Final answer:
The probabilities for the various events on the standard normal distribution were calculated using z-scores and a z-table, with several options being close to or exactly the calculated values after comparing area under the curve.
Step-by-step explanation:
The probability is represented by the area under the normal curve. To find the probability, we calculate the z-score and look up the z-score in the z-table under the z-column. Most z-tables show the area under the normal curve to the left of z. If you have a z-table that shows the area to the right, you can subtract the area to the left from 1 to find the area to the right. This can be used to answer probability questions related to standard normal distribution.
- P(z > 1.38) = 1 - Area to the left of z = 1 - 0.9162 = 0.0838 (Option A is close but seems to be a typo).
- P(1.23 < z < 3.24) = Area to the left of z = 3.24 - Area to the left of z = 1.23 = 0.9993 - 0.8907 = 0.1086 (None of the given options are correct).
- P(z < -0.42) = Area to the left of z = -0.42 = 0.3372 (Option A is close but also seems to be incorrect).
- P(-2.64 < z < 1.64) = Area to the left of z = 1.64 - Area to the left of z = -2.64 = 0.9495 - 0.0041 = 0.9454 (Option D is the closest).
- P(z < 3.04) = Area to the left of z = 3.04 = 0.9988 (Option D is correct).
- P(0 < z < 2) = Area to the left of z = 2 - Area to the left of z = 0 = 0.9772 - 0.5 = 0.4772 (None of the given options are correct).
- P(z > -2.43) = 1 - Area to the left of z = -2.43 = 1 - 0.0075 = 0.9925 (Option A is closest to the correct value).