Z Score Review Question
TEK S(3)(D) describe and model variability using population and sampling distributions.
10. A consumer testing laboratory is evaluating the battery lifespan of a newly released wireless headphone model during continuous streaming. The battery life for this model is known to possess a normal distribution with a mean of 450 minutes and a standard deviation of 50 minutes. Find the z-score that a randomly selected pair of headphones lasts 335 minutes.
Z Score Answer Key
TEKS S(3)(D) Teacher Reference • Answer Key
Review Question Solution & Scoring Guide
Official Key
Question 10 Prompt Standardized Test Item
10. A consumer testing laboratory is evaluating the battery lifespan of a newly released wireless headphone model during continuous streaming. The battery life for this model is known to possess a normal distribution with a mean of 450 minutes and a standard deviation of 50 minutes. Find the z-score that a randomly selected pair of headphones lasts 335 minutes.
Step-by-Step Mathematical Solution
1
Identify Given Parameters from Problem Text:
Population Mean (\(\mu\)) = 450 min Standard Deviation (\(\sigma\)) = 50 min Observed Value (\(x\)) = 335 min
2
State the Standard Normal Score (z-score) Formula:
\[ z = \frac{x - \mu}{\sigma} \]
3
Substitute Values and Simplify:
\[ z = \frac{335 - 450}{50} = \frac{-115}{50} = -2.30 \]
Correct Final Answer
\( z = -2.30 \) (or \( -2.3 \))
Contextual Meaning: A battery duration of 335 minutes is 2.30 standard deviations below the average battery life.
Common Student Errors
- Sign Reversal (\(+2.30\)): Student computes \(\mu - x\) instead of \(x - \mu\). Remind students that values below the mean must produce negative z-scores.
- Arithmetic Error: Miscalculating \(335 - 450\) as \(-125\) or \(-105\).
- Units on z-score: Adding "minutes" to \(z\). Note that z-scores are dimensionless standardized quantities.
Suggested Scoring Rubric (3 Pts)
1 Pt: Correct formula setup & substitution 1 Point
1 Pt: Accurate numerator calculation (\(-115\)) 1 Point
1 Pt: Correct quotient with negative sign (\(-2.30\)) 1 Point
TEKS S(3)(D) Review Question Key • Population & Sampling Variability Secondary Statistics Assessment Bank