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 1.31 .
(Round to two decimal places as needed.)
The $z$-score for 1200 is -0.89
(Round to two decimal places as needed.)
The $z$-score for 2170 is 2.16 .
(Round to two decimal places as needed.)
The $z$-score for 1380 is
(Round to two decimal places as needed.)
Answer
Final Answer: The z-scores for the values 1900, 1200, 2170, and 1380 are approximately \(\boxed{1.31}\), \(\boxed{-0.89}\), \(\boxed{2.16}\), and \(\boxed{-0.32}\) respectively. The value 2170 could be considered unusual as its z-score is above 2.
Step 1 ：The z-score is a measure of how many standard deviations an element is from the mean. To calculate the z-score of a value, we subtract the mean from the value and then divide by the standard deviation. In this case, we are given the mean (1483) and the standard deviation (318), and we are asked to calculate the z-scores for the values 1900, 1200, 2170, and 1380.
Step 2 ：Calculate the z-scores for each value using the formula \(z = \frac{x - \mu}{\sigma}\), where \(x\) is the value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
Step 3 ：The z-scores for the values 1900, 1200, 2170, and 1380 are approximately 1.31, -0.89, 2.16, and -0.32 respectively. These values indicate how many standard deviations each value is from the mean. A z-score of 1.31 means that the value 1900 is 1.31 standard deviations above the mean, a z-score of -0.89 means that the value 1200 is 0.89 standard deviations below the mean, and so on.
Step 4 ：In terms of whether any of the values are unusual, it is generally considered that a z-score above 2 or below -2 is unusual, as it means the value is more than 2 standard deviations from the mean. In this case, the value 2170 has a z-score of 2.16, which is above 2, so it could be considered unusual.
Step 5 ：Final Answer: The z-scores for the values 1900, 1200, 2170, and 1380 are approximately \(\boxed{1.31}\), \(\boxed{-0.89}\), \(\boxed{2.16}\), and \(\boxed{-0.32}\) respectively. The value 2170 could be considered unusual as its z-score is above 2.
