Z-table (Standard Normal Distribution Table)

In potato chip manufacturing, the normal distribution (ND) finds application in ensuring the consistency of product weights. The goal is to meet customer expectations regarding the weight of chips in each package. Here’s how ND is applied:

Quality Control in Potato Chip Production: Manufacturers use ND to model the distribution of weights of individual potato chip bags. The assumption is that the weights of the bags follow a normal distribution, allowing for statistical analysis of the production process.

Example Scenario: Suppose a potato chip company aims to pack chips in 150-gram bags. Due to variations in the manufacturing process, the actual weights of the bags may deviate slightly, and let’s assume a standard deviation of 7-gram. The company can use the normal distribution to estimate the percentage of bags falling below or above the target weight.

In this scenario:

• Target weight (μ) (mean) of potato chip bags is 150 grams.
• Standard deviation (σ) is 7 grams.

Let’s consider two scenarios:

1. Bags falling below the target weight (e.g., X =140 grams).
2. Bags falling above the target weight (e.g., X =160 grams).

Z = X−μ / σ

For scenario 1: Z = 140−150​ / 7 = −1.43

For scenario 2: Z = 160−150 / 7​ = 1.43

Now, we can interpret these Z-scores in terms of standard deviations from the mean:

1. A Z-score of -1.43 (for a bag weighing 140 grams) means that the weight of this bag is 1.43 standard deviations below the average or target weight of 150 grams. This bag is lighter than the average, and the negative sign indicates it falls below the mean.
2. A Z-score of 1.43 (for a bag weighing 160 grams) means that the weight of this bag is 1.43 standard deviations above the average or target weight of 150 grams. This bag is heavier than the average, and the positive sign indicates it falls above the mean.

These Z-scores can be used to estimate the percentage of bags falling below or above specified weights, based on the standard normal distribution table or a calculator. For example, you can use the above given Z-table to find the percentage of values below Z ​=−1.43 and above Z ​=1.43 to assess the quality and consistency of the chip weights.

Assuming a standard normal distribution table, we find the probability associated with a Z-score of -1.43. The table typically provides the cumulative probability up to a certain Z-score. Let’s say the probability is P (Z < −1.43).

For a more precise result, consult the above Z-table or use a statistical calculator. Therefore, the probability that a randomly selected bag weighs less than 140 grams = (0.5^* – 0.4236) = is approximately 0.0764, or 7.64%. This means there’s a 7.64% chance of randomly selecting a bag that is lighter than 140 grams based on the normal distribution assumption.

* Please note that the total area under the normal distribution curve is equal to 1 because the normal distribution is a probability distribution. In a probability distribution, the entire set of possible outcomes covers the entire probability space, and the sum of all probabilities must be equal to 1. If we consider only half of it, we have 0.5. For example, if we’re interested in bags lighter than a certain weight (e.g., 150 grams), we focus on the left side of the curve. In this scenario, we might say, ‘Let’s look at the area to the left of 150 grams.’ This represents the half (0.5) we are referring to, and the other half pertains to the opposite side. It’s essentially a matter of directing our attention to the specific part of the distribution that is relevant to our question or hypothesis.

Application:

• Quality Assurance: By employing statistical tools based on the normal distribution, the company can estimate the percentage of bags that fall below the specified weight threshold. This insight is crucial for quality assurance and helps in fine-tuning production processes to minimize deviations.
• Customer Satisfaction: Ensuring that a high percentage of bags meet the target weight reduces the likelihood of customer dissatisfaction. Consistency in product quality, in this case, the quantity of chips per bag, is essential for maintaining customer trust and loyalty.
• Process Optimization: The normal distribution analysis allows the manufacturer to identify areas of the production process that may contribute to weight variations. This information can be used to optimize machinery, refine ingredient measurements, and enhance overall production efficiency.

In summary, the normal distribution is a valuable tool in the food processing industry. It helps manufacturers maintain product consistency, meet quality standards, and ultimately enhance customer satisfaction.