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# A population of values has a normal distribution with \mu=99.6 and \sigma=35.1. # A population of values has a normal distribution with \mu=99.6 and \sigma=35.1.

Question
Random variables asked 2020-12-07

A population of values has a normal distribution with $$\displaystyle\mu={99.6}$$ and $$\displaystyle\sigma={35.1}$$. You intend to draw a random sample of size $$\displaystyle{n}={84}$$.
Find the probability that a single randomly selected value is between 98.5 and 100.7.
$$\displaystyle{P}{\left({98.5}{<}{X}{<}{100.7}\right)}=$$?
Write your answers as numbers accurate to 4 decimal places.

## Answers (1) 2020-12-08

Step 1
From the provided information,
Mean $$\displaystyle{\left(\mu\right)}={99.6}$$
Standard deviation $$\displaystyle{\left(\sigma\right)}={35.1}$$
$$\displaystyle{X}\sim{N}{\left({99.6},{35.1}\right)}$$
Sample size $$\displaystyle{\left({n}\right)}={84}$$
Step 2
The required probability that a single randomly selected value is between 98.5 and 100.7 can be obtained as:
$$\displaystyle{P}{\left({98.5}{<}{X}{<}{100.7}\right)}={P}{\left({\frac{{{98.5}-{99.6}}}{{{35.1}}}}{<}{\frac{{{x}-\mu}}{{\sigma}}}{<}{\frac{{{100.7}-{99.6}}}{{{35.1}}}}\right)}$$
$$\displaystyle={P}{\left(-{0.031}{<}{Z}{<}{0.031}\right)}$$
$$\displaystyle={P}{\left({Z}{<}{0.031}\right)}-{P}{\left({Z}{<}-{0.031}\right)}$$
$$\displaystyle={P}{\left({Z}{<}{0.031}\right)}-{\left[{1}-{P}{\left({z}{<}{0.031}\right)}\right]}$$
$$\displaystyle={2}{P}{\left({Z}{<}{0.031}\right)}-{1}$$
$$\displaystyle={2}{\left({0.5124}\right)}-{1}={0.0248}$$ (Using standard normal table)
Thus, the required probability is 0.0248.

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