For a standard normal distribution, which of the following expressions must always be equal to 1?
P(z ≤ -a) - P(-a ≤ z ≤ a) - P(z ≥ a)
P(z ≤ -a) - P(-a ≤ z ≤ a) + P(z ≥ a)
P(z ≤ -a) + P(-a ≤ z ≤ a) - P(z ≥ a)
P(z ≤ -a) + P(-a ≤ z ≤ a) + P(z ≥ a)
Answer (2)
The expression that must always equal 1 in a standard normal distribution is P ( z ≤ − a ) + P ( − a ≤ z ≤ a ) + P ( z ≥ a ) . The sum of the probabilities of these three intervals represents the total probability, which is always 1. Therefore, the correct choice is the fourth option.
;
The total probability for any distribution is 1.
For a standard normal distribution, we have a) = 1"> P ( z ≤ − a ) + P ( − a < z ≤ a ) + P ( z > a ) = 1 .
Comparing the given expressions, only a)"> P ( z ≤ − a ) + P ( − a < z ≤ a ) + P ( z > a ) matches this.
Therefore, the expression that must always be equal to 1 is: a)}"> P ( z ≤ − a ) + P ( − a < z ≤ a ) + P ( z > a ) .
Explanation
Understanding the problem We are given a standard normal distribution and need to identify which expression equals 1. The key property of any probability distribution is that the total probability over its entire range is 1. For a standard normal distribution, this means that the probability of the random variable taking any value within its range (-infinity to +infinity) is 1. We can divide this range into disjoint intervals and sum their probabilities.
Expressing total probability Let's denote the standard normal random variable as z. We are given a real number a. The expressions involve probabilities related to z being less than or equal to -a, between -a and a, and greater than a. We know that:
a) = 1"> P ( z ≤ − a ) + P ( − a < z ≤ a ) + P ( z > a ) = 1
This equation represents the total probability over the entire range, split into three disjoint intervals.
Comparing the expressions Now, let's examine the given expressions and compare them to the equation above:
a)"> P ( z ≤ − a ) − P ( − a < z ≤ a ) − P ( z > a )
a)"> P ( z ≤ − a ) − P ( − a < z ≤ a ) + P ( z > a )
a)"> P ( z ≤ − a ) + P ( − a < z ≤ a ) − P ( z > a )
a)"> P ( z ≤ − a ) + P ( − a < z ≤ a ) + P ( z > a )
Only the fourth expression matches the equation representing the total probability of 1.
Examples
Consider a quality control process where 'a' represents a tolerance level. We want to know the probability that a manufactured part falls within acceptable limits. If 'z' represents the deviation from the ideal measurement, then P(z <= -a) is the probability of being too small, P(-a < z <= a) is the probability of being within tolerance, and P(z > a) is the probability of being too large. The sum of these probabilities must equal 1, representing all possible outcomes.
