Question
A set of data items is normally distributed with a mean of 900 and a standard deviation of 30. Find the data item in this distribution that corresponds to the given \( z \)-score. \( z=-2.5 \) The data item that corresponds to \( z=-2.5 \) is (Type an integer or a decimal.)
Ask by Huang Mckenzie. in the United States
Feb 26,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The data item corresponding to \( z = -2.5 \) is 825.
Solution
The formula to convert a z-score to the original data item is:
Data item = Mean + (z-score × Standard Deviation)
Plugging in the values:
Data item = 900 + (-2.5 × 30)
Data item = 900 - 75
Data item = 825
Thus, the data item corresponding to z = -2.5 is 825.
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Simplify this solution Extra Insights
To find the data item that corresponds to the given \( z \)-score of -2.5, you can use the formula: \[ X = \mu + (z \times \sigma) \] Where: - \( X \) is the data item we want to find, - \( \mu \) is the mean (which is 900), - \( z \) is the z-score (-2.5), - \( \sigma \) is the standard deviation (30). Substituting in the values: \[ X = 900 + (-2.5 \times 30) \] \[ X = 900 - 75 \] \[ X = 825 \] The data item that corresponds to \( z = -2.5 \) is **825**.
