1. **Problem:** A worker randomly selects one out of every 7 sets from the 3,500 sets of headphones produced. The sample size is 500 sets. We need to find the ratio of defective headphones in the sample and estimate the number of defective headphones in the entire population. 2. **Step A:** Identify population and sample. - Population: Total headphones produced = 3,500 sets. - Sample: Selected for testing = 500 sets. 3. **Step B:** Write the ratio of defective headphones to total headphones in the sample. - Suppose the number of defective headphones in the sample is $d$ (not given explicitly, so assume $d$). - Ratio = $\frac{d}{500}$. 4. **Step C:** Convert the ratio to decimal and percent. - Decimal form = $\frac{d}{500}$. - Percent form = $\frac{d}{500} \times 100$%. 5. **Step D:** Write and solve an equation to estimate defective headphones in the population. - Let $y$ = estimated defective headphones in population. - Equation: $$y = \frac{d}{500} \times 3500$$ 6. **Step E:** Calculate the estimated defective headphones. - Simplify: $$y = d \times \frac{3500}{500} = d \times 7$$ 7. **Explanation:** The estimate is found by scaling the defective ratio in the sample to the entire population. **Note:** Since the exact number of defective headphones $d$ in the sample is not provided, the answer is expressed in terms of $d$.