Question

find the mean of the data summarized in the given frequency distribution compare the computed mean to the actual mean of 46.6 miles per hour
speed (miles per hour)
42 - 45
46 - 49
50 - 53
54 - 57
58 - 61
frequency
24
12
7
3
1
the mean of the frequency distribution is
miles per hour
(type an integer or decimal rounded to one decimal place as needed.)

Explanation:

Step1: Calculate the mid - points of each class

For the class \(42 - 45\), the mid - point \(x_1=\frac{42 + 45}{2}=43.5\).
For the class \(46 - 49\), the mid - point \(x_2=\frac{46+49}{2}=47.5\).
For the class \(50 - 53\), the mid - point \(x_3=\frac{50 + 53}{2}=51.5\).
For the class \(54 - 57\), the mid - point \(x_4=\frac{54+57}{2}=55.5\).
For the class \(58 - 61\), the mid - point \(x_5=\frac{58 + 61}{2}=59.5\).

Step2: Calculate the sum of \(f\times x\)

Let \(f_1 = 24\), \(f_2=12\), \(f_3 = 7\), \(f_4=3\), \(f_5 = 1\).
\(\sum(f\times x)=f_1x_1+f_2x_2+f_3x_3+f_4x_4+f_5x_5\)
\(=24\times43.5+12\times47.5 + 7\times51.5+3\times55.5+1\times59.5\)
\(=24\times43.5+12\times47.5+7\times51.5 + 3\times55.5+59.5\)
\(=1044+570+360.5+166.5+59.5\)
\(=1044+570+(360.5+166.5)+59.5\)
\(=1044+570+527+59.5\)
\(=1614+527+59.5\)
\(=2141+59.5\)
\(=2200.5\)

Step3: Calculate the sum of frequencies \(\sum f\)

\(\sum f=f_1 + f_2+f_3+f_4+f_5=24 + 12+7+3+1=47\)

Step4: Calculate the mean \(\bar{x}\)

The formula for the mean of a frequency distribution is \(\bar{x}=\frac{\sum(f\times x)}{\sum f}\)
\(\bar{x}=\frac{2200.5}{47}\approx46.8\)

Answer:

\(46.8\)