1. **Problem Statement:** We have a frequency distribution of lifetimes (in days) for 500 light bulbs. We need to: - a) Sketch a frequency histogram and frequency polygon. - b) Estimate the mean lifetime. - c) Add a relative frequency column. - d) Find the probability a bulb fails in 400 days or fewer. - e) Decide how often to replace all bulbs to avoid burnout. 2. **Frequency Histogram and Frequency Polygon:** - The histogram has intervals on the x-axis (300–325, 325–350, ..., 575–600). - The y-axis shows frequencies. - Bars heights correspond to frequencies. - The frequency polygon connects midpoints of intervals to frequencies. - Midpoint formula: $$\text{midpoint} = \frac{\text{lower limit} + \text{upper limit}}{2}$$ 3. **Calculate Midpoints:** - 300–325: $\frac{300+325}{2} = 312.5$ - 325–350: $337.5$ - 350–375: $362.5$ - 375–400: $387.5$ - 400–425: $412.5$ - 425–450: $437.5$ - 450–475: $462.5$ - 475–500: $487.5$ - 500–525: $512.5$ - 525–550: $537.5$ - 550–575: $562.5$ - 575–600: $587.5$ 4. **Estimate Mean Lifetime:** - Use formula: $$\text{mean} = \frac{\sum (\text{midpoint} \times \text{frequency})}{\sum \text{frequency}}$$ - Calculate numerator: $$2\times312.5 + 15\times337.5 + 38\times362.5 + 55\times387.5 + 91\times412.5 + 94\times437.5 + 73\times462.5 + 68\times487.5 + 40\times512.5 + 14\times537.5 + 9\times562.5 + 1\times587.5$$ - Calculate each term: $$625 + 5062.5 + 13775 + 21312.5 + 37537.5 + 41187.5 + 33712.5 + 33150 + 20500 + 7525 + 5062.5 + 587.5 = 218935$$ - Total frequency: 500 - Mean: $$\frac{218935}{500} = 437.87$$ days 5. **Add Relative Frequency Column:** - Relative frequency = $$\frac{\text{frequency}}{500}$$ - For example, for 300–325: $$\frac{2}{500} = 0.004$$ - Similarly for others: 0.004, 0.03, 0.076, 0.11, 0.182, 0.188, 0.146, 0.136, 0.08, 0.028, 0.018, 0.002 6. **Probability Bulb Fails in 400 Days or Fewer:** - Sum frequencies for intervals up to 400 days: $$2 + 15 + 38 + 55 = 110$$ - Probability: $$\frac{110}{500} = 0.22$$ 7. **Replacement Frequency to Avoid Burnout:** - Since mean lifetime is about 438 days, and 22% fail by 400 days, to be reasonably sure no bulb burns out, replace bulbs before 400 days. - This ensures most bulbs are replaced before failure.