Assume that the random variable (Z) is normally distributed with a mean of 0 and a standard deviation of 1.

If P(-b<Z<b) find b= 0.581. Please round your answer to 2 decimal places.

 b=


To find the value of \( b \) for which \( P(-b < Z < b) = 0.581 \), we need to use the properties of the standard normal distribution, where \( Z \) has a mean of 0 and a standard deviation of 1.

1. First, we know that \( P(-b < Z < b) = 0.581 \).
2. The standard normal distribution is symmetric, so \( P(-b < Z < b) = 2 \times P(0 < Z < b) \).
   \[
   P(0 < Z < b) = \frac{0.581}{2} = 0.2905
   \]
3. Now, we need to find the value of \( b \) such that the probability to the right of 0 and up to \( b \) is 0.2905. This can be done by using the standard normal distribution table (Z-table) or by using the inverse cumulative distribution function (also known as the Z-inverse or the quantile function).

Using the Z-table or a statistical calculator, we find that the Z-value corresponding to a cumulative probability of \( 0.5 + 0.2905 = 0.7905 \) is approximately \( b = 0.81 \).

Thus, the value of \( b \) is approximately:

\[
\boxed{0.81}
\]

Would you like further clarification on the process?

Here are 5 related questions to expand on this concept:
1. What is the value of \( P(Z < 0.81) \) for a standard normal distribution?
2. How would the value of \( b \) change if \( P(-b < Z < b) \) was 0.90?
3. How do we interpret the standard normal distribution in real-life applications?
4. What does the symmetry of the normal distribution imply about probabilities for positive and negative Z-scores?
5. How do you calculate the Z-score for any normal distribution, not just the standard normal?

**Tip:** The standard normal distribution always has a mean of 0 and a standard deviation of 1, which allows easy lookup of probabilities using Z-tables.

Assume that the random variable (Z) is normally distributed with a mean of 0 and a standard deviation of 1. If P(-b < Z < b) = 0.581, find b. Please round your answer to 2 decimal places.

Learn how to find the value of b for a standard normal distribution where P(-b < Z < b) = 0.581. This solution involves understanding the properties of the normal distribution, using Z-tables, and applying the inverse cumulative distribution function. Ideal for students in Grades 10-12 studying statistics and probability.

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