Question

The Procter & Gamble Company is an American multinational consumer goods corporation headquartered in Cincinnati, Ohio. One of Procter & Gamble’s products is Cascade Complete Dishwasher Detergent. The weight of boxes of Cascade Complete Dishwasher Detergent is normally distributed, and has a mean of 75 ounces and a standard deviation of 2.6 ounces.

What is the probability that a single box of Cascade Complete Dishwasher Detergent selected at random will weigh more than 80 ounces?

What is the probability that a single box of Cascade Complete Dishwasher Detergent selected at random will weigh less than 73 ounces?

What percent of the boxes of Cascade Complete Dishwasher Detergent will weigh between 71.8 ounces and 74.4 ounces?

Sixty-seven percent of the boxes of Cascade Complete Dishwasher Detergent should be above what weight?

Answer 1

Solution :

Given that mean μ = 75 , standard deviation σ = 2.6

=> P(x > 80) = P((x - μ)/σ > (80 - 75)/2.6)

= P(Z > 1.9231)

= 1 - P(Z < 1.9231)

= 1 - 0.9726

= 0.0274

=> P(Z < 73) = P((x - μ)/σ < (73 - 75)/2.6)

= P(Z < -0.7692)

= 1 - P(Z < 0.7692)

= 1 - 0.7794

= 0.2206

=> P(71.8 < x < 74.4) = P((71.8 - 75)/2.6 < (x - μ)/σ < (74.4 - 75)/2.6)

= P(-1.2308 < Z < -0.2308)

= P(Z < -0.2308) - P(Z < -1.2308)

= 0.4090 - 0.1093

= 0.2997

= 29.97%

=> P(Z < z) = 0.67

=> z = 0.4399

=> X = μ + z*σ

= 75 + 0.4399*2.6

= 76.1437 ounces