Answer:
a)
b)
c)
Step-by-step explanation:
1) 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".
Let X the random variable that represent interest on this case, and for this case we know the distribution for X is given by:
And let represent the sample mean, the distribution for the sample mean is given by:
On this case
Part a
We want this probability:
The best way to solve this problem is using the normal standard distribution and the z score given by:
If we apply this formula to our probability we got this:
Part b
We want this probability:
If we apply the formula for the z score to our probability we got this:
Part c
We are interested on this probability
If we apply the Z score formula to our probability we got this:
And we can find this probability on this way:
And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.