Let Z~N(0, 1). Find P(Z < 1.3).
Answer (2)
To find P(Z < 1.3) for a standard normal distribution Z ~ N(0, 1), we use a Z-table or calculator. The probability is approximately 0.9032, indicating that about 90.32% of values fall below 1.3. Thus, P(Z < 1.3) ≈ 0.9032.
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We have a standard normal distribution Z ~ N(0, 1).
We want to find P(Z < 1.3), the probability that Z is less than 1.3.
Using a Z-table or calculator, we find the cumulative probability for Z = 1.3.
The probability P(Z < 1.3) is approximately 0.9032 .
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 1.3, which is written as P(Z < 1.3).
Find the probability using Z-table or calculator To find P(Z < 1.3), 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, which is the area under the standard normal curve to the left of a given Z-value.
Look up the value and find the probability Looking up the value 1.3 in the Z-table (or using a calculator), we find the corresponding probability. The value represents the area under the standard normal curve to the left of z = 1.3. The result of this operation is approximately 0.9032.
State the final answer Therefore, the probability that Z is less than 1.3 is approximately 0.9032.
Examples
Imagine you are analyzing the performance of students on a standardized test where the scores are normally distributed. If the average score is 0 and the standard deviation is 1, then finding P(Z < 1.3) tells you the proportion of students who scored below 1.3 standard deviations above the mean. This kind of calculation is crucial in many fields, including finance (analyzing stock returns), engineering (assessing system reliability), and healthcare (evaluating treatment effectiveness).
