To find the probability that a standard normal variable [tex]\(Z\)[/tex] is greater than 2.09, denoted as [tex]\(P(Z > 2.09)\)[/tex], let's follow these steps:
1. Identify the Z-Score:
We are given that [tex]\( z \)[/tex] is 2.09.
2. Understand the concept:
The standard normal distribution is symmetric about the mean (which is 0 for a standard normal distribution), and the total area under the curve is 1. The cumulative distribution function (CDF) gives us the probability that a standard normal variable is less than or equal to a given z-score.
3. Use the Cumulative Distribution Function (CDF):
The CDF of the standard normal distribution, [tex]\( P(Z \leq \text{z}) \)[/tex], gives us the area to the left of the z-score.
4. Convert to Required Probability:
Since we need the probability [tex]\( P(Z > 2.09) \)[/tex], we can use the relationship:
[tex]\[
P(Z > 2.09) = 1 - P(Z \leq 2.09)
\][/tex]
5. Look up or calculate the CDF value:
By using standard normal distribution tables or statistical software, we find that the cumulative probability up to [tex]\( z = 2.09 \)[/tex] is approximately 0.9816911.
6. Subtract the CDF from 1:
[tex]\[
P(Z > 2.09) = 1 - P(Z \leq 2.09) = 1 - 0.9816911 = 0.0183089
\][/tex]
Therefore, the probability that [tex]\( Z \)[/tex] is greater than 2.09 is approximately [tex]\( 0.018 \)[/tex] when rounded to at least three decimal places.
So,
[tex]\[
P(Z > 2.09) \approx 0.018
\][/tex]
