Probability of an event is the measure of its chance of occurrence. The event out of the listed events whose probability is 0.2957 is given by : Option C: ![P(0.25 \leq Z \leq 1.25)](https://tex.z-dn.net/?f=P%280.25%20%5Cleq%20Z%20%5Cleq%201.25%29)
<h3>How to get the z scores?</h3>
If we've got a normal distribution, then we can convert it to standard normal distribution and its values will give us the z-score.
If we have
![X \sim N(\mu, \sigma)](https://tex.z-dn.net/?f=X%20%5Csim%20N%28%5Cmu%2C%20%5Csigma%29)
(X is following normal distribution with mean
and standard deviation
)
then it can be converted to standard normal distribution as
![Z = \dfrac{X - \mu}{\sigma}, \\\\Z \sim N(0,1)](https://tex.z-dn.net/?f=Z%20%3D%20%5Cdfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D%2C%20%5C%5C%5C%5CZ%20%5Csim%20N%280%2C1%29)
(Know the fact that in continuous distribution, probability of a single point is 0, so we can write
![P(Z \leq z) = P(Z < z) )](https://tex.z-dn.net/?f=P%28Z%20%5Cleq%20z%29%20%3D%20P%28Z%20%3C%20z%29%20%29)
Also, know that if we look for Z = z in z tables, the p-value we get is
![P(Z \leq z) = \rm p \: value](https://tex.z-dn.net/?f=P%28Z%20%5Cleq%20z%29%20%3D%20%5Crm%20p%20%5C%3A%20value)
Using the z-table, we get the needed probabilities as:
Case 1:
![P(-1.25 \leq Z \leq 0.25) = P(Z \leq 0.25) - P(Z \leq -1.25) \approx 0.5987 - 0.1056 = 0.4931](https://tex.z-dn.net/?f=P%28-1.25%20%5Cleq%20Z%20%5Cleq%200.25%29%20%3D%20P%28Z%20%5Cleq%200.25%29%20-%20P%28Z%20%5Cleq%20-1.25%29%20%5Capprox%200.5987%20-%200.1056%20%3D%200.4931)
Case 2:
![P(-1.25 \leq Z \leq 0.75) = P(Z \leq 0.75) - P(Z \leq -1.25) \approx 0.7734- 0.1056=0.6678](https://tex.z-dn.net/?f=P%28-1.25%20%5Cleq%20Z%20%5Cleq%200.75%29%20%3D%20P%28Z%20%5Cleq%200.75%29%20-%20P%28Z%20%5Cleq%20-1.25%29%20%5Capprox%200.7734-%200.1056%3D0.6678)
Case 3:
![P(0.25 \leq Z \leq 1.25) = P(Z \leq 1.25) - P(Z \leq 0.25) \approx 0.8944 - 0.5987=0.2957](https://tex.z-dn.net/?f=P%280.25%20%5Cleq%20Z%20%5Cleq%201.25%29%20%3D%20P%28Z%20%5Cleq%201.25%29%20-%20P%28Z%20%5Cleq%200.25%29%20%5Capprox%200.8944%20-%200.5987%3D0.2957)
Case 4:
![P(0.75 \leq Z \leq 1.25) = P(Z \leq 1.25) - P(Z \leq 0.75) \approx 0.8944 - 0.7734 =0.1210](https://tex.z-dn.net/?f=P%280.75%20%5Cleq%20Z%20%5Cleq%201.25%29%20%3D%20P%28Z%20%5Cleq%201.25%29%20-%20P%28Z%20%5Cleq%200.75%29%20%5Capprox%200.8944%20-%200.7734%20%3D0.1210)
Thus, the event out of the listed events whose probability is 0.2957 is given by : Option C: ![P(0.25 \leq Z \leq 1.25)](https://tex.z-dn.net/?f=P%280.25%20%5Cleq%20Z%20%5Cleq%201.25%29)
Learn more about z-scores here:
brainly.com/question/13299273