Let z be a standard normal random variable with mean 𝜇 = 0 and standard deviation 𝜎 = 1. Find the value c that satisfies the inequality. (Round your answer to two decimal places.)
P(z > c) = 0.0268

To find the value \( c \) such that \( P(z > c) = 0.0268 \), where \( z \) is a standard normal random variable (with mean \( \mu = 0 \) and standard deviation \( \sigma = 1 \)), we follow these steps:

### Step 1: Understand the probability relation
The standard normal distribution is symmetric about 0. We are looking for the value of \( c \) such that the probability of \( z \) being greater than \( c \) equals 0.0268:

\[
P(z > c) = 0.0268
\]

This means we need to find \( c \) such that the area to the right of \( c \) under the standard normal curve is 0.0268.

### Step 2: Use the cumulative distribution function (CDF)
The cumulative distribution function \( P(z \leq c) \) gives the probability that \( z \) is less than or equal to a certain value. The relationship between the cumulative probability and the given probability is:

\[
P(z > c) = 1 - P(z \leq c)
\]

So, we can rewrite the equation as:

\[
1 - P(z \leq c) = 0.0268
\]

Solving for \( P(z \leq c) \):

\[
P(z \leq c) = 1 - 0.0268 = 0.9732
\]

### Step 3: Find the z-score corresponding to the cumulative probability
Now, we need to find the z-score that corresponds to a cumulative probability of 0.9732. This can be done by looking up the value in the standard normal distribution table or using an inverse cumulative distribution function (also called the percentile function or inverse CDF).

From the standard normal distribution table or using a calculator, the z-score that corresponds to a cumulative probability of 0.9732 is approximately **1.94**.

Thus, \( c = 1.94 \).

### Final Answer:
The value of \( c \) that satisfies \( P(z > c) = 0.0268 \) is approximately **1.94**.

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Would you like any further explanation or details?

Here are 5 related questions to explore further:
1. How do you find the z-score corresponding to a given probability without a table?
2. What is the significance of the standard normal distribution in statistics?
3. How does the symmetry of the normal distribution affect probability calculations?
4. Can we use the same method to find the value of \( c \) for \( P(z < c) = 0.0268 \)?
5. How would the process differ if \( \mu \) and \( \sigma \) were not standard values?

**Tip**: When using z-tables, always remember that they provide the cumulative probability for values less than or equal to a given z-score.

Let z be a standard normal random variable with mean 𝜇 = 0 and standard deviation 𝜎 = 1. Find the value c that satisfies the inequality. P(z > c) = 0.0268 (Round your answer to two decimal places.)

This problem involves finding the value of c for which P(z > c) = 0.0268 in a standard normal distribution, with mean 0 and standard deviation 1. The solution uses cumulative distribution functions and the inverse CDF to determine the correct z-score. Suitable for high school students (Grades 10-12).

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