To find the probability that [tex]\(z\)[/tex] is greater than 3 in a standard normal distribution, denoted as [tex]\(P(z > 3)\)[/tex], follow these steps:
1. Understand the standard normal distribution: The standard normal distribution has a mean ([tex]\(\mu\)[/tex]) of 0 and a standard deviation ([tex]\(\sigma\)[/tex]) of 1.
2. Locate the z-value: We are asked to find the probability that [tex]\(z\)[/tex] is greater than 3, so our z-value (3) is to the right of the mean on the standard normal curve.
3. Look up the cumulative distribution function (CDF): The cumulative distribution function [tex]\(P(z \leq z_0)\)[/tex] provides the probability that a random variable [tex]\(z\)[/tex] takes on a value less than or equal to [tex]\(z_0\)[/tex].
4. Determine the complement: Since we want to find [tex]\(P(z > 3)\)[/tex], and we have the cumulative distribution function [tex]\(P(z \leq 3)\)[/tex], we use the complement rule. The rule states that:
[tex]\[
P(z > 3) = 1 - P(z \leq 3)
\][/tex]
5. Reference cumulative probabilities: From standard normal distribution tables or software, find the cumulative probability up to [tex]\(z = 3\)[/tex]. The CDF value for [tex]\(z = 3\)[/tex] is very close to 1 (as most of the distribution lies to the left of 3).
6. Calculate the specific probability: The probability [tex]\(P(z \leq 3)\)[/tex] is commonly found in tables or via computational tools to be approximately [tex]\(0.9987\)[/tex].
7. Subtract from 1: To find [tex]\(P(z > 3)\)[/tex]:
[tex]\[
P(z > 3) = 1 - 0.9987 = 0.0013
\][/tex]
Thus, the probability that a value from a standard normal distribution is greater than 3, expressed as a decimal rounded to four decimal places, is [tex]\(0.0013\)[/tex].
