Answer:
First part
![P(X< 3.4-0.6*3.1) = P(X](https://tex.z-dn.net/?f=%20P%28X%3C%203.4-0.6%2A3.1%29%20%3D%20P%28X%3C1.54%29)
And for this case we can use the z score formula given by:
![z = \frac{x- \mu}{\sigma}](https://tex.z-dn.net/?f=%20z%20%3D%20%5Cfrac%7Bx-%20%5Cmu%7D%7B%5Csigma%7D)
And using this formula we got:
![P(X](https://tex.z-dn.net/?f=%20P%28X%3C1.54%29%20%3D%20P%28Z%3C%5Cfrac%7B1.54%20-3.4%7D%7B3.1%7D%29%3D%20P%28Z%3C-0.6%29)
And we can use the normal standard table or excel and we got:
![P(Z](https://tex.z-dn.net/?f=P%28Z%3C-0.6%29%20%3D%200.274)
Second part
For the other part of the question we want to find the following probability:
![P(-1.715](https://tex.z-dn.net/?f=%20P%28-1.715%20%3CX%3C%207.12%29)
And using the score we got:
![P(-1.715](https://tex.z-dn.net/?f=%20P%28-1.715%20%3CX%3C%207.12%29%3DP%28%5Cfrac%7B-1.715-3.4%7D%7B3.1%7D%20%3C%20Z%3C%20%5Cfrac%7B7.15-3.4%7D%7B3.1%7D%29%20%3D%20P%28-1.65%3C%20Z%3C%201.210%29)
And we can find this probability with this difference:
![P(-1.65< Z< 1.210)=P(Z](https://tex.z-dn.net/?f=P%28-1.65%3C%20Z%3C%201.210%29%3DP%28Z%3C1.210%29-P%28z%3C-1.65%29%20%3D%200.887-0.049%3D0.837)
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".
Solution to the problem
Let X the random variable that represent the data of a population, and for this case we know the distribution for X is given by:
Where
and
First part
And for this case we want this probability:
![P(X< 3.4-0.6*3.1) = P(X](https://tex.z-dn.net/?f=%20P%28X%3C%203.4-0.6%2A3.1%29%20%3D%20P%28X%3C1.54%29)
And for this case we can use the z score formula given by:
![z = \frac{x- \mu}{\sigma}](https://tex.z-dn.net/?f=%20z%20%3D%20%5Cfrac%7Bx-%20%5Cmu%7D%7B%5Csigma%7D)
And using this formula we got:
![P(X](https://tex.z-dn.net/?f=%20P%28X%3C1.54%29%20%3D%20P%28Z%3C%5Cfrac%7B1.54%20-3.4%7D%7B3.1%7D%29%3D%20P%28Z%3C-0.6%29)
And we can use the normal standard table or excel and we got:
![P(Z](https://tex.z-dn.net/?f=P%28Z%3C-0.6%29%20%3D%200.274)
Second part
For the other part of the question we want to find the following probability:
![P(-1.715](https://tex.z-dn.net/?f=%20P%28-1.715%20%3CX%3C%207.12%29)
And using the score we got:
![P(-1.715](https://tex.z-dn.net/?f=%20P%28-1.715%20%3CX%3C%207.12%29%3DP%28%5Cfrac%7B-1.715-3.4%7D%7B3.1%7D%20%3C%20Z%3C%20%5Cfrac%7B7.15-3.4%7D%7B3.1%7D%29%20%3D%20P%28-1.65%3C%20Z%3C%201.210%29)
And we can find this probability with this difference:
![P(-1.65< Z< 1.210)=P(Z](https://tex.z-dn.net/?f=P%28-1.65%3C%20Z%3C%201.210%29%3DP%28Z%3C1.210%29-P%28z%3C-1.65%29%20%3D%200.887-0.049%3D0.837)