Final answer:
The probability of a value falling more than z standard deviations from the mean is calculated using the z-score and the standard normal distribution, with tables or software aiding in finding the exact probability.
Step-by-step explanation:
The probability that falls more than z standard deviations from the mean can be determined by using the standard normal distribution. For instance, a z-score of 1.645 corresponds to the probability of 0.05 (5%) for falling more than 1.645 standard deviations above the mean because it leaves 95% within the central part of the curve (as a critical value for a 90% confidence interval would). To calculate the precise probability related to a specific z-score, you would use a z-table or software that computes the area under the standard normal curve.
If a value is z standard deviations away from the mean, we calculate the z-score using the formula z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation of the original distribution before standardization. For a normalized distribution with a mean of 0 and standard deviation of 1, you find the probability by looking at how far away the z-score falls from the mean (0) and referring to the standard normal distribution table or software to find the area under the curve.