Question

the histogram represents the distribution of lengths, in inches, of 25 catfish caught in a lake. a. if possible, find the mean. if not possible, explain why not.

Explanation:

Step1: Determine the mid - points of each class interval

For the interval 3 - 6, mid - point $x_1=\frac{3 + 6}{2}=4.5$; for 6 - 9, $x_2=\frac{6+9}{2}=7.5$; for 9 - 12, $x_3=\frac{9 + 12}{2}=10.5$; for 12 - 15, $x_4=\frac{12+15}{2}=13.5$; for 15 - 18, $x_5=\frac{15 + 18}{2}=16.5$.

Step2: Determine the frequencies

From the histogram, the frequencies $f_1 = 4$, $f_2=9$, $f_3 = 5$, $f_4=6$, $f_5 = 1$.

Step3: Calculate the sum of the products of mid - points and frequencies

$\sum_{i = 1}^{5}f_ix_i=f_1x_1+f_2x_2+f_3x_3+f_4x_4+f_5x_5=(4\times4.5)+(9\times7.5)+(5\times10.5)+(6\times13.5)+(1\times16.5)=18 + 67.5+52.5 + 81+16.5=235.5$.

Step4: Calculate the total frequency

$n=\sum_{i = 1}^{5}f_i=4 + 9+5+6+1=25$.

Step5: Calculate the mean

The mean $\bar{x}=\frac{\sum_{i = 1}^{5}f_ix_i}{n}=\frac{235.5}{25}=9.42$.

Answer:

The mean is 9.42 inches.