You work in quality control at a rod manufacturing plant. You want to analyze the quality of rod batches using the following probability rules and descriptive statistics.

Core Formulas

Union Rule

Conditional Probability

Sample Statistics

• Sample Mean:

  $$\bar{x} = \frac{\sum x_i}{n}$$

• Sample Variance:

  $$s^2 = \frac{\sum (x_i - \bar{x})^2}{n}$$

• Sample Standard Deviation:

  $$s = \sqrt{s^2}$$

Given Data

• Total rods: 100;
• Defective rods: 20;
• Rods longer than 50 cm: 30;
• Rods that are both defective and long: 10;
• Population mean rod length: 50 cm;
• Population standard deviation of rod lengths: 0.5 cm;
• Sample size for analysis: 10 rods.