Answer:
a)
And we can use excel or the normal standard table to find this probability:
b)
And we can use excel or the normal standard table to find this probability:
c)
And we can use excel or the normal standard table to find this probability:
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".
Part a
Let X the random variable that represent the scores for critical reading of a population, and for this case we know the distribution for X is given by:
Where and
We select a sample of size n=90, since the distribution for X is normal then the distribution for the sample size is also normal
And for this case we want this probability:
And for this case we can use the z score given by:
And if we use this formula we got:
And we can use excel or the normal standard table to find this probability:
Part b
Let X the random variable that represent the scores for Math of a population, and for this case we know the distribution for X is given by:
Where and
We select a sample of size n=90, since the distribution for X is normal then the distribution for the sample size is also normal
And for this case we want this probability:
And for this case we can use the z score given by:
And if we use this formula we got:
And we can use excel or the normal standard table to find this probability:
Part c
Let X the random variable that represent the scores for Writing of a population, and for this case we know the distribution for X is given by:
Where and
We select a sample of size n=100, since the distribution for X is normal then the distribution for the sample size is also normal
And for this case we want this probability:
And for this case we can use the z score given by:
And if we use this formula we got:
And we can use excel or the normal standard table to find this probability: