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The question asks to find \( a \) such that \( P(Z > a) = 0.24 \), where \( Z \sim N(0, 1) \) is a standard normal distribution.

This is essentially asking for the value of \( a \) such that the probability that \( Z \) (a standard normal random variable) is greater than \( a \) is 0.24. In terms of the cumulative distribution function (CDF) \( \Phi(a) \), this can be written as:

\[
P(Z > a) = 1 - \Phi(a) = 0.24
\]

Therefore, \( \Phi(a) = 1 - 0.24 = 0.76 \). Now, we need to find the value of \( a \) such that \( \Phi(a) = 0.76 \), which corresponds to finding the inverse of the standard normal distribution for a probability of 0.76.

Let me calculate that for you.The value of \( a \) such that \( P(Z > a) = 0.24 \) is approximately \( a = 0.71 \).

Would you like a detailed explanation of the steps, or do you have any questions?

Here are 5 related questions to expand your understanding:
1. What is the significance of the cumulative distribution function (CDF) in statistics?
2. How do you calculate probabilities for non-standard normal distributions?
3. What is the relationship between percentiles and standard scores (z-scores)?
4. How would the calculation change if the probability given was \( P(Z < a) \)?
5. How does the area under the standard normal curve help in understanding probabilities?

**Tip:** When working with standard normal distributions, remember that the total area under the curve is 1, and the curve is symmetric about the mean, \( Z = 0 \).

Let Z ~ N(0,1). Find a so that P(Z > a) = 0.24.

Learn how to calculate the value of 'a' in a standard normal distribution where P(Z > a) = 0.24. This involves using the cumulative distribution function (CDF) and finding the inverse to determine the critical value of 'a'. Suitable for undergraduate statistics students.

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