A random sample of 36 observations has been drawn from a normal distribution with mean 50 and standard deviation 12. Find the probability that the sample mean is in the interval (Express the result to four significant digits.)

And using a calculator, excel ir the normal standard table we have that:

And we can calculate the probability like this:Step-by-step explanation:

A random sample of 36 observations has been drawn from a normal distribution with mean 50 and standard deviation 12. Find the probability that the sample mean is in the interval 47<=X<53. Is the assumption of normality important. Why?

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".  

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 we know that the distribution for the sample mean is given by:

We can find the probability required like this:

And using a calculator, excel ir the normal standard table we have that:

And we can calculate the probability like this:

what the question so we can you out

the answer is 31.27mph