Question

# A population of values has a normal distribution with \mu = 13.7 and \sigma = 22.You intend

Random variables

A population of values has a normal distribution with $$\displaystyle\mu={13.7}$$ and $$\displaystyle\sigma={22}$$.
You intend to draw a random sample of size $$\displaystyle{n}={78}$$.
Find the probability that a sample of size $$\displaystyle{n}={78}$$ is randomly selected with a mean less than 11.5.
$$\displaystyle{P}{\left({M}{<}{11.5}\right)}=$$?

2020-12-26

Step 1
From the provided information,
Mean $$\displaystyle{\left(\mu\right)}={13.7}$$
Standard deviation $$\displaystyle{\left(\sigma\right)}={22}$$
Let X be a random variable which represents the score.
$$\displaystyle{X}\sim{N}{\left({13.7},{22}\right)}$$
Sample size $$\displaystyle{\left({n}\right)}={78}$$
Step 2
The required probability that a sample size $$\displaystyle{n}={78}$$ is randomly selected with a mean less than 11.5 can be obtained as:
$$\displaystyle{P}{\left({M},{11.5}\right)}={P}{\left({\frac{{{x}-\mu}}{{{\frac{{\sigma}}{{\sqrt{{{n}}}}}}}}}{<}{\frac{{{11.5}-{13.7}}}{{{\frac{{{22}}}{{{s}{q}{r}{\left\lbrace{78}\right\rbrace}}}}}}}\right)}$$
$$\displaystyle={P}{\left({Z}{<}-{0.8832}\right)}={0.1886}$$ (Using standard normal table)
Thus, the required probability is 0.1886.