Applying Cumulative Distributions to Real-World Decisions Quiz

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| Questions: 20 | Updated: Nov 19, 2025
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1) What is the z-score for a rod measuring 117 mm?

Explanation

z = (117 − 120) / 2 = −3 / 2 = −1.5

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About This Quiz
Applying Cumulative Distributions To Real-world Decisions Quiz - Quiz

Here, you'll apply your understanding of cumulative distributions to make real-world decisions based on probability data. The quiz will use practical examples such as quality control, standardized test scores, battery life, and factory production to help you assess cumulative probabilities and make data-driven decisions.

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2) Approximately what proportion of rods are shorter than 117 mm?

Explanation

z = −1.5 → from the z-table, P(Z

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3) What percentage of rods fail for being too short (below 116 mm)?

Explanation

z = (116 − 120)/2 = −4/2 = −2 → P(Z

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4) What percentage of rods are too long (exceeding 123 mm)?

Explanation

z = (123 − 120)/2 = 3/2 = 1.5 → P(Z > 1.5) = 1 − 0.9332 ≈ 0.0668 = 6.68%

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5) Approximately what proportion of rods pass quality control (between 116 mm and 123 mm)?

Explanation

P(116 ≤ X ≤ 123) = Φ(1.5) − Φ(−2) ≈ 0.9332 − 0.0228 = 0.9104 = 91.04%

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6) For a box of 100 rods, what is the expected number of rods that pass quality control?

Explanation

Expected passes = 0.9104 × 100 ≈ 91 rods

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7) Management wants to adjust the process so that 99% of rods pass within symmetric limits around 120 mm. Approximately what half-width (in mm) should the limits be, assuming normality?

Explanation

99% in the center → z ≈ 2.58 → half-width = 2.58 × 2 ≈ 5.16 ≈ 5.15 mm

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8) What is the z-score for the 75th percentile?

Explanation

The 75th percentile corresponds to z ≈ 0.67.

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9) What is College A’s cutoff score (nearest whole number)?

Explanation

Cutoff = 500 + 0.67 × 100 = 567

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10) What is the z-score that corresponds to the top 10%?

Explanation

Top 10% means 90th percentile → z ≈ 1.28.

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11) What is College B’s scholarship cutoff score (nearest whole number)?

Explanation

Cutoff = 500 + 1.28 × 100 = 628

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12) An applicant scores 590. Which is true?

Explanation

590 > 567 (A’s cutoff) but 590

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13) What proportion of batteries last less than 8 hours?

Explanation

z = (8 − 10) / 1.5 ≈ −1.33 → P(Z

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14) What is the probability that a randomly chosen battery passes the field test (≥ 8 hours)?

Explanation

P(X ≥ 8) = 1 − 0.0918 = 0.9082

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15) What is the probability that all 3 batteries pass the field test (each ≥ 8 hours)?

Explanation

P(all 3 pass) = (0.9082)³ ≈ 0.75

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16) The company wants only 5% of batteries to fail the 8-hour requirement. What minimum mean would achieve this?

Explanation

Want P(X

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17) What z-score corresponds to the 2.5th percentile?

Explanation

2.5th percentile corresponds to z ≈ −1.96.

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18) What mean μ should the factory set so only 2.5% are underfilled (below 500 g)?

Explanation

(500 − μ)/6 = −1.96 → μ ≈ 511.8 g

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19) If μ = 512 g and σ = 6 g, what proportion of jars are underfilled (< 500 g)?

Explanation

z = (500 − 512)/6 = −2 → P(Z

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20) The factory wants at most 1% underfilled. What μ is needed?

Explanation

1% tail → z ≈ −2.33 → μ ≈ 515.6 g

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What is the z-score for a rod measuring 117 mm?
Approximately what proportion of rods are shorter than 117 mm?
What percentage of rods fail for being too short (below 116 mm)?
What percentage of rods are too long (exceeding 123 mm)?
Approximately what proportion of rods pass quality control (between...
For a box of 100 rods, what is the expected number of rods that pass...
Management wants to adjust the process so that 99% of rods pass within...
What is the z-score for the 75th percentile?
What is College A’s cutoff score (nearest whole number)?
What is the z-score that corresponds to the top 10%?
What is College B’s scholarship cutoff score (nearest whole number)?
An applicant scores 590. Which is true?
What proportion of batteries last less than 8 hours?
What is the probability that a randomly chosen battery passes the...
What is the probability that all 3 batteries pass the field test (each...
The company wants only 5% of batteries to fail the 8-hour requirement....
What z-score corresponds to the 2.5th percentile?
What mean μ should the factory set so only 2.5% are underfilled...
If μ = 512 g and σ = 6 g, what proportion of jars are underfilled...
The factory wants at most 1% underfilled. What μ is needed?
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