Sure, let's find the mean given the heights and their respective frequencies. The formula for calculating the mean in a grouped frequency distribution is:
[tex]\[ \text{Mean} = \frac{\sum (x \cdot f)}{\sum f} \][/tex]
Where [tex]\( x \)[/tex] represents the height values, and [tex]\( f \)[/tex] represents their corresponding frequencies.
First, let's tabulate the given data:
[tex]\[
\begin{array}{|c|c|c|}
\hline
\text{Height (below)} & x & f \\
\hline
10 & 15 \\
20 & 35 \\
30 & 60 \\
40 & 84 \\
50 & 96 \\
\hline
\end{array}
\][/tex]
Step 1: Calculate the total number of observations, which is the sum of the frequencies ([tex]\( \sum f \)[/tex]).
[tex]\[
\sum f = 15 + 35 + 60 + 84 + 96 = 290
\][/tex]
Step 2: Calculate [tex]\( \sum (x \cdot f) \)[/tex], which is the sum of the product of each height and its corresponding frequency.
[tex]\[
\sum (x \cdot f) = (10 \times 15) + (20 \times 35) + (30 \times 60) + (40 \times 84) + (50 \times 96)
\][/tex]
Let's break this down:
[tex]\[
10 \times 15 = 150
\][/tex]
[tex]\[
20 \times 35 = 700
\][/tex]
[tex]\[
30 \times 60 = 1800
\][/tex]
[tex]\[
40 \times 84 = 3360
\][/tex]
[tex]\[
50 \times 96 = 4800
\][/tex]
Now, summing these products:
[tex]\[
150 + 700 + 1800 + 3360 + 4800 = 10810
\][/tex]
Step 3: Calculate the mean using the formula:
[tex]\[
\text{Mean} = \frac{\sum (x \cdot f)}{\sum f} = \frac{10810}{290} \approx 37.275862068965516
\][/tex]
Therefore, the total number of observations is 290 and the mean height is approximately 37.2759.
