Question: You work for a large farm with many fields of corn. You are investigating the mass in grams of a sample of ears of corn. You gather the following data: Mass(g) 480.8 689.3 496.7 556.3 574.6 479.4 531.1 824.4 510.5 422.4 416.3 488.3 651.9 Some of the masses in the sample seem much larger than the rest. You decide to make several calculations to help in the search for outliers. Report your answer with all the decimal places. Find the following: a. The 32nd percentile = b. IQR = c. Apply the 1.5 IQR rule to search for outliers. Report the lower and upper cutoffs limits (fences). Lower: Upper: d. Are there any outliers using the 1.5 IQR rule?

1. **State the problem:** We have a sample of masses (in grams) of ears of corn and need to find the 32nd percentile, the interquartile range (IQR), the lower and upper fences using the 1.5 IQR rule, and determine if there are any outliers. 2. **Organize the data in ascending order:** $$416.3, 422.4, 479.4, 480.8, 488.3, 496.7, 510.5, 531.1, 556.3, 574.6, 651.9, 689.3, 824.4$$ 3. **Find the 32nd percentile ($P_{32}$):** - Number of data points $n=13$. - Position of the 32nd percentile: $$L = \frac{32}{100} \times (n+1) = 0.32 \times 14 = 4.48$$ - This means $P_{32}$ lies between the 4th and 5th data points. - 4th data point = $480.8$, 5th data point = $488.3$. - Interpolate: $$P_{32} = 480.8 + 0.48 \times (488.3 - 480.8) = 480.8 + 0.48 \times 7.5 = 480.8 + 3.6 = 484.4$$ 4. **Find the quartiles $Q_1$ and $Q_3$ to calculate IQR:** - $Q_1$ is the 25th percentile, position: $$L_1 = 0.25 \times 14 = 3.5$$ - Between 3rd and 4th data points: 479.4 and 480.8 - Interpolate: $$Q_1 = 479.4 + 0.5 \times (480.8 - 479.4) = 479.4 + 0.7 = 480.1$$ - $Q_3$ is the 75th percentile, position: $$L_3 = 0.75 \times 14 = 10.5$$ - Between 10th and 11th data points: 574.6 and 651.9 - Interpolate: $$Q_3 = 574.6 + 0.5 \times (651.9 - 574.6) = 574.6 + 38.65 = 613.25$$ 5. **Calculate IQR:** $$IQR = Q_3 - Q_1 = 613.25 - 480.1 = 133.15$$ 6. **Apply the 1.5 IQR rule to find fences:** - Lower fence: $$Q_1 - 1.5 \times IQR = 480.1 - 1.5 \times 133.15 = 480.1 - 199.725 = 280.375$$ - Upper fence: $$Q_3 + 1.5 \times IQR = 613.25 + 199.725 = 812.975$$ 7. **Determine outliers:** - Any data point below $280.375$ or above $812.975$ is an outlier. - Minimum data point is $416.3$ (above lower fence). - Maximum data point is $824.4$ (above upper fence). - So, $824.4$ is an outlier. **Final answers:** - a. The 32nd percentile = $484.4$ - b. IQR = $133.15$ - c. Lower fence = $280.375$, Upper fence = $812.975$ - d. Outliers? Yes, $824.4$ is an outlier.