Ask question

# Let X \sim N(6,4).Find the probabilities P(X<3). # Let X \sim N(6,4).Find the probabilities P(X<3).

Question
Random variables asked 2021-01-31

Let $$\displaystyle{X}\sim{N}{\left({6},{4}\right)}$$.Find the probabilities $$\displaystyle{P}{\left({X}{<}{3}\right)}$$.

## Answers (1) 2021-02-01

Step 1
Introduction:
The normal probability is a type of continuous probability distribution that can take random values. The normal distribution is determined by the two parameters - the population mean $$(\mu)$$ and population variance $$\displaystyle{\left(\sigma^{{{2}}}\right)}$$. It is symmetric with respect to its mean.
Given information:
$$\displaystyle{X}\sim{N}{\left({6},{4}\right)}$$
Therefore,
$$\displaystyle\mu={6}$$
$$\displaystyle\sigma^{{{2}}}={4}$$

Step 2
$$\displaystyle{P}{\left({X}{<}{3}\right)}$$ is computed as follows:
$$\displaystyle{P}{\left({X}{<}{3}\right)}={P}{\left({\frac{{{X}-\mu}}{{\sqrt{{\sigma^{{{2}}}}}}}}{<}{\frac{{{3}-\mu}}{{\sqrt{{\sigma^{{{2}}}}}}}}\right)}$$
$$\displaystyle={P}{\left({Z}{<}{\frac{{{3}-{6}}}{{\sqrt{{{4}}}}}}\right)}$$
$$\displaystyle={P}{\left({Z}{<}-{1.5}\right)}$$
$$\displaystyle={1}-{P}{\left({Z}{<}{1.5}\right)}$$
$$\displaystyle={1}-{0.93319}={0.06681}$$
Therefore,
$$\displaystyle{P}{\left({X}{<}{3}\right)}={0.0668}$$

### Relevant Questions asked 2020-11-16

Let $$\displaystyle{X}\sim{N}{\left({6},{4}\right)}$$.Find the probabilities $$\displaystyle{P}{\left({5}{<}{X}{<}{7}\right)}$$. asked 2021-01-19

A distribution of values is normal with a mean of 79.5 and a standard deviation of 7.4.
Find the probability that a randomly selected value is between 71.4 and 78.
$$\displaystyle{P}{\left({71.4}{<}{x}{<}{78}\right)}={P}{\left({<}{z}{<}\right)}=$$? asked 2020-11-17

If you have $$\displaystyle\lambda={4.2}$$, what is the probability that $$P(x < 2)$$? asked 2021-01-15

The random variable X follows a normal distribution $$\displaystyle?{\left({20},{102}\right)}$$.
Find $$\displaystyle{P}{\left({10}{<}?{<}{35}\right)}$$, asked 2021-02-08

A population of values has a normal distribution with $$\displaystyle\mu={204.3}$$ and $$\displaystyle\sigma={43}$$. You intend to draw a random sample of size $$\displaystyle{n}={111}$$.
Find the probability that a single randomly selected value is less than 191.2.
$$\displaystyle{P}{\left({X}{<}{191.2}\right)}=$$?
Find the probability that a sample of size $$\displaystyle{n}={111}$$ is randomly selected with a mean less than 191.2.
$$\displaystyle{P}{\left({M}{<}{191.2}\right)}=$$?
Write your answers as numbers accurate to 4 decimal places. asked 2021-02-25

A population of values has a normal distribution with $$\displaystyle\mu={192.6}$$ and $$\displaystyle\sigma={34.4}$$. You intend to draw a random sample of size $$\displaystyle{n}={173}$$.
Find the probability that a sample of size $$\displaystyle{n}={173}$$ is randomly selected with a mean less than 186.1.
$$\displaystyle{P}{\left({M}{<}{186.1}\right)}=$$?
Write your answers as numbers accurate to 4 decimal places. asked 2020-11-30

A population of values has a normal distribution with $$\displaystyle\mu={29.3}$$ and $$\displaystyle\sigma={65.1}$$. You intend to draw a random sample of size $$\displaystyle{n}={142}$$.
Find the probability that a sample of size n=142 is randomly selected with a mean between 27.7 and 35.3.
$$\displaystyle{P}{\left({27.7}{<}\overline{{{X}}}{<}{35.3}\right)}=$$?
Write your answers as numbers accurate to 4 decimal places. asked 2021-01-04

A population of values has a normal distribution with $$\displaystyle\mu={198.8}$$ and $$\displaystyle\sigma={69.2}$$. You intend to draw a random sample of size $$\displaystyle{n}={147}$$.
Find the probability that a sample of size $$\displaystyle{n}={147}$$ is randomly selected with a mean between 184 and 205.1.
$$\displaystyle{P}{\left({184}{<}{M}{<}{205.1}\right)}=$$?
Write your answers as numbers accurate to 4 decimal places. asked 2020-11-29

A population of values has a normal distribution with $$\displaystyle\mu={116.5}$$ and $$\displaystyle\sigma={63.7}$$. You intend to draw a random sample of size $$\displaystyle{n}={244}$$.
Find the probability that a sample of size $$\displaystyle{n}={244}$$ is randomly selected with a mean between 104.7 and 112.8.
$$\displaystyle{P}{\left({104.7}{<}{M}{<}{112.8}\right)}=$$?
Write your answers as numbers accurate to 4 decimal places. asked 2021-01-05

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 sample of size $$\displaystyle{n}={84}$$ is randomly selected with a mean between 98.5 and 100.7.
$$\displaystyle{P}{\left({98.5}{<}\overline{{{X}}}{<}{100.7}\right)}=$$?
Write your answers as numbers accurate to 4 decimal places.

...