Question

An intern at a wine bottling plant is tasked with measuring the total weight and average pH of each wine shipment comprising 50 bottles of wine.

(a) The shipping & handling team typically measures the individual weight of each bottle (in grams) using an analytical weighing balance that is correct to the nearest microgram. However, the intern chooses to round off the actual measured weight to the nearest integer (grams) before summing individual weights to estimate the net weight of the pack of fifty bottles. What is the probability that the resultant total weight differs from the actual weight by more than ? ??

(b) A high precision pH meter is available to the intern to estimate the acidity/alkalinity of wine. As usual, the intern chooses to round off the actual measured pH to the nearest integer before taking an average of pH measurements across ?0 independent wine samples. What is the probability that the resultant average pH differs from the actual average by more than ?.? pH units?

Answer 1

Given,

Sample size, n = 50

(a)

Let

x denote the resultant estimate of total weight of the pack of fifty bottles

X denote the resultant total actual weight of overall shipment

Assuming that X ~ Normal(0,1), implies that x ~ Normal(0,1).

Since sample wine bottles were picked from the shipment

Hence,

The probability that the resultant total weight differs from the actual weight by more than ? ?

(b)

Let

mx denotes the resultant average pH measured across 50 independent wine samples

mX denotes the resultant actual average pH measured across the wine shipment

Assuming mX ~ Normal(0,1), implies that mx ~ Normal(0,1).

Since sample wine bottles were picked from the shipment

Hence,

The probability that the resultant average pH differs from the actual average by more than ?.? pH units