displaystyle\sigma={69.2}$$. You intend to draw a random sample of size $$\displaystyle{n}={147}$$.
Find the probability that a single randomly selected value is between 184 and 205.1.
$$\displaystyle{P}{\left({184}{<}{X}{<}{205.1}\right)}=$$</span>?

2020-10-27
The mean is 198.8, standard deviation is 69.2, and sample size is 147.
The probability that a single randomly selected value is between 184 and 205.1 is,
$$\displaystyle{P}{\left({184}{<}{X}{<}{205.1}\right)}={P}{\left({\frac{{{184}-{198.8}}}{{{69.2}}}}{<}{\frac{{{X}-\mu}}{{\sigma}}}{<}{\frac{{{205.1}-{198.8}}}{{{69.2}}}}\right)}$$</span>
$$\displaystyle={P}{\left(-{0.214}{<}{z}{<}{0.091}\right)}$$</span>
$$\displaystyle={P}{\left({z}{<}{0.091}\right)}-{P}{\left({z}{<}-{0.214}\right)}$$</span>
The probability of z less than 0.091 can be obtained using the excel formula “=NORM.S.DIST(0.091,TRUE)”. The probability value is 0.5363.
The probability of z less than –0.214 can be obtained using the excel formula “=NORM.S.DIST(–0.214,TRUE)”. The probability value is 0.4153.
The required probability value is,
$$\displaystyle{P}{\left({184}{<}{X}{<}{205.1}\right)}={P}{\left({z}{<}{0.091}\right)}-{P}{\left({z}{<}-{0.214}\right)}$$</span>
$$\displaystyle={0.5363}−{0.4153}={0.1210}$$
Thus, the probability that a single randomly selected value is between 184 and 205.1 is 0.1210.