A standardized exam's scores are normally distributed. In a recent year, the mean test score was 1483 and the standard deviation was 318. The test scores of four students selected at random are 1900, 1200, 2170, and 1380. Find the $z$-scores that correspond to each value and determine whether any of the values are unusual.
The $z$-score for 1900 is (Round to two decimal places as needed.)
Answer
Final Answer: The z-score for 1900 is \(\boxed{1.31}\).
Step 1 ：Given that the mean score is 1483 and the standard deviation is 318, we need to find the z-score for the score 1900.
Step 2 ：The z-score is a measure of how many standard deviations an element is from the mean. To find the z-score of a value, we subtract the mean from the value and then divide by the standard deviation.
Step 3 ：Using the formula for z-score, \(z = \frac{x - \mu}{\sigma}\), where \(x\) is the value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation, we substitute the given values into the formula.
Step 4 ：So, \(z = \frac{1900 - 1483}{318} = 1.3113207547169812\).
Step 5 ：Rounding to two decimal places, we get \(z = 1.31\).
Step 6 ：Final Answer: The z-score for 1900 is \(\boxed{1.31}\).
