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?

Where and

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

So then is appropiate use the normal distribution to find the probabilities for

Step-by-step 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 is used to denote the cumulative area for a b quantile on the normal standard distribution, or in other words:

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:

Where and

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

So then is appropiate use the normal distribution to find the probabilities for