Discover answers to your most pressing questions at Westonci.ca, the ultimate Q&A platform that connects you with expert solutions. Explore in-depth answers to your questions from a knowledgeable community of experts across different fields. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
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.
### 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 appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
