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

Random variables
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asked 2020-12-06

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 single randomly selected value is less than 11.5.
$$\displaystyle{P}{\left({X}{<}{11.5}\right)}=$$?
Write your answers as numbers accurate to 4 decimal places.

## Expert Answers (1)

2020-12-07

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 single randomly selected value is less than 11.5
can be obtained as:
$$\displaystyle{P}{\left({X},{11.5}\right)}={P}{\left({\frac{{{x}-\mu}}{{\sigma}}}{<}{\frac{{{11.5}-{13.7}}}{{{22}}}}\right)}$$
$$\displaystyle={P}{\left({Z}{<}-{0.1}\right)}={0.4602}$$ (Using standard normal table)
Thus, the required probability is 0.4602.

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