Question

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

Random variables
ANSWERED
asked 2020-12-25

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.
\(\displaystyle{P}{\left({M}{<}{11.5}\right)}=\)?
Write your answers as numbers accurate to 4 decimal places.

Expert Answers (1)

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.

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