Question 20: Given the mean of a data set is 15 and the coefficient of variation is 48%, find the standard deviation. Question 21: Given the time taken (in minutes) to complete homework by 8 students: 38, 40, 47, 44, 46, 43, 49, 53. Find the standard deviation.

1. Problem Description

Question 20: Given the mean of a data set is 15 and the coefficient of variation is 48%, find the standard deviation.

Question 21: Given the time taken (in minutes) to complete homework by 8 students: 38, 40, 47, 44, 46, 43, 49,

5

3. Find the standard deviation.

2. Solution Steps

Question 20:

The coefficient of variation (CV) is defined as the ratio of the standard deviation ($\sigma$) to the mean ($\mu$):

$CV = \frac{\sigma}{\mu}$

We are given $CV = 48\% = 0.48$ and $\mu = 15$. We need to find $\sigma$.

$\sigma = CV \times \mu$

$\sigma = 0.48 \times 15$

$\sigma = 7.2$

Question 21:

First, calculate the mean of the data set:

$\mu = \frac{38 + 40 + 47 + 44 + 46 + 43 + 49 + 53}{8}$

$\mu = \frac{360}{8} = 45$

Next, calculate the squared differences from the mean:

$(38-45)^2 = (-7)^2 = 49$

$(40-45)^2 = (-5)^2 = 25$

$(47-45)^2 = (2)^2 = 4$

$(44-45)^2 = (-1)^2 = 1$

$(46-45)^2 = (1)^2 = 1$

$(43-45)^2 = (-2)^2 = 4$

$(49-45)^2 = (4)^2 = 16$

$(53-45)^2 = (8)^2 = 64$

Sum of squared differences:

$49 + 25 + 4 + 1 + 1 + 4 + 16 + 64 = 164$

Calculate the sample standard deviation ($s$):

$s = \sqrt{\frac{\sum (x_i - \mu)^2}{n-1}}$

$s = \sqrt{\frac{164}{8-1}}$

$s = \sqrt{\frac{164}{7}}$

$s = \sqrt{23.42857}$

$s \approx 4.84$

Since none of the answers match, let's calculate the population standard deviation ($\sigma$):

$\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{n}}$

$\sigma = \sqrt{\frac{164}{8}}$

$\sigma = \sqrt{20.5}$

$\sigma \approx 4.53$

3. Final Answer

Question 20: B. 7.2

Question 21: A. 4.53