The mean of exam scores in a math class was 85 with a standard deviation of 8.
(a) Lisa had a score of 95, find her z-score.
(b) Jose had a [tex]$z$-score[/tex] of -1.25, find his exam score.
Answer (2)
Lisa's z-score is calculated to be 1.25, indicating she scored above the mean. Jose's exam score, derived from his z-score of -1.25, is 75, indicating he scored below the mean. Both scores help understand their performance relative to the class average.
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Calculate Lisa's z-score: z = 8 95 − 85 = 1.25 .
Calculate Jose's exam score: x = ( − 1.25 ) ( 8 ) + 85 = 75 .
Lisa's z-score indicates her position above the mean.
Jose's score is derived from his z-score relative to the mean: 75 .
Explanation
Understanding the Problem Let's break down this problem step by step. We're given the mean and standard deviation of exam scores, and we need to find Lisa's z-score and Jose's actual exam score.
Finding Lisa's Z-score (a) To find Lisa's z-score, we'll use the z-score formula, which tells us how many standard deviations away from the mean her score is. The formula is: z = σ x − μ where:
x is Lisa's score
μ is the mean of the exam scores
σ is the standard deviation
Calculating Lisa's Z-score Now, let's plug in the values we know:
x = 95
μ = 85
σ = 8 So, the calculation becomes: z = 8 95 − 85 = 8 10 = 1.25 Lisa's z-score is 1.25. This means her score is 1.25 standard deviations above the mean.
Finding Jose's Exam Score (b) Next, we need to find Jose's actual exam score, given his z-score. We'll rearrange the z-score formula to solve for x (Jose's score): x = z σ + μ where:
z is Jose's z-score
μ is the mean of the exam scores
σ is the standard deviation
Calculating Jose's Exam Score Now, let's plug in the values we know:
z = − 1.25
μ = 85
σ = 8 So, the calculation becomes: x = ( − 1.25 ) ( 8 ) + 85 = − 10 + 85 = 75 Jose's exam score is 75. This means his score is 1.25 standard deviations below the mean.
Final Answer Therefore, Lisa's z-score is 1.25 and Jose's exam score is 75.
Examples
Understanding z-scores is incredibly useful in many real-world scenarios. For instance, in finance, z-scores can help assess the creditworthiness of a company by measuring its financial stability relative to its peers. In healthcare, z-scores are used to track a child's growth relative to the average growth patterns. In education, like in this problem, z-scores help to quickly understand how a student performed compared to the rest of the class, adjusting for the class's average performance and the spread of scores.
