Question

a study recorded the time (in minutes) spent on daily exercise by a group of 50 individuals. the collected data was summarized into the following frequency distribution:

compute the mean of the data summarized in the frequency distribution.
compare the computed mean to the actual mean obtained from the original list of data values, which is 31.2 minutes.

a. the computed mean is 34.5 minutes, which is higher than the actual mean of 31.2 minutes.

b. the computed mean is 31.2 minutes, which matches the actual mean exactly.

c. the computed mean is 28.9 minutes, which is lower than the actual mean of 31.2 minutes.

d. the computed mean is 26.1 minutes, which is lower than the actual mean of 31.2 minutes.

Explanation:

Step1: Find mid - points of each class interval

For the interval \(10 - 19\), the mid - point \(x_1=\frac{10 + 19}{2}=14.5\)
For the interval \(20 - 29\), the mid - point \(x_2=\frac{20+29}{2}=24.5\)
For the interval \(30 - 39\), the mid - point \(x_3=\frac{30 + 39}{2}=34.5\)
For the interval \(40 - 49\), the mid - point \(x_4=\frac{40+49}{2}=44.5\)
For the interval \(50 - 59\), the mid - point \(x_5=\frac{50 + 59}{2}=54.5\)

Step2: Calculate the product of mid - points and frequencies

Let \(f_1 = 6\), \(f_2=12\), \(f_3 = 15\), \(f_4=10\), \(f_5 = 7\)
\(f_1x_1=6\times14.5 = 87\)
\(f_2x_2=12\times24.5=294\)
\(f_3x_3=15\times34.5 = 517.5\)
\(f_4x_4=10\times44.5=445\)
\(f_5x_5=7\times54.5 = 381.5\)

Step3: Sum up the products and the frequencies

\(\sum f_ix_i=87 + 294+517.5+445+381.5=1725\)
\(\sum f_i=6 + 12+15+10+7=50\)

Step4: Calculate the mean

The formula for the mean of a frequency distribution is \(\bar{x}=\frac{\sum f_ix_i}{\sum f_i}\)
\(\bar{x}=\frac{1725}{50}=34.5\)

Answer:

A. The computed mean is 34.5 minutes, which is higher than the actual mean of 31.2 minutes.