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Answer:
If the distribution is bell shaped we can approximate the probability with high accuracy using the z score formula.
a)
And for this case we can use the z score given by:
And if we use it we got:
b)
And for this case we can use the z score given by:
And if we use it we got:
c)
And for this case we can use the z score given by:
And if we use it we got:
If the distribution is NOT bell shaped the approximation with the z score NOT works and we need to have the distribution for X in order to find the probabilities.
Step-by-step explanation:
Previous concepts
If the distribution is bell shaped we can approximate the probability with high accuracy using the z score formula.
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 of interest
We assume for this case that and
We are interested on this probability
And for this case we can use the z score given by:
And if we use it we got:
Part b
And for this case we can use the z score given by:
And if we use it we got:
Part c
And for this case we can use the z score given by:
And if we use it we got:
If the distribution is NOT bell shaped the approximation with the z score NOT works and we need to have the distribution for X in order to find the probabilities.