What is $P (0.6 \leq z \leq 2.12)$?
Answer (2)
Find the CDF value at 2.12: Φ ( 2.12 ) ≈ 0.9830 .
Find the CDF value at 0.6: Φ ( 0.6 ) ≈ 0.7257 .
Calculate the difference: 0.9830 − 0.7257 = 0.2573 .
Convert to percentage: 0.2573 ≈ 26% . The answer is 26% .
Explanation
Understanding the problem We are asked to find the probability that a standard normal random variable z lies between 0.6 and 2.12. This is equivalent to finding the area under the standard normal curve between these two values.
Finding the CDF values To find P ( 0.6 ≤ z ≤ 2.12 ) , we need to calculate the difference between the cumulative distribution function (CDF) values at 2.12 and 0.6. The CDF, denoted by Φ ( z ) , gives the probability that a standard normal random variable is less than or equal to z . Therefore, we need to find Φ ( 2.12 ) − Φ ( 0.6 ) .
Calculating CDF values Using a standard normal table or a calculator, we find the following values: Φ ( 2.12 ) ≈ 0.9830 Φ ( 0.6 ) ≈ 0.7257
Calculating the probability Now, we subtract the CDF values to find the desired probability: P ( 0.6 ≤ z ≤ 2.12 ) = Φ ( 2.12 ) − Φ ( 0.6 ) ≈ 0.9830 − 0.7257 = 0.2573
Converting to percentage and finding the answer Converting this to a percentage, we get: 0.2573 × 100 ≈ 25.73% which is approximately 26% .
Examples
This type of probability calculation is used in many fields, such as finance, to assess the risk associated with investments. For example, if you know the average return and standard deviation of a stock, you can calculate the probability of the return falling within a certain range. This helps investors make informed decisions about their investments. Another example is in quality control, where you can calculate the probability that a product's measurement falls within acceptable limits.
To find P ( 0.6 ≤ z ≤ 2.12 ) , we use the CDF values for the standard normal distribution: Φ ( 2.12 ) ≈ 0.9830 and Φ ( 0.6 ) ≈ 0.7257 . The probability is calculated as 0.9830 − 0.7257 ≈ 0.2573 , which is about 26%.
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