Question

An unknown distribution has a mean of 45 and a standard deviation of eight. Samples of size $n=30$are drawn randomly from the population. Find the probability that the sample mean is between $42$ and $50$.

Short answer

the probability that the sample mean is $P(42<\overline{X}<50)=0.9797$

Step-by-step solution

Given information

Given :

Mean value$=45$

Standard deviation$=8$

$n=30$

Formula used:

$X~N\left({\mu }_x,\frac{\alpha x}{\sqrt{n}}\right)$

The chance that the sample mean is between 42 and 50 is calculated.

The probability that the sample mean is between 42 and 50 is

$$\begin{gathered} X~N\left(45,\frac{8}{\sqrt{30}}\right) \\ P(42<X<50)=\text{normalcdf}\left(42,50,45,\frac{8}{\sqrt{30}}\right) \\ P(42<\overline{X}<50)=0.9797 \end{gathered}$$