Question

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?

Answer 1

Answer:

If the distribution is bell shaped we can approximate the probability with high accuracy using the z score formula.

a) $P(75$

And for this case we can use the z score given by:

$z = \frac{X-\mu}{\sigma}$

And if we use it we got:

$P(75$

b) $P(60$

And for this case we can use the z score given by:

$z = \frac{X-\mu}{\sigma}$

And if we use it we got:

$P(60$

c) $P(45$

And for this case we can use the z score given by:

$z = \frac{X-\mu}{\sigma}$

And if we use it we got:

$P(45$

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 $\mu=90$ and $\sigma=15$

We are interested on this probability

$P(75$

And for this case we can use the z score given by:

$z = \frac{X-\mu}{\sigma}$

And if we use it we got:

$P(75$

Part b

$P(60$

And for this case we can use the z score given by:

$z = \frac{X-\mu}{\sigma}$

And if we use it we got:

$P(60$

Part c

$P(45$

And for this case we can use the z score given by:

$z = \frac{X-\mu}{\sigma}$

And if we use it we got:

$P(45$

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.