![]() If the return is $0.10, then \(x = 0.1\) (this is our observed value). The probability that the return is less than $1 is closest to: Because the normal distribution is symmetrical, if x1 x 1 were the same distance to the left of the mean the area. This is the unshaded part of the graph from the mean to x1 x 1. The returns on ABC stock are normally distributed, where the mean is $0.60 with a standard deviation of $0.20. Basically, this conversion forces the mean and stddev to be standardized to 0 and 1 respectively, which enables a standard defined set of Z-values (from the Normal Distribution Table) to be used. Normal tables provide the probability between the mean, zero for the standard normal distribution, and a specific value such as x1 x 1. Positive z-values Example: Using the z-score Table Simply put, if an examiner asks you to find the probability behind a given positive z-value, you will have to look it up directly on the table, knowing that \(P(Z ≤ z) = θ(z)\) when \(z\) is positive. To produce outputs from a standard normal distribution with this calculator, set the mean equal to 0 and the standard deviation equal to 1. ![]() ![]() However, the table does this only when we have positive values of \(z\). Note: The normal distribution table, found in the appendix of most statistics texts, is based on the standard normal distribution, which has a mean of 0 and a standard deviation of 1. Using the standard normal distribution table, we can confirm that a normally distributed random variable \(Z\), with a mean equal to 0 and variance equal to 1, is less than or equal to \(z\), i.e., \(P(Z ≤ z)\). ![]()
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