# A population of values has a normal distribution with \mu = 49 and \sigma = 79.5. You intend to draw a random sample of size n=84. Find the probability that a single randomly selected value is greater than 72.4. P(X>72.4)=? Write your answers as numbers accurate to 4 decimal places.

Random variables
A population of values has a normal distribution with $$\displaystyle\mu={49}$$ and $$\displaystyle\sigma={79.5}$$. You intend to draw a random sample of size $$\displaystyle{n}={84}$$.
Find the probability that a single randomly selected value is greater than 72.4.
$$\displaystyle{P}{\left({X}{>}{72.4}\right)}=$$?
Write your answers as numbers accurate to 4 decimal places.

2020-11-03

Step 1
Given,
Mean $$= 49$$
Standard deviation $$= 79.5$$
Sample size $$= 84$$
Step 2
Consider,
$$\displaystyle{P}{\left({X}{>}{72.4}\right)}={P}{\left({\frac{{{X}-\mu}}{{\sigma}}}{>}{\frac{{{72.4}-\mu}}{{\sigma}}}\right)}$$
$$\displaystyle={P}{\left({Z}{>}{\frac{{{72.4}-{49}}}{{{79.5}}}}\right)}$$
$$\displaystyle={P}{\left({Z}{>}{0.294}\right)}$$
$$\displaystyle={1}-{P}{\left({Z}\leq{0.294}\right)}$$
$$\displaystyle={1}-{0.6156}={0.3844}$$ (From the standard normal table)
The probability that a single randomly selected value is greater than 72.4 is, 0.3844.