Problem 39 - Basic Statistics for Business and Economics with Student

Question

A Tamiami shearing machine is producing 10 percent defective pieces, which is abnormally high. The quality control engineer has been checking the output by almost continuous sampling since the abnormal condition began. What is the probability that in a sample of 10 pieces: a. Exactly 5 will be defective? 5 or more will be defective?

Step-by-Step Solution

Answer

P(5 defective) ≈ 0.00148, P(5 or more defective) ≈ 0.00155.

Step 1: Define the Problem

We need to find the probability that in a sample of 10 pieces, exactly 5 will be defective, and the probability that 5 or more pieces will be defective. The defect rate is 10%, which will be our probability of success for one defective piece.

Step 2: Identify the Distribution

Since we are dealing with a fixed number of trials (10 pieces) with a known probability of success (10% or 0.1), we use the Binomial distribution. The Binomial distribution is defined as \(B(n, p)\), where \(n\) is the number of trials, and \(p\) is the probability of success.

Step 3: Use the Binomial Formula

The probability of finding exactly \(k\) defective pieces in \(n\) trials is given by the binomial probability formula: \[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\] For exactly 5 defective pieces: \( n = 10, \, k = 5, \, p = 0.1 \).

Step 4: Calculate for Exactly 5 Defective Pieces

Substitute the values into the formula:\[P(X = 5) = \binom{10}{5} (0.1)^5 (0.9)^5\] Compute \( \binom{10}{5} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252 \). Calculate the expression:\[P(X = 5) = 252 \times (0.1)^5 \times (0.9)^5 \approx 0.00148 \]

Step 5: Calculate for 5 or More Defective Pieces

We calculate the probability for \(X \geq 5\) by summing probabilities from \(X = 5\) to \(X = 10\): \[P(X \geq 5) = P(X = 5) + P(X = 6) + \, \dots \, + P(X = 10)\] Calculate these using the binomial formula for each case, then sum the probabilities.

Step 6: Calculate Further Probabilities

Compute the necessary probabilities:\[P(X = 6) = \binom{10}{6} (0.1)^6 (0.9)^4, \, ..., \, P(X = 10) = \binom{10}{10} (0.1)^{10} (0.9)^0\] Add them to find \(P(X \geq 5)\).

Step 7: Summarize the Results

After computation: \[P(X = 5) \approx 0.00148\] \[P(X \geq 5) \approx 0.00155\]

Key Concepts

Probability

Probability plays a fundamental role in understanding chance and uncertainty, especially in quality control scenarios like the one presented. Probability is essentially the likelihood of an event occurring and is expressed as a number between 0 and 1. A probability of 0 means the event will never happen, whereas a probability of 1 means the event will always happen.

To calculate the probability of finding exactly 5 defective pieces in a sample of 10 from a machine with a defect rate of 10%, we use the Binomial distribution. This method is chosen because we have a set number of trials (10 pieces) and a fixed chance of success (finding a defective piece, which is considered a 'success' in this context, though it's a bit counterintuitive).

Using the binomial formula, \[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\] it allows us to compute the exact likelihood of having 5 defective pieces. This structured approach helps quality engineers quantify the probability of rare or uncommon events occurring, making it easier to implement effective control measures.

Defective Rate

The defective rate is an essential measure in manufacturing and quality control, representing the proportion of defective items in a batch. It's crucial for assessing the efficiency of a production process and determining whether the current process meets desired quality standards.

In this exercise, the defective rate is high at 10%, which flags an abnormality since it leads to significant financial and operational consequences. This rate is used as the probability of success for one defective piece in our Binomial distribution calculations.

Understanding the defective rate allows engineers to identify whether improvements in the manufacturing process are needed. This involves continuous sampling and calculating probabilities to prevent defective pieces from reaching customers. Tracking these rates provides a baseline for quality improvement projects and can highlight when there is a significant deviation from typical production standards.

Sampling

Sampling is a statistical process used to test a portion of a lot or population to infer information about the whole. In the context of quality control, sampling is an efficient way to assess product quality without needing to check every single item. Continuous sampling, like the near-constant checking described in the exercise, ensures ongoing monitoring of the product's quality over time.

The method of sampling can vary, but the key is to select samples that are representative of the entire batch. In this problem, the engineer uses a sample size of 10 to monitor defects, which provides insights without overburdening with exhaustive checks.

Proper sampling techniques help companies maintain control over production processes and deliver consistent quality. It prevents defective products from impacting consumer satisfaction and retains manufacturing standards, even under conditions indicating higher defect rates. Whether through statistical sampling plans or random checks, the goal is to ensure decision-making based on sufficient, accurate data.