Final answer:
The value of a random variable X for a z-score of 1 is found using the formula X = μ + (z)(σ), where μ is the mean and σ is the standard deviation of the normal distribution X. For example, if X~N(5, 6), then X equals 11 when the z-score is 1.
Step-by-step explanation:
The student has posed a question regarding the z-score and its application to normally distributed variables. Specifically, the student is asking what the value of the realization of a random variable X is when the z-score is equal to 1.
To determine this, we use the formula for finding the value of X from its z-score:
X = μ + (z)(σ)
Where μ is the mean of the normal distribution X and σ is the standard deviation. If X~N(μ, σ), and z is the z-score for a value x from this distribution, then the z-score tells how many standard deviations x is above (if positive) or below (if negative) the mean μ. For z = 1, the value of x would be one standard deviation above the mean, which can be calculated using the provided formula. As an example, if the distribution is X~N(5, 6), then:
X = 5 + (1)(6) = 11
So, in this case, x equals 11 when the z-score is 1.