Question

In a certain sporting goods manufacturing company, a quality control expert tests a randomly selected group of 1,000 tennis balls in order to determine how many contain defects. If this quality control expert discovered that 13 of the randomly selected tennis balls were defective, which of the following inferences would be most supported? A) \(98.7 \%\) of the company's tennis balls are defective. B) \(98.7 \%\) of the company's tennis balls are not defective. C) \(9.87 \%\) of the company's tennis balls are defective. D) \(9.87 \%\) of the company's tennis balls are not defective.

Short answer

The correct inference is option B) 98.7% of the company's tennis balls are not defective.

Step-by-step solution

Understand the given information

We are given a sample of 1,000 tennis balls, and out of which 13 balls are found to be defective.

Calculate the percentage of defective tennis balls

To find the percentage of defective tennis balls, we need to divide the number of defective balls by the total number of balls tested and then multiply by 100. Defective Percentage = \(\frac{13}{1000} * 100\)

Perform the calculation

Defective Percentage = \(\frac{13}{1000} * 100 = 1.3\%\) of tennis balls are defective.

Calculate the percentage of non-defective tennis balls

Since we know the percentage of defective tennis balls, we can subtract it from 100% to find the percentage of non-defective tennis balls. Non-defective Percentage = 100% - Defective Percentage = 100% - 1.3% = 98.7% Thus, 98.7% of tennis balls are non-defective.

Identify the correct inference

Using our calculations, we can choose the correct answer: A) 98.7% of the company's tennis balls are defective. (Incorrect) B) 98.7% of the company's tennis balls are not defective. (Correct) C) 9.87% of the company's tennis balls are defective. (Incorrect) D) 9.87% of the company's tennis balls are not defective. (Incorrect) The correct inference is option B) 98.7% of the company's tennis balls are not defective.

Key concepts

Defective Rate Calculation

When you're dealing with quality control in manufacturing, understanding the defective rate is crucial. In our example, a company tested 1,000 tennis balls, finding 13 of them to be defective. To calculate the defective rate, you simply take the number of defective items and divide it by the total tested samples. This gives you the defective fraction.For calculation:- In this case: \[ \text{Defective Rate} = \frac{13}{1000} = 0.013 \]Next, convert this fraction to a percentage to make it more understandable. Multiply by 100 to get the defective rate percentage:- Defective Percentage = \(0.013 \times 100 = 1.3\%\).Understanding this percentage tells you that out of every 1000 tennis balls produced, about 13 will likely have defects. This defective rate is a key performance indicator for the company's quality control processes.

Sample Analysis

Sample analysis is a cornerstone of statistics, especially when assessing product quality. Rather than evaluating each individual product, you select a representative subset or sample. In our scenario, the expert examined 1,000 tennis balls out of a larger batch, which is a practical approach to infer the quality of the entire batch. Why choose a sample? - It saves time. - It reduces costs. The accuracy of your conclusion depends on choosing a sufficiently large and random sample, which should represent the entire population's characteristics. This involves testing them under consistent conditions to avoid skewing the results. In our situation, by evaluating 1,000 randomly selected tennis balls, we gain a reliable insight into the overall quality, helping the company identify and address potential quality issues efficiently.

Percentage Calculation

Calculating percentages is a fundamental aspect of statistics, useful in various scenarios including assessing quality control as in our example. Here’s how you effectively perform a percentage calculation:1. **Find the ratio**: Start by determining the fraction of interest, such as the defective tennis balls' ratio: \[ \text{Fraction} = \frac{13}{1000} \] 2. **Convert Fraction to Percentage**: Multiply the ratio by 100 to convert it into a percentage. This transformation makes the data more intuitive: \[ \text{Percentage} = \left(\frac{13}{1000}\right) \times 100 = 1.3\% \]Remember, percentages express data in a form that is easy for most people to understand and compare. Furthermore, when calculating the non-defective percentage in this context, subtract the defective percentage from 100%: \[ \text{Non-defective Percentage} = 100\% - 1.3\% = 98.7\% \]Percentages simplify complex numbers into easily digestible figures, which is essential in making informed decisions or evaluations.