Question

find the mean of the data summarized in the given frequency distribution. compare the computed mean to the actual mean of 56.8 degrees. low temperature (°f) 40 - 44 45 - 49 50 - 54 55 - 59 60 - 64 frequency 2 6 9 4 2 the mean of the frequency distribution is degrees. (type an integer or decimal rounded to one decimal place as needed.)

Explanation:

Step1: Find the mid - points

For the interval $40 - 44$, the mid - point $x_1=\frac{40 + 44}{2}=42$.
For $45 - 49$, $x_2=\frac{45+49}{2}=47$.
For $50 - 54$, $x_3=\frac{50 + 54}{2}=52$.
For $55 - 59$, $x_4=\frac{55+59}{2}=57$.
For $60 - 64$, $x_5=\frac{60 + 64}{2}=62$.

Step2: Calculate the product of mid - points and frequencies

$f_1 = 2$, $f_1x_1=2\times42 = 84$.
$f_2 = 6$, $f_2x_2=6\times47 = 282$.
$f_3 = 9$, $f_3x_3=9\times52 = 468$.
$f_4 = 4$, $f_4x_4=4\times57 = 228$.
$f_5 = 2$, $f_5x_5=2\times62 = 124$.

Step3: Calculate the sum of frequencies and the sum of products

$\sum f_i=2 + 6+9 + 4+2=23$.
$\sum f_ix_i=84 + 282+468+228+124 = 1186$.

Step4: Calculate the mean

The mean $\bar{x}=\frac{\sum f_ix_i}{\sum f_i}=\frac{1186}{23}\approx51.6$.

Answer:

$51.6$