Final answer:
The probabilities for the various ranges of z can be found using a z-table or statistical software, with each probability value representing the area under the normal distribution curve for that specific range. These areas are cumulative probabilities up to the z-score provided.
Step-by-step explanation:
To compute the probabilities for a standard normal random variable z, you need to use a z-table, calculator, or software that provides the cumulative probability up to a given z-score. The probabilities reflect the area under the standard normal curve corresponding to the given z-score ranges or thresholds.
- a. P(0 ≤ z ≤ 0.59): Refer to the z-table to find the cumulative probability for z = 0.59, and then subtract the cumulative probability for z = 0 to find the area between these two z-scores.
- b. P(-1.59 ≤ z ≤ 0): This probability can be found by looking up the cumulative probability for z = -1.59 and then subtracting that from 0.5 (since the total area to the left of z = 0 is 0.5).
- c. P(z > 0.30): To find this probability, look up the cumulative probability for z = 0.30 and subtract this value from 1.
- d. P(z ≥ -0.46): Find the cumulative probability for z = -0.46; since the question asks for the area to the right, this value would represent the desired probability directly.
- e. P(z < 2.12): The cumulative probability for z = 2.12 straight from the z-table gives you the probability that z is less than 2.12.
- f. P(z ≤ -0.62): The cumulative probability for z = -0.62 gives you the probability that z is less than or equal to -0.62.
It is important to understand that the areas under the normal distribution curve equate to probabilities, and the total area under the curve is always 1.0.