Answer:
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".
Solution to the problem
Let X the random variable that represent the mass of a population, and for this case we know the distribution for X is given by:
And the z score is defined as:
We know that for some amount m we have this:
(a)
(b)
We can use the z table or excel in order to find a quantile that satisfy the two conditions. The excel code would be:
"=NORM.INV(0.6,0,1)" and using condition (b) we have that Z=0.253
So we have this:
If we solve for m from the last equation we got:
And let represent the sample mean, the distribution for the sample mean is given by:
And we want this probability:
We can apply the z score formula for this case given by:
If we use the expression obtained for m we got: