Westonci.ca is the best place to get answers to your questions, provided by a community of experienced and knowledgeable experts. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our 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.
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.