A population with μ = 59 and s = 8 is standardized to create a new distribution with μ = 100 and s = 20. After the transformation, an individual receives a new score of X = 90. The original score for this individual was X = 51.
True or False
For this question I got true
Disagree.
Z = (score-mean)/SD = (51-59)/8 = -1
Z = (score-mean)/SD = (90-100)/20 = -.5
true
In order to determine whether the statement is true or false, we need to consider the process of standardization and how it affects the scores.
Standardization involves transforming the original scores of a population into z-scores using the formula:
z = (X - μ) / s
Where:
X: Original score
μ: Mean of the population
s: Standard deviation of the population
Through standardization, the new distribution will have a mean of 0 (μ' = 0) and a standard deviation of 1 (s' = 1), regardless of the original mean and standard deviation.
To create a new distribution with μ = 100 and s = 20, we need to use the formula for standardization:
z' = (X' - μ') / s'
Where:
X': New score
μ': Desired mean (100 in this case)
s': Desired standard deviation (20 in this case)
Now, let's calculate the z-score for the original score (X = 51) using the given information:
z = (X - μ) / s
z = (51 - 59) / 8
z = -8 / 8
z = -1
Since we know the z-score for the original score, we can calculate the new score (X') using the formula for standardization:
z' = (X' - μ') / s'
-1 = (90 - 100) / 20
Let's solve for X':
-1 * 20 = 90 - 100
-20 = -10
X' = 90 + 20
X' = 110
Now, we can determine whether the statement is true or false.
Statement: After the transformation, an individual receives a new score of X = 90. The original score for this individual was X = 51.
False.
After the transformation, the new score for this individual would be X' = 110, not X = 90.