Let Z~N(0, 1). Find P(Z < -0.8).
Answer (2)
The probability that Z is less than -0.8 in a standard normal distribution is approximately 0.2119, or 21.19%. This value is obtained using a Z-table or statistical software, which indicates the area under the curve to the left of the given Z-score. Overall, it tells us how likely it is to get a value less than -0.8 from the standard normal distribution.
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Z follows a standard normal distribution: Z ~ N(0, 1).
We want to find P(Z < -0.8).
Using a calculator or statistical software, we determine the probability.
The probability that Z is less than -0.8 is: 0.2119 .
Explanation
Understand the problem and provided data We are given that Z follows a standard normal distribution, denoted as Z ~ N(0, 1). This means Z has a mean of 0 and a standard deviation of 1. Our goal is to find the probability that Z is less than -0.8, which is written as P(Z < -0.8).
Finding the probability using Z-table or calculator To find P(Z < -0.8), we can use the standard normal distribution table (also known as the Z-table) or a calculator with statistical functions. The Z-table provides the cumulative probability for values less than a given Z-score. Alternatively, we can use software or programming languages with statistical libraries to compute this probability.
Calculate the probability Using a calculator or statistical software, we find that the probability P(Z < -0.8) is approximately 0.2119. This value represents the area under the standard normal curve to the left of Z = -0.8.
State the final answer Therefore, the probability that Z is less than -0.8 is approximately 0.2119.
Examples
In finance, the standard normal distribution is used to model stock prices. For example, if Z represents the daily return of a stock, P(Z < -0.8) would give the probability of the stock's daily return being less than -0.8 standard deviations from the mean, which can help investors assess risk.
