A tool and die company manufactures a precise pin that is supposed to measure 10.25 mm in diameter. For quality control purposes, ten pins are selected at random and measured to determine how closely they match the specified diameter. The results are provided. Complete parts (a) and (b) below. \( 10.28,10.31,10.24,10.24,10.24,10.23,10.23,10.31,10.24,10.22 \), Find the range of the measurements. The range is 0.09 mm . (Simplify your answer.) (b) Find the standard deviation of the measurements. The standard deviation is (Round to three decimal places as needed.)

To find the range of the measurements, we need to subtract the smallest measurement from the largest measurement. Given measurements: \( 10.28, 10.31, 10.24, 10.24, 10.24, 10.23, 10.23, 10.31, 10.24, 10.22 \) The largest measurement is 10.31 mm and the smallest measurement is 10.22 mm. Range = Largest measurement - Smallest measurement Range = 10.31 mm - 10.22 mm Range = 0.09 mm Therefore, the range of the measurements is 0.09 mm. To find the standard deviation of the measurements, we can use the formula for the standard deviation of a set of data: \[ \text{Standard Deviation} = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}} \] where: - \( x_i \) is each individual measurement - \( \bar{x} \) is the mean of the measurements - \( n \) is the number of measurements First, we need to calculate the mean of the measurements: Mean = \( \frac{\sum_{i=1}^{n} x_i}{n} \) Let's calculate the mean and then the standard deviation. Calculate the value by following steps: - step0: Calculate: \(10.28+10.31+10.24+10.24+10.24+10.23+10.23+10.31+10.24+10.22\) - step1: Write as a multiplication: \(10.28+10.31+10.24\times 4+10.23+10.23+10.31+10.22\) - step2: Multiply the numbers: \(10.28+10.31+40.96+10.23+10.23+10.31+10.22\) - step3: Add the numbers: \(102.54\) The sum of the measurements is 102.54 mm. Next, we calculate the mean of the measurements: Mean = \( \frac{102.54}{10} \) Mean = 10.254 mm Now, we can calculate the standard deviation using the formula provided earlier. Calculate the value by following steps: - step0: Calculate: \(\left(10.28-10.254\right)^{2}+\left(10.31-10.254\right)^{2}+\left(10.24-10.254\right)^{2}+\left(10.24-10.254\right)^{2}+\left(10.24-10.254\right)^{2}+\left(10.23-10.254\right)^{2}+\left(10.23-10.254\right)^{2}+\left(10.31-10.254\right)^{2}+\left(10.24-10.254\right)^{2}+\left(10.22-10.254\right)^{2}\) - step1: Write as a multiplication: \(\left(10.28-10.254\right)^{2}+\left(10.31-10.254\right)^{2}+\left(10.24-10.254\right)^{2}\times 4+\left(10.23-10.254\right)^{2}+\left(10.23-10.254\right)^{2}+\left(10.31-10.254\right)^{2}+\left(10.22-10.254\right)^{2}\) - step2: Subtract the numbers: \(0.026^{2}+\left(10.31-10.254\right)^{2}+\left(10.24-10.254\right)^{2}\times 4+\left(10.23-10.254\right)^{2}+\left(10.23-10.254\right)^{2}+\left(10.31-10.254\right)^{2}+\left(10.22-10.254\right)^{2}\) - step3: Subtract the numbers: \(0.026^{2}+0.056^{2}+\left(10.24-10.254\right)^{2}\times 4+\left(10.23-10.254\right)^{2}+\left(10.23-10.254\right)^{2}+\left(10.31-10.254\right)^{2}+\left(10.22-10.254\right)^{2}\) - step4: Subtract the numbers: \(0.026^{2}+0.056^{2}+\left(-0.014\right)^{2}\times 4+\left(10.23-10.254\right)^{2}+\left(10.23-10.254\right)^{2}+\left(10.31-10.254\right)^{2}+\left(10.22-10.254\right)^{2}\) - step5: Subtract the numbers: \(0.026^{2}+0.056^{2}+\left(-0.014\right)^{2}\times 4+\left(-0.024\right)^{2}+\left(10.23-10.254\right)^{2}+\left(10.31-10.254\right)^{2}+\left(10.22-10.254\right)^{2}\) - step6: Subtract the numbers: \(0.026^{2}+0.056^{2}+\left(-0.014\right)^{2}\times 4+\left(-0.024\right)^{2}+\left(-0.024\right)^{2}+\left(10.31-10.254\right)^{2}+\left(10.22-10.254\right)^{2}\) - step7: Subtract the numbers: \(0.026^{2}+0.056^{2}+\left(-0.014\right)^{2}\times 4+\left(-0.024\right)^{2}+\left(-0.024\right)^{2}+0.056^{2}+\left(10.22-10.254\right)^{2}\) - step8: Subtract the numbers: \(0.026^{2}+0.056^{2}+\left(-0.014\right)^{2}\times 4+\left(-0.024\right)^{2}+\left(-0.024\right)^{2}+0.056^{2}+\left(-0.034\right)^{2}\) - step9: Convert the expressions: \(\left(\frac{13}{500}\right)^{2}+0.056^{2}+\left(-0.014\right)^{2}\times 4+\left(-0.024\right)^{2}+\left(-0.024\right)^{2}+0.056^{2}+\left(-0.034\right)^{2}\) - step10: Convert the expressions: \(\left(\frac{13}{500}\right)^{2}+\left(\frac{7}{125}\right)^{2}+\left(-0.014\right)^{2}\times 4+\left(-0.024\right)^{2}+\left(-0.024\right)^{2}+0.056^{2}+\left(-0.034\right)^{2}\) - step11: Convert the expressions: \(\left(\frac{13}{500}\right)^{2}+\left(\frac{7}{125}\right)^{2}+\left(-\frac{7}{500}\right)^{2}\times 4+\left(-0.024\right)^{2}+\left(-0.024\right)^{2}+0.056^{2}+\left(-0.034\right)^{2}\) - step12: Convert the expressions: \(\left(\frac{13}{500}\right)^{2}+\left(\frac{7}{125}\right)^{2}+\left(-\frac{7}{500}\right)^{2}\times 4+\left(-\frac{3}{125}\right)^{2}+\left(-0.024\right)^{2}+0.056^{2}+\left(-0.034\right)^{2}\) - step13: Convert the expressions: \(\left(\frac{13}{500}\right)^{2}+\left(\frac{7}{125}\right)^{2}+\left(-\frac{7}{500}\right)^{2}\times 4+\left(-\frac{3}{125}\right)^{2}+\left(-\frac{3}{125}\right)^{2}+0.056^{2}+\left(-0.034\right)^{2}\) - step14: Convert the expressions: \(\left(\frac{13}{500}\right)^{2}+\left(\frac{7}{125}\right)^{2}+\left(-\frac{7}{500}\right)^{2}\times 4+\left(-\frac{3}{125}\right)^{2}+\left(-\frac{3}{125}\right)^{2}+\left(\frac{7}{125}\right)^{2}+\left(-0.034\right)^{2}\) - step15: Convert the expressions: \(\left(\frac{13}{500}\right)^{2}+\left(\frac{7}{125}\right)^{2}+\left(-\frac{7}{500}\right)^{2}\times 4+\left(-\frac{3}{125}\right)^{2}+\left(-\frac{3}{125}\right)^{2}+\left(\frac{7}{125}\right)^{2}+\left(-\frac{17}{500}\right)^{2}\) - step16: Multiply the terms: \(\left(\frac{13}{500}\right)^{2}+\left(\frac{7}{125}\right)^{2}+4\times \frac{7^{2}}{500^{2}}+\left(-\frac{3}{125}\right)^{2}+\left(-\frac{3}{125}\right)^{2}+\left(\frac{7}{125}\right)^{2}+\left(-\frac{17}{500}\right)^{2}\) - step17: Rewrite the expression: \(\left(\frac{13}{500}\right)^{2}+\left(\frac{7}{125}\right)^{2}+\left(\frac{7}{125}\right)^{2}+4\times \frac{7^{2}}{500^{2}}+\left(-\frac{3}{125}\right)^{2}+\left(-\frac{3}{125}\right)^{2}+\left(-\frac{17}{500}\right)^{2}\) - step18: Add the numbers: \(\left(\frac{13}{500}\right)^{2}+\frac{98}{15625}+4\times \frac{7^{2}}{500^{2}}+\left(-\frac{3}{125}\right)^{2}+\left(-\frac{3}{125}\right)^{2}+\left(-\frac{17}{500}\right)^{2}\) - step19: Add the numbers: \(\left(\frac{13}{500}\right)^{2}+\frac{98}{15625}+4\times \frac{7^{2}}{500^{2}}+\frac{18}{15625}+\left(-\frac{17}{500}\right)^{2}\) - step20: Rewrite the expression: \(\frac{13^{2}}{500^{2}}+\frac{98}{15625}+4\times \frac{7^{2}}{500^{2}}+\frac{18}{15625}+\left(-\frac{17}{500}\right)^{2}\) - step21: Rewrite the expression: \(\frac{13^{2}}{500^{2}}+\frac{98}{15625}+4\times \frac{7^{2}}{500^{2}}+\frac{18}{15625}+\frac{17^{2}}{500^{2}}\) - step22: Rewrite the expression: \(\frac{13^{2}}{500^{2}}+\frac{98}{15625}+\frac{196}{500^{2}}+\frac{18}{15625}+\frac{17^{2}}{500^{2}}\) - step23: Reduce fractions to a common denominator: \(\frac{13^{2}\times 15625\times 16}{500^{2}\times 15625\times 16}+\frac{98\times 16\times 500^{2}}{15625\times 16\times 500^{2}}+\frac{196\times 15625\times 16}{500^{2}\times 15625\times 16}+\frac{18\times 16\times 500^{2}}{15625\times 16\times 500^{2}}+\frac{17^{2}\times 500^{2}}{500^{2}\times 500^{2}}\) - step24: Multiply the terms: \(\frac{13^{2}\times 15625\times 16}{500^{4}}+\frac{98\times 16\times 500^{2}}{15625\times 16\times 500^{2}}+\frac{196\times 15625\times 16}{500^{2}\times 15625\times 16}+\frac{18\times 16\times 500^{2}}{15625\times 16\times 500^{2}}+\frac{17^{2}\times 500^{2}}{500^{2}\times 500^{2}}\) - step25: Multiply the terms: \(\frac{13^{2}\times 15625\times 16}{500^{4}}+\frac{98\times 16\times 500^{2}}{500^{4}}+\frac{196\times 15625\times 16}{500^{2}\times 15625\times 16}+\frac{18\times 16\times 500^{2}}{15625\times 16\times 500^{2}}+\frac{17^{2}\times 500^{2}}{500^{2}\times 500^{2}}\) - step26: Multiply the terms: \(\frac{13^{2}\times 15625\times 16}{500^{4}}+\frac{98\times 16\times 500^{2}}{500^{4}}+\frac{196\times 15625\times 16}{500^{4}}+\frac{18\times 16\times 500^{2}}{15625\times 16\times 500^{2}}+\frac{17^{2}\times 500^{2}}{500^{2}\times 500^{2}}\) - step27: Multiply the terms: \(\frac{13^{2}\times 15625\times 16}{500^{4}}+\frac{98\times 16\times 500^{2}}{500^{4}}+\frac{196\times 15625\times 16}{500^{4}}+\frac{18\times 16\times 500^{2}}{500^{4}}+\frac{17^{2}\times 500^{2}}{500^{2}\times 500^{2}}\) - step28: Multiply the terms: \(\frac{13^{2}\times 15625\times 16}{500^{4}}+\frac{98\times 16\times 500^{2}}{500^{4}}+\frac{196\times 15625\times 16}{500^{4}}+\frac{18\times 16\times 500^{2}}{500^{4}}+\frac{17^{2}\times 500^{2}}{500^{4}}\) - step29: Transform the expression: \(\frac{13^{2}\times 15625\times 16+98\times 16\times 500^{2}+196\times 15625\times 16+18\times 16\times 500^{2}+17^{2}\times 500^{2}}{500^{4}}\) - step30: Multiply the terms: \(\frac{42250000+98\times 16\times 500^{2}+196\times 15625\times 16+18\times 16\times 500^{2}+17^{2}\times 500^{2}}{500^{4}}\) - step31: Multiply the terms: \(\frac{42250000+1568\times 500^{2}+196\times 15625\times 16+18\times 16\times 500^{2}+17^{2}\times 500^{2}}{500^{4}}\) - step32: Multiply the terms: \(\frac{42250000+1568\times 500^{2}+49000000+18\times 16\times 500^{2}+17^{2}\times 500^{2}}{500^{4}}\) - step33: Multiply the terms: \(\frac{42250000+1568\times 500^{2}+49000000+288\times 500^{2}+17^{2}\times 500^{2}}{500^{4}}\) - step34: Multiply the terms: \(\frac{42250000+1568\times 500^{2}+49000000+288\times 500^{2}+8500^{2}}{500^{4}}\) - step35: Add the numbers: \(\frac{555250000+8500^{2}}{500^{4}}\) - step36: Rewrite the expression: \(\frac{500^{2}\left(2221+289\right)}{500^{4}}\) - step37: Reduce the fraction: \(\frac{2221+289}{500^{2}}\) - step38: Simplify: \(\frac{2510}{500^{2}}\) - step39: Reduce the fraction: \(\frac{251}{25000}\) The sum of the squared differences from the mean is 0.01004. Now, we can calculate the standard deviation: Standard Deviation = \( \sqrt{\frac{0.01004}{10-1}} \) Standard Deviation = \( \sqrt{\frac{0.01004}{9}} \) Standard Deviation = \( \sqrt{0.001112} \) Standard Deviation = 0.033 mm Therefore, the standard deviation of the measurements is 0.033 mm.