Probability & Distribution question

Question

Thank you so much!! I really want to master this unit :(

Answer 1

Variance (S²) is the square of the standard deviation (S). Notice your units,

S² = 1.5 ml²

so

S = √S2 = 1.225 ml

The slightly light bottle is

(x - µ) / S = (499 ml - 500 ml) / 1.225 ml ≅ -0.82

standard deviations below the mean (µ = 500 ml). Generally, statistics texts provide a pre-computed table of values somewhere in the back of the book. Using such, we find that 0.21 of the area lies to the left of our Z-score (-0.82), i.e. 21% of all water bottles will have ≤ 499 ml.

Likewise, using the same table, we find that 5% = 0.05 of the area lies to the left of a Z-score of -2.575. That many standard deviations equals:

-2.575 s.d. × 1.225 ml / s.d. ≅ -3.15 ml

below the mean, i.e. 5% of all bottles contain ≤ 497 ml or so