Which is equivalent to [tex]$P (z \geq 1.4) ?$[/tex]
[tex]$\begin{array}{l}
P(z \leq 1.4) \
1-P(z \leq 1.4) \
P(z \geq-1.4)
\end{array}$[/tex]
Answer (2)
The problem requires finding an expression equivalent to P ( z g e 1.4 ) .
Using the property that P ( z l e a ) + P ( z g e a ) = 1 , we rewrite P ( z g e 1.4 ) as 1 − P ( z l e 1.4 ) .
We verify that P ( z g e 1.4 ) is not equal to P ( z l e 1.4 ) or P ( z g e − 1.4 ) .
The equivalent expression is 1 − P ( z l e 1.4 ) .
Explanation
Understand the problem and provided data We are given the probability P ( z g e 1.4 ) and asked to find an equivalent expression from the options:
P ( z l e 1.4 )
1 − P ( z l e 1.4 )
P ( z g e − 1.4 )
Here, z represents a standard normal random variable.
Express P(z >= a) in terms of P(z <= a) Recall that the total probability under the standard normal curve is 1. Therefore, for any value a , we have
P ( z < a ) + P ( z g e a ) = 1
Since z is a continuous random variable, P ( z < a ) = P ( z l e a ) . Thus,
P ( z l e a ) + P ( z g e a ) = 1
Rearranging this equation, we get
P ( z g e a ) = 1 − P ( z l e a )
Identify the equivalent expression Using the result from the previous step, we can write
P ( z g e 1.4 ) = 1 − P ( z l e 1.4 )
This matches option 2.
Analyze the other options The standard normal distribution is symmetric about 0. This means that P ( z l e − a ) = P ( z g e a ) and P ( z g e − a ) = P ( z l e a ) .
Therefore, P ( z g e − 1.4 ) = P ( z l e 1.4 ) .
Since P ( z g e 1.4 ) = 1 − P ( z l e 1.4 ) , we can see that P ( z g e 1.4 ) e qP ( z l e 1.4 ) and P ( z g e 1.4 ) e qP ( z g e − 1.4 ) .
Final Answer Therefore, the expression equivalent to P ( z g e 1.4 ) is 1 − P ( z l e 1.4 ) .
Examples
In statistical analysis, understanding probabilities related to the standard normal distribution is crucial. For instance, if you're analyzing test scores that follow a normal distribution, you might want to find the probability that a student scores above a certain value. Knowing that P ( z g e 1.4 ) = 1 − P ( z l e 1.4 ) allows you to use standard normal distribution tables or software to easily find this probability.
The expression equivalent to P ( z ≥ 1.4 ) is 1 − P ( z ≤ 1.4 ) . This is derived from the property of probabilities in a standard normal distribution. The choice that matches this is option 2, 1 − P ( z ≤ 1.4 ) .
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