On a standardized exam, the scores are normally distributed with a mean of 77 and a...
Check the final answer first, then review the worked steps.
Problem
On a standardized exam, the scores are normally distributed with a mean of 77 and a standard deviation of 25. Find the z-score of a person who scored 27 on the exam.
Answer
\(-2\)
Step-by-step solution
- Identify the given information: The problem provides the mean ($\\mu$) and the standard deviation ($\\sigma$) of a normal distribution, as well as a specific score ($x$). Here, $\\mu = 77$, $\\sigma = 25$, and $x = 27$.
- State the z-score formula: The z-score is calculated by subtracting the mean from the raw score and dividing the result by the standard deviation: $$z = \frac{x - \mu}{\\sigma}$$
- Substitute the values into the formula: Plug the given values into the equation: $$z = \frac{27 - 77}{25}$$
- Perform the calculation: First, subtract the mean from the score: $27 - 77 = -50$. Then, divide by the standard deviation: $z = \frac{-50}{25} = -2$. The z-score represents how many standard deviations the score is from the mean.