A population of values has a normal distribution with

A population of values has a normal distribution with

Question
Random variables
asked 2021-01-19

A population of values has a normal distribution with \(\displaystyle\mu={200.7}\) and \(\displaystyle\sigma={10}\). You intend to draw a random sample of size \(\displaystyle{n}={178}\)
Find the probability that a sample of size \(\displaystyle{n}={178}\) is randomly selected with a mean less than 198.6.
\(\displaystyle{P}{\left({M}{<}{198.6}\right)}=\)?

Answers (1)

2021-01-20

Step 1
From the provided information,
Mean \(\displaystyle{\left(\mu\right)}={200.7}\)
Standard deviation \(\displaystyle{\left(\sigma\right)}={10}\)
Let X be a random variable which represents the value.
\(\displaystyle{X}\sim{N}{\left({200.7},{10}\right)}\)
Sample size \(\displaystyle{\left({n}\right)}={178}\)
Step 2
The required probability that a sample of size \(\displaystyle{n}={178}\) is randomly selected with a mean less than 198.6 can be obtained as:
\(\displaystyle{P}{\left({M}{<}{198.6}\right)}={P}{\left({\frac{{{M}-\mu}}{{{\frac{{\sigma}}{{\sqrt{{{n}}}}}}}}}{<}{\frac{{{198.6}-{200.7}}}{{{\frac{{{10}}}{{\sqrt{{{178}}}}}}}}}\right)}\)
\(\displaystyle={P}{\left({Z}{<}-{2.802}\right)}={0.0025}\) (Using standard normal table)
Thus, the required probability is 0.0025.

0

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