Use the following distribution to complete parts (a) through (d) below. \( 3,5,9,12,11 \) a a) Compute the mean and standard deviation of the distribution. The mean is 8 . The standard deviation is 3.87 . Round to the nearest hundredth as needed.) b) Multiply each number in the distribution by 2 and compute the mean and the standard deviation of this new distribution. The mean is

To complete parts (a) through (d), we need to follow the instructions provided. ### Part (a) - Compute the mean and standard deviation of the distribution: Given distribution: \( 3, 5, 9, 12, 11 \) The mean of the distribution is given as 8. To find the standard deviation, we can use the formula: \[ \text{Standard Deviation} = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \text{Mean})^2}{n}} \] Where: - \( x_i \) represents each number in the distribution - \( \text{Mean} \) is the mean of the distribution - \( n \) is the total number of values in the distribution Let's calculate the standard deviation using the given mean of 8. ### Part (b) - Multiply each number in the distribution by 2 and compute the mean and the standard deviation of this new distribution: To find the new distribution, we multiply each number in the original distribution by 2. New distribution: \( 6, 10, 18, 24, 22 \) Now, we need to calculate the mean and standard deviation of this new distribution. Let's calculate the mean and standard deviation of the new distribution. Calculate the value by following steps: - step0: Calculate: \(\frac{\left(3+5+9+12+11\right)}{5}\) - step1: Remove the parentheses: \(\frac{3+5+9+12+11}{5}\) - step2: Add the numbers: \(\frac{40}{5}\) - step3: Reduce the numbers: \(\frac{8}{1}\) - step4: Calculate: \(8\) Calculate or simplify the expression \( \sqrt{\frac{(3-8)^2+(5-8)^2+(9-8)^2+(12-8)^2+(11-8)^2}{5}} \). Calculate the value by following steps: - step0: Calculate: \(\sqrt{\frac{\left(3-8\right)^{2}+\left(5-8\right)^{2}+\left(9-8\right)^{2}+\left(12-8\right)^{2}+\left(11-8\right)^{2}}{5}}\) - step1: Subtract the numbers: \(\sqrt{\frac{\left(-5\right)^{2}+\left(5-8\right)^{2}+\left(9-8\right)^{2}+\left(12-8\right)^{2}+\left(11-8\right)^{2}}{5}}\) - step2: Subtract the numbers: \(\sqrt{\frac{\left(-5\right)^{2}+\left(-3\right)^{2}+\left(9-8\right)^{2}+\left(12-8\right)^{2}+\left(11-8\right)^{2}}{5}}\) - step3: Subtract the numbers: \(\sqrt{\frac{\left(-5\right)^{2}+\left(-3\right)^{2}+1^{2}+\left(12-8\right)^{2}+\left(11-8\right)^{2}}{5}}\) - step4: Subtract the numbers: \(\sqrt{\frac{\left(-5\right)^{2}+\left(-3\right)^{2}+1^{2}+4^{2}+\left(11-8\right)^{2}}{5}}\) - step5: Subtract the numbers: \(\sqrt{\frac{\left(-5\right)^{2}+\left(-3\right)^{2}+1^{2}+4^{2}+3^{2}}{5}}\) - step6: Evaluate the power: \(\sqrt{\frac{\left(-5\right)^{2}+\left(-3\right)^{2}+1+4^{2}+3^{2}}{5}}\) - step7: Add the numbers: \(\sqrt{\frac{60}{5}}\) - step8: Divide the terms: \(\sqrt{12}\) - step9: Write as a product: \(\sqrt{4\times 3}\) - step10: Write in exponential form: \(\sqrt{2^{2}\times 3}\) - step11: Use the properties of radicals: \(\sqrt{2^{2}}\times \sqrt{3}\) - step12: Simplify the root: \(2\sqrt{3}\) The mean of the original distribution is 8, and the standard deviation is approximately 3.87. Now, let's calculate the mean and standard deviation of the new distribution after multiplying each number by 2. Calculate the value by following steps: - step0: Calculate: \(\frac{\left(6+10+18+24+22\right)}{5}\) - step1: Remove the parentheses: \(\frac{6+10+18+24+22}{5}\) - step2: Add the numbers: \(\frac{80}{5}\) - step3: Reduce the numbers: \(\frac{16}{1}\) - step4: Calculate: \(16\) Calculate or simplify the expression \( \sqrt{\frac{(6-16)^2+(10-16)^2+(18-16)^2+(24-16)^2+(22-16)^2}{5}} \). Calculate the value by following steps: - step0: Calculate: \(\sqrt{\frac{\left(6-16\right)^{2}+\left(10-16\right)^{2}+\left(18-16\right)^{2}+\left(24-16\right)^{2}+\left(22-16\right)^{2}}{5}}\) - step1: Subtract the numbers: \(\sqrt{\frac{\left(-10\right)^{2}+\left(10-16\right)^{2}+\left(18-16\right)^{2}+\left(24-16\right)^{2}+\left(22-16\right)^{2}}{5}}\) - step2: Subtract the numbers: \(\sqrt{\frac{\left(-10\right)^{2}+\left(-6\right)^{2}+\left(18-16\right)^{2}+\left(24-16\right)^{2}+\left(22-16\right)^{2}}{5}}\) - step3: Subtract the numbers: \(\sqrt{\frac{\left(-10\right)^{2}+\left(-6\right)^{2}+2^{2}+\left(24-16\right)^{2}+\left(22-16\right)^{2}}{5}}\) - step4: Subtract the numbers: \(\sqrt{\frac{\left(-10\right)^{2}+\left(-6\right)^{2}+2^{2}+8^{2}+\left(22-16\right)^{2}}{5}}\) - step5: Subtract the numbers: \(\sqrt{\frac{\left(-10\right)^{2}+\left(-6\right)^{2}+2^{2}+8^{2}+6^{2}}{5}}\) - step6: Add the numbers: \(\sqrt{\frac{240}{5}}\) - step7: Divide the terms: \(\sqrt{48}\) - step8: Write as a product: \(\sqrt{16\times 3}\) - step9: Write in exponential form: \(\sqrt{4^{2}\times 3}\) - step10: Use the properties of radicals: \(\sqrt{4^{2}}\times \sqrt{3}\) - step11: Simplify the root: \(4\sqrt{3}\) The mean of the new distribution is 16, and the standard deviation is approximately 6.93.