Question

the results of a random sample of the number of pets per household in a region are shown in the histogram. estimate the sample mean and the sample standard deviation of the data set. the sample mean is $\bar{x}=$ (round to one decimal place as needed.)

Explanation:

Step1: Identify mid - points and frequencies

Let the number of pets be grouped as follows: For the first bar with number of pets from 0 - 1, assume mid - point $x_1 = 0.5$ and frequency $f_1=3$; for the second bar (1 - 2) with mid - point $x_2 = 1.5$ and frequency $f_2 = 12$; for the third bar (2 - 3) with mid - point $x_3=2.5$ and frequency $f_3 = 6$; for the fourth bar (3 - 4) with mid - point $x_4 = 3.5$ and frequency $f_4=1$; for the fifth bar (4 - 5) with mid - point $x_5 = 4.5$ and frequency $f_5 = 5$.

Step2: Calculate the sum of the products of mid - points and frequencies

The formula for the sample mean $\bar{x}=\frac{\sum_{i = 1}^{n}f_ix_i}{\sum_{i=1}^{n}f_i}$. First, calculate $\sum_{i = 1}^{n}f_ix_i=f_1x_1 + f_2x_2+f_3x_3+f_4x_4+f_5x_5=(3\times0.5)+(12\times1.5)+(6\times2.5)+(1\times3.5)+(5\times4.5)=1.5 + 18+15 + 3.5+22.5 = 60.5$. And $\sum_{i=1}^{n}f_i=3 + 12+6 + 1+5=27$.

Step3: Calculate the sample mean

$\bar{x}=\frac{60.5}{27}\approx2.2$.

Answer:

$2.2$