Question

A sample of size 6 will be drawn from a normal population with mean 61 and standard deviation 14. Use the TI-84 Plus calculator. Is it appropriate to use the normal distribution to find probabilities for x?

Answer 1

`X \sim N(61,14)`

Where
`\mu=61` and
`\sigma=14`

Since the distribution for X is normal then the distribution for the sample mean is also normal and given by:

`\bar X \sim N(\mu, (\sigma)/(√(n)))`

`\mu_(\bar X) = 61`

`\sigma_(\bar X)= (14)/(√(6))= 5.715`

So then is appropiate use the normal distribution to find the probabilities for
`\bar X`

Explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean". The letter
`\phi(b)` is used to denote the cumulative area for a b quantile on the normal standard distribution, or in other words:
`\phi(b)=P(z<b)`

Solution to the problem

Let X the random variable that represent the variable of interest of a population, and for this case we know the distribution for X is given by:

`X \sim N(61,14)`

Where
`\mu=61` and
`\sigma=14`

Since the distribution for X is normal then the distribution for the sample mean
`\bar X` is also normal and given by:

`\bar X \sim N(\mu, (\sigma)/(√(n)))`

`\mu_(\bar X) = 61`

`\sigma_(\bar X)= (14)/(√(6))= 5.715`

So then is appropiate use the normal distribution to find the probabilities for
`\bar X`