Suppose Z follows the standard normal distribution. Use the calculator provided, or this table, to determine the value of e so that the following is true. P(0.52≤Z≤c)=0.2813

Carry your intermediate computations to at least four decimal places. Round your answer to two decimal places.

To find the value of \( c \) such that \( P(0.52 \leq Z \leq c) = 0.2813 \), where \( Z \) follows the standard normal distribution, we can follow these steps:

1. **Understand the Problem:**
   - We know the probability between 0.52 and \( c \) is 0.2813.
   - First, we'll find the cumulative probability \( P(Z \leq 0.52) \) using the standard normal distribution.
   - Then, we'll add 0.2813 to this cumulative probability to find \( P(Z \leq c) \).
   - Finally, we'll find the corresponding \( c \)-value for this cumulative probability.

2. **Steps to Calculate \( c \):**
   - Calculate \( P(Z \leq 0.52) \).
   - Add 0.2813 to this probability to find \( P(Z \leq c) \).
   - Find the \( c \)-value corresponding to this cumulative probability.

Let's calculate this step by step.The value of \( c \) such that \( P(0.52 \leq Z \leq c) = 0.2813 \) is \( c = 2.05 \) (rounded to two decimal places).

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. How would you find \( c \) if \( P(-0.52 \leq Z \leq c) = 0.2813 \)?
2. What is \( P(Z \leq 2.05) \) for the standard normal distribution?
3. How would the value of \( c \) change if \( P(0.52 \leq Z \leq c) \) were 0.35 instead of 0.2813?
4. What is the interpretation of a z-value of 2.05 in a standard normal distribution?
5. How would you solve for \( c \) if the probability was given as \( P(0 \leq Z \leq c) = 0.2813 \)?

**Tip:** Understanding how to manipulate probabilities and z-values is essential when working with the standard normal distribution, especially for hypothesis testing and confidence intervals.

Learn how to find the value of c in a standard normal distribution problem where P(0.52 ≤ Z ≤ c) = 0.2813. Understand the steps involved, including calculating cumulative probabilities and determining the corresponding c-value. Suitable for advanced high school students studying probability and statistics.

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