Which of the following statements is equivalent to $P (z<-2.1)$?
A. $P(z>-2.1)$
B. $1-P(z<2.1)$
C. $P(z<2.1)$
D. $1-P(z>2.1)$
Answer (2)
Use the symmetry property of the standard normal distribution: a)"> P ( z < − a ) = P ( z > a ) .
Apply the complement rule: a) = 1 - P(z < a)"> P ( z > a ) = 1 − P ( z < a ) .
Combine these properties: 2.1) = 1 - P(z < 2.1)"> P ( z < − 2.1 ) = P ( z > 2.1 ) = 1 − P ( z < 2.1 ) .
The equivalent statement is: 1 − P ( z < 2.1 ) .
Explanation
Understand the problem We are given the probability P ( z < − 2.1 ) and need to find an equivalent expression from the provided options. Here, z represents a random variable following the standard normal distribution, which is symmetric around 0.
Apply the symmetry property The key property we'll use is the symmetry of the standard normal distribution. This means that the probability of z being less than a negative value is equal to the probability of z being greater than the corresponding positive value. Mathematically, this is expressed as: a)"> P ( z < − a ) = P ( z > a ) In our case, a = 2.1 , so we have: 2.1)"> P ( z < − 2.1 ) = P ( z > 2.1 )
Use the complement rule Another important property is the complement rule, which states that the probability of an event occurring is equal to 1 minus the probability of the event not occurring. In terms of z , this can be written as: a) = 1 - P(z < a)"> P ( z > a ) = 1 − P ( z < a ) Applying this to our case, we have: 2.1) = 1 - P(z < 2.1)"> P ( z > 2.1 ) = 1 − P ( z < 2.1 )
Combine the properties Now, we can combine the symmetry property and the complement rule to find an expression equivalent to P ( z < − 2.1 ) . From step 2, we have 2.1)"> P ( z < − 2.1 ) = P ( z > 2.1 ) . From step 3, we have 2.1) = 1 - P(z < 2.1)"> P ( z > 2.1 ) = 1 − P ( z < 2.1 ) . Therefore: P ( z < − 2.1 ) = 1 − P ( z < 2.1 )
Identify the correct option Comparing this result with the given options, we see that 1 − P ( z < 2.1 ) is one of the options.
State the final answer Therefore, the statement equivalent to P ( z < − 2.1 ) is 1 − P ( z < 2.1 ) .
Examples
Understanding probabilities related to the standard normal distribution is crucial in many fields. For instance, in quality control, it helps determine the likelihood of a product's measurements falling within acceptable limits. In finance, it's used to assess the risk associated with investments by estimating the probability of returns falling below a certain threshold. These probabilities are also used in hypothesis testing to determine the statistical significance of results.
The expression equivalent to P ( z < − 2.1 ) is 1 − P ( z < 2.1 ) , using properties of symmetry and the complement rule in probability. Therefore, the correct choice is option B: 1 − P ( z < 2.1 ) .
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