Answer:
a)
b)
c)
d)
And using a calculator, excel ir the normal standard table we have that:
So for this case we expect anyone with a heigth higher than 190 cm in a random sample of 1000.
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".
Part a
Let X the random variable that represent the heights of a population, and for this case we know the distribution for X is given by:
Where and
We are interested on this probability
And 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:
And we can find this probability with the normal standard table or with excel:
Part b
And we can find this probability with the normal standard table or with excel: using the complement rule
Part c
And 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:
And we can find this probability on this way:
And in order to find these probabilities we can use tables for the normal standard distribution, excel or a calculator.
Part d
Since the distributiion for X is normal then the distribution for the sample mean is also normal and given by:
And we eant this probability:
And using a calculator, excel ir the normal standard table we have that:
So for this case we expect anyone with a heigth higher than 190 cm in a random sample of 1000.