A statistics practitioner determined that the mean and stan-dard deviation of a data set were 90 and 15, respectively. What can yousay about the proportions of observations that lie between each of thefollowing intervals if you know that the distribution is bell-shaped? Whatcan you say if the distribution is not bell-shaped?

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.