Problem 30 - Algebra 2

Question

Quality Control Tubs of Better Butter weigh 1.0 each, with a standard deviation of 0.06 Ib. At a quality control checkpoint, 12 of the tubs taken as samples weighed less than 0.88 b. Assume that the weights of the samples were normally distributed. How many tubs of butter were taken as samples?

Step-by-Step Solution

Answer

There were approximately 526 tubs of butter taken as samples.

Step 1: Calculation of Z Score

First, find out the Z Score using the following formula: \( Z = \frac{X - \mu}{\sigma} \) Here, X = 0.88, \(\mu\) (mean) = 1.0 and \(\sigma\) (standard deviation) = 0.06. Substituting the values into the formula gives: \( Z = \frac{0.88 - 1.0}{0.06} \). After calculating, you'll find that Z ≈ -2.

Step 2: Use Z Score in Standard Normal Distribution

The Z score found in step 1 corresponds to the area to the left under the standard normal distribution curve. Looking up -2 in the Z table or using a calculator function, the area can be approximated to 0.0228 (or 2.28% when expressed as a percentage).

Step 3: Determine the Total Number of Tubs

Knowing that 12 tubs represent the 2.28% of the total, the total number of tubs sampled can be found by dividing 12 by 2.28% (or 0.0228 when not in percentage form): \( n = \frac{12}{0.0228} \). After doing the calculation, you'll find that \(n \approx 526\), so approximately 526 tubs of butter were taken as a sample.

Key Concepts

Quality Control

Quality control is a vital process in manufacturing and production that ensures that products meet specified quality standards. This process helps maintain consistency and reliability, reducing the chances of defective items reaching consumers. In the context of the butter tubs, quality control involves regularly checking the weights to ensure they meet the desired specifications.

Implementing quality control involves several steps:

• Collecting samples: Selecting a random group of items to check.
• Measuring attributes: Weighing the tubs to test their compliance with set standards.
• Analyzing data: Using statistical methods, such as standard deviation and mean, to determine if the product quality is consistent.
• Making decisions: Deciding actions to take if items do not meet the set criteria, like halting production or adjusting processes.

Effective quality control aligns with consumer expectations and legal requirements, maintaining the company’s reputation by ensuring the products are reliable and safe.

Normal Distribution

Normal distribution is a key concept in statistics, often referred to as a "bell curve" due to its distinct shape. It describes how data points are expected to be spread around a central value (mean). This distribution is symmetric, with most observations clustering around the central peak and fewer observations at the tails.

Characteristics of normal distribution include:

• Symmetry: The left and right sides of the curve are mirror images.
• Mean, median, and mode are equal and located at the center.
• 68% of data falls within one standard deviation from the mean, 95% within two, and 99.7% within three.

In practice, normal distribution allows us to make inferences about a population based on sample observations. For the butter tubs, assuming normal distribution helps predict the likelihood of any given tub meeting the weight standard.

Z Score

The Z score is a statistical measure that indicates how many standard deviations a data point is from the mean. It is a key component in standardizing data, making it easier to compare results from different data sets.

Z Score Calculation
Using the formula:
\[ Z = \frac{X - \mu}{\sigma} \]
where:

• \( X \) is the data point (e.g., weight of a tub).
• \( \mu \) is the mean of the data set.
• \( \sigma \) is the standard deviation.

This measure assists in determining the position of a value within a distribution, allowing quality control managers to assess the probability of an observed value occurring. For example, a Z score of -2 in the butter tub analysis signifies that the tub weight is two standard deviations below the mean, indicating a small likelihood of occurrence in a normal distribution.

Mean

The mean, or average, is a fundamental concept in statistics that represents the central tendency of a data set. It is calculated by summing all the data points and dividing by the count of data points.

Mean Calculation
The formula for mean is:
\[ \mu = \frac{\sum X}{N} \]
where:

• \( \sum X \) is the sum of all data points.
• \( N \) is the number of data points.

In the context of the butter tubs, the mean weight, 1.0 Ib, serves as a benchmark to evaluate individual tub weights. It helps in identifying how typical or atypical a sample is by comparing it against this average. Observations can be analyzed for variations, helping in establishing weight consistency across production batches.