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:
![z = \frac{X-\mu}{\sigma}](https://tex.z-dn.net/?f=%20z%20%3D%20%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D)
And if we use it we got:
![P(75](https://tex.z-dn.net/?f=%20P%2875%3CX%3C105%29%20%3DP%28%5Cfrac%7B75-90%7D%7B15%7D%20%3CZ%3C%5Cfrac%7B105-90%7D%7B15%7D%29%20%3D%20P%28-1%3C%20Z%3C1%29%3D%20P%28Z%3C1%29-P%28Z%3C-1%29%20%3D%200.841-0.159%3D0.683)
b)
And for this case we can use the z score given by:
![z = \frac{X-\mu}{\sigma}](https://tex.z-dn.net/?f=%20z%20%3D%20%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D)
And if we use it we got:
![P(60](https://tex.z-dn.net/?f=%20P%2860%3CX%3C120%29%20%3DP%28%5Cfrac%7B60-90%7D%7B15%7D%20%3CZ%3C%5Cfrac%7B120-90%7D%7B15%7D%29%20%3D%20P%28-2%3C%20Z%3C2%29%3D%20P%28Z%3C2%29-P%28Z%3C-2%29%20%3D%200.977-0.0228%3D0.955)
c)
And for this case we can use the z score given by:
![z = \frac{X-\mu}{\sigma}](https://tex.z-dn.net/?f=%20z%20%3D%20%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D)
And if we use it we got:
![P(45](https://tex.z-dn.net/?f=%20P%2845%3CX%3C135%29%20%3DP%28%5Cfrac%7B45-90%7D%7B15%7D%20%3CZ%3C%5Cfrac%7B135-90%7D%7B15%7D%29%20%3D%20P%28-3%3C%20Z%3C3%29%3D%20P%28Z%3C3%29-P%28Z%3C-3%29%20%3D%200.999-0.0014%3D0.997)
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:
![z = \frac{X-\mu}{\sigma}](https://tex.z-dn.net/?f=%20z%20%3D%20%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D)
And if we use it we got:
![P(75](https://tex.z-dn.net/?f=%20P%2875%3CX%3C105%29%20%3DP%28%5Cfrac%7B75-90%7D%7B15%7D%20%3CZ%3C%5Cfrac%7B105-90%7D%7B15%7D%29%20%3D%20P%28-1%3C%20Z%3C1%29%3D%20P%28Z%3C1%29-P%28Z%3C-1%29%20%3D%200.841-0.159%3D0.683)
Part b
And for this case we can use the z score given by:
![z = \frac{X-\mu}{\sigma}](https://tex.z-dn.net/?f=%20z%20%3D%20%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D)
And if we use it we got:
![P(60](https://tex.z-dn.net/?f=%20P%2860%3CX%3C120%29%20%3DP%28%5Cfrac%7B60-90%7D%7B15%7D%20%3CZ%3C%5Cfrac%7B120-90%7D%7B15%7D%29%20%3D%20P%28-2%3C%20Z%3C2%29%3D%20P%28Z%3C2%29-P%28Z%3C-2%29%20%3D%200.977-0.0228%3D0.955)
Part c
And for this case we can use the z score given by:
![z = \frac{X-\mu}{\sigma}](https://tex.z-dn.net/?f=%20z%20%3D%20%5Cfrac%7BX-%5Cmu%7D%7B%5Csigma%7D)
And if we use it we got:
![P(45](https://tex.z-dn.net/?f=%20P%2845%3CX%3C135%29%20%3DP%28%5Cfrac%7B45-90%7D%7B15%7D%20%3CZ%3C%5Cfrac%7B135-90%7D%7B15%7D%29%20%3D%20P%28-3%3C%20Z%3C3%29%3D%20P%28Z%3C3%29-P%28Z%3C-3%29%20%3D%200.999-0.0014%3D0.997)
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.