Answer:
a)
b)
c)
d)
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".
Let X the random variable that represent the scores for the East population, and for this case we know the distribution for X is given by:
Where and
Let Y the random variable that represent the scores for the West population, and for this case we know the distribution for Y is given by:
Where and
Part a
For this case we want this probability:
And we can use the z score given by:
And if we replace we got:
Part b
For this case we need to define the following random variable R = X-Y and we know that the distribution of R is given by:
And we want this probability:
We can use the z score given by:
If we use this formula we got:
Part c
For this case we select a sample size of n =3 for the Y distribution, the sample mean have the following distribution:
And we want this probability:
Part d
For this case we define the following random variable and the distribution for H is given by:
And the z score would be given by:
And if we find the probability required we got: