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## Sample Mean

The sample mean is a statistic that is used to estimate a population's mean. Even with smaller samples, if the population is normally distributed, the sample means will be normally distributed.

An average of a group of data is referred to as a sample mean. A data set's central tendency, standard deviation, and variance can all be calculated using the sample mean. The sample mean has a wide range of applications, including determining population averages.

** Probability of sample means**

For example, **The height of 25 students is collected in a class. Notably, the data has a $\delta =10and\mu =40$. Calculate the probability that a sample mean will be greater than 45.**

This means that $P(\stackrel{-}{X}>45)\phantom{\rule{0ex}{0ex}}$

Recall that; $z=\frac{{\displaystyle \stackrel{-}{X}}-\mu}{{\displaystyle \frac{\delta}{\sqrt{n}}}}$

$z=\frac{{\displaystyle 45-40}}{{\displaystyle \frac{10}{\sqrt{25}}}}$

$=\frac{5}{2}=2.5$

Therefore, **the probability that the sample mean** will be greater than 45 can be said to be the **probability that z is greater than** 2.5.

Using the z-tables to find the probability;

We have: $P(\stackrel{-}{X}>45)=P(z2.5)$

$=0.99379$$=1-0.99379=0.00621$

$=0.621\%$

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