Suppose Z follows the standard normal distribution. Use the calculator provided, or this table, to determine the value of a so that the following is true.

P(-c≤Z≤c)=0.9500

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

To solve for the value of \( c \) such that \( P(-c \leq Z \leq c) = 0.9500 \) where \( Z \) follows the standard normal distribution, we follow these steps:

1. **Understand the Problem:**
   - The probability \( P(-c \leq Z \leq c) = 0.9500 \) implies that the middle 95% of the standard normal distribution lies between \(-c\) and \( c \).
   - Since the standard normal distribution is symmetric about 0, the probability to the left of \(-c\) and to the right of \( c \) is equal, each accounting for \( (1 - 0.9500)/2 = 0.0250 \).

2. **Use the Standard Normal Distribution:**
   - We need to find the z-value \( c \) such that the cumulative probability to the left of \( c \) is \( 0.9750 \) (since 0.9500 + 0.0250 = 0.9750).

Let's calculate the value of \( c \).

The value of \( c \) such that \( P(-c \leq Z \leq c) = 0.9500 \) is \( c = 1.96 \) (rounded to two decimal places).

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. What is the value of \( c \) for \( P(-c \leq Z \leq c) = 0.99 \)?
2. How would you interpret the value \( c = 1.96 \) in a confidence interval context?
3. What is \( P(Z \leq -1.96) \) in the standard normal distribution?
4. How does \( c \) change if the probability \( P(-c \leq Z \leq c) \) decreases to 0.90?
5. How do you use \( c \) to find critical values for a two-tailed test?

**Tip:** The value \( c = 1.96 \) is often used in statistics for constructing 95% confidence intervals, a common scenario in hypothesis testing.

Learn how to calculate the value of c for the standard normal distribution, ensuring that P(-c ≤ Z ≤ c) = 0.9500. Step-by-step solution with detailed computations and interpretation. Suitable for advanced high school and undergraduate students.

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