Question

use the appropriate formulas and methods to answer the following questions. do not use technology as an aid, unless otherwise stated. approximate the measures of center for following gfdt. data frequency 30 - 34 2 35 - 39 1 40 - 44 4 45 - 49 3 50 - 54 6 55 - 59 12 60 - 64 12 65 - 69 21 70 - 74 12 mode = median = mean = report mode and median accurate to one decimal place. report the mean accurate to two decimal places (or enter as a fraction).

Explanation:

Step1: Find the mode

The mode is the class with the highest frequency. The class 65 - 69 has the highest frequency of 21. To approximate the mode, we use the formula for the mode of a grouped - data: $L+\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\times w$, where $L$ is the lower limit of the modal class ($L = 65$), $f_1$ is the frequency of the modal class ($f_1=21$), $f_0$ is the frequency of the class before the modal class ($f_0 = 12$), $f_2$ is the frequency of the class after the modal class ($f_2 = 12$), and $w$ is the class width ($w = 5$).
\[

$$\begin{align*} \text{Mode}&=65+\frac{21 - 12}{2\times21-12 - 12}\times5\\ &=65+\frac{9}{42 - 24}\times5\\ &=65+\frac{9}{18}\times5\\ &=65 + 2.5\\ &=67.5 \end{align*}$$

\]

Step2: Find the median

First, find the total frequency $n=\sum f=2 + 1+4 + 3+6+12+12+21+12=73$. The median class is the class where $\frac{n + 1}{2}=37$th value lies. Cumulative frequencies: $2,2 + 1=3,3 + 4 = 7,7+3 = 10,10 + 6=16,16+12 = 28,28+12 = 40,40+21 = 61,61+12 = 73$. The median class is 60 - 64. The formula for the median of grouped - data is $L+\frac{\frac{n}{2}-F}{f}\times w$, where $L$ is the lower limit of the median class ($L = 60$), $n$ is the total frequency ($n = 73$), $F$ is the cumulative frequency of the class before the median class ($F = 28$), $f$ is the frequency of the median class ($f = 12$), and $w$ is the class width ($w = 5$).
\[

$$\begin{align*} \text{Median}&=60+\frac{\frac{73}{2}-28}{12}\times5\\ &=60+\frac{36.5 - 28}{12}\times5\\ &=60+\frac{8.5}{12}\times5\\ &=60+\frac{42.5}{12}\\ &\approx60 + 3.5\\ &=63.5 \end{align*}$$

\]

Step3: Find the mean

Find the mid - points of each class:
For 30 - 34, mid - point $x_1=\frac{30 + 34}{2}=32$; for 35 - 39, $x_2=\frac{35 + 39}{2}=37$; for 40 - 44, $x_3=\frac{40 + 44}{2}=42$; for 45 - 49, $x_4=\frac{45 + 49}{2}=47$; for 50 - 54, $x_5=\frac{50 + 54}{2}=52$; for 55 - 59, $x_6=\frac{55 + 59}{2}=57$; for 60 - 64, $x_7=\frac{60 + 64}{2}=62$; for 65 - 69, $x_8=\frac{65 + 69}{2}=67$; for 70 - 74, $x_9=\frac{70 + 74}{2}=72$.
The mean $\bar{x}=\frac{\sum_{i = 1}^{k}f_ix_i}{\sum_{i = 1}^{k}f_i}$, where $f_i$ are the frequencies and $x_i$ are the mid - points.
\[

$$\begin{align*} \sum_{i = 1}^{9}f_ix_i&=2\times32+1\times37 + 4\times42+3\times47+6\times52+12\times57+12\times62+21\times67+12\times72\\ &=64+37+168+141+312+684+744+1407+864\\ &=4421 \end{align*}$$

\]
$\bar{x}=\frac{4421}{73}\approx60.56$

Answer:

Mode = 67.5
Median = 63.5
Mean = 60.56