To find the mean length of the steel bolts from supplier [tex]\( A \)[/tex], we need to perform a series of calculations. Here is the detailed, step-by-step solution:
1. List of Bolt Lengths and Frequencies:
The data from Table Q1a shows:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Length (mm)} & \text{Frequency} \\
\hline
44.4 & 1 \\
\hline
44.7 & 14 \\
\hline
45.0 & 15 \\
\hline
45.1 & 17 \\
\hline
45.2 & 13 \\
\hline
45.5 & 13 \\
\hline
45.8 & 2 \\
\hline
\end{array}
\][/tex]
2. Total Number of Bolts:
We sum up the frequencies to find the total number of bolts:
[tex]\[
1 + 14 + 15 + 17 + 13 + 13 + 2 = 75
\][/tex]
There are 75 bolts in total.
3. Mean Length Calculation:
To calculate the mean length, we need to take into account the frequency of each length:
First, multiply each length by its corresponding frequency:
[tex]\[
\begin{align*}
44.4 \times 1 &= 44.4 \\
44.7 \times 14 &= 625.8 \\
45.0 \times 15 &= 675.0 \\
45.1 \times 17 &= 766.7 \\
45.2 \times 13 &= 587.6 \\
45.5 \times 13 &= 591.5 \\
45.8 \times 2 &= 91.6 \\
\end{align*}
\][/tex]
4. Sum of All Lengths (Weighted by Frequency):
Add up all these products:
[tex]\[
44.4 + 625.8 + 675.0 + 766.7 + 587.6 + 591.5 + 91.6 = 3382.6
\][/tex]
5. Mean Length:
Finally, divide the total sum of the lengths by the total number of bolts to get the mean length:
[tex]\[
\text{Mean length} = \frac{3382.6}{75} \approx 45.10133333333333
\][/tex]
So, the mean length of the steel bolts from supplier [tex]\( A \)[/tex] is approximately [tex]\( 45.101 \)[/tex] mm.
