PHI

Application:

Quality Control of Coffee Bean Weight

A coffee company, "Aroma Roasters," packages its ground coffee into bags. The company's quality control department wants to ensure that the weight of each bag is within an acceptable range. They have found that the weight of the coffee in the bags is normally distributed with a mean (μ) of 500 grams and a standard deviation (σ) of 5 grams.

The company sets a standard that bags with weights below 495 grams are considered underweight and must be repackaged. They want to know what percentage of their bags are expected to be underweight.

Step 1: Standardize the value

To use the $\Phi$ function, we first need to convert our value (495 grams) into a z-score. The z-score tells us how many standard deviations away from the mean a data point is. The formula for the z-score is:

$z=\frac{x-\mu }{\sigma }$

Where:

• x = the value we are interested in (495 grams)
• μ = the mean (500 grams)
• σ = the standard deviation (5 grams)

$z=\frac{495-500}{5}=\frac{-5}{5}=-1.00$

So, a bag weighing 495 grams is exactly one standard deviation below the mean.

Step 2: Use the PHI function to find the probability

The probability that a bag weighs less than 495 grams is the same as the probability that its z-score is less than -1.00. This is expressed using the $\Phi$ function as $\Phi$(−1.00).

Standard Normal Distribution Table