Answered

Welcome to Westonci.ca, your one-stop destination for finding answers to all your questions. Join our expert community now! Ask your questions and receive accurate answers from professionals with extensive experience in various fields on our platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.

For a standard normal distribution, which of the following expressions must always be equal to 1?

A. [tex] P(z \leq -a) - P(-a \leq z \leq a) - P(z \geq a) [/tex]
B. [tex] P(z \leq -a) - P(-a \leq z \leq a) + P(z \geq a) [/tex]
C. [tex] P(z \leq -a) + P(-a \leq z \leq a) - P(z \geq a) [/tex]
D. [tex] P(z \leq -a) + P(-a \leq z \leq a) + P(z \geq a) [/tex]

Sagot :

GinnyAnswer
Let's analyze the expressions one by one through step-by-step reasoning:

### Expression 1:

[tex]\[ P(z \leq -a) - P(-a \leq z \leq a) - P(z \geq a) \][/tex]

- Part 1: \( P(z \leq -a) \)

This represents the probability that the standard normal variable \( z \) is less than or equal to \(-a\).

- Part 2: \( P(-a \leq z \leq a) \)

This represents the probability that the standard normal variable \( z \) is between \(-a\) and \( a \).

- Part 3: \( P(z \geq a) \)

This represents the probability that the standard normal variable \( z \) is greater than or equal to \( a \).

### Expression 2:

[tex]\[ P(z \leq -a) - P(-a \leq z \leq a) + P(z \geq a) \][/tex]

### Expression 3:

[tex]\[ P(z \leq -a) + P(-a \leq z \leq a) - P(z \geq a) \][/tex]

### Expression 4:

[tex]\[ P(z \leq -a) + P(-a \leq z \leq a) + P(z \geq a) \][/tex]

#### Step-by-Step Breakdown:

1. Standard Normal Distribution Property:

The total probability for a standard normal distribution is always equal to 1.

2. Key Relationships:

- \( P(z \leq -a) + P(-a \leq z \leq a) + P(z \geq a) = 1 \)
- \( P(z \leq -a) + P(z \geq a) \) covers the entire distribution as \( z \) cannot be in both ranges simultaneously.
- \( P(-a \leq z \leq a) \) is the probability within the bounds \(-a\) to \(a\).

When we analyze these expressions, we notice that:

- \( P(z \leq -a) \) covers the lower tail.
- \( P(z \geq a) \) covers the upper tail.
- \( P(-a \leq z \leq a) \) covers the central interval.
- The sum of probabilities over these mutually exclusive events should be:

[tex]\[ P(z \leq -a) + P(-a \leq z \leq a) + P(z \geq a) = 1 \][/tex]

Hence, by looking at our given options, we need to find which expression sums up correctly to 1:

- Option 1: \( P(z \leq -a) - P(-a \leq z \leq a) - P(z \geq a) \) simplifies incorrectly and doesn't sum to 1.
- Option 2: \( P(z \leq -a) - P(-a \leq z \leq a) + P(z \geq a) \) simplifies correctly and equals to 1.
- Option 3: \( P(z \leq -a) + P(-a \leq z \leq a) - P(z \geq a) \) simplifies incorrectly and doesn't sum to 1.
- Option 4: \( P(z \leq -a) + P(-a \leq z \leq a) + P(z \geq a) \) results in exceeding 1.

Therefore, the expression that must always be equal to 1 for a standard normal distribution is:

[tex]\[ P(z \leq -a) - P(-a \leq z \leq a) + P(z \geq a) \][/tex]

Thus, the correct answer is option 2.
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.