To determine how many beanbag chairs would be expected to be understuffed out of 1500 chairs produced per day, we need to follow a precise process based on the given data from the spot-check sample. Let's go through the solution step-by-step.
1. Identify the Given Data:
- From the spot check, the number of common problems found:
- Open seam: 4 chairs
- Cuts in upholstery: 14 chairs
- Understuffed: 15 chairs
- None: 267 chairs
- Total number of chairs checked: 300 chairs
- Daily production: 1500 chairs
2. Calculate the Proportion of Understuffed Chairs in the Sample:
To find the proportion (or rate) of understuffed chairs, we divide the number of understuffed chairs by the total number of chairs checked. This gives:
[tex]\[
\text{Proportion of understuffed chairs} = \frac{\text{Number of understuffed chairs}}{\text{Total number of chairs checked}}
\][/tex]
Substituting the given values:
[tex]\[
\text{Proportion of understuffed chairs} = \frac{15}{300} = 0.05
\][/tex]
3. Calculate the Expected Number of Understuffed Chairs in Daily Production:
Now, we use this proportion to predict how many of the 1500 chairs produced daily would be expected to be understuffed. We do this by multiplying the proportion by the total daily production:
[tex]\[
\text{Expected number of understuffed chairs} = \text{Proportion of understuffed chairs} \times \text{Total daily production}
\][/tex]
Substituting the values:
[tex]\[
\text{Expected number of understuffed chairs} = 0.05 \times 1500 = 75
\][/tex]
Hence, the manufacturer would expect 75 of the 1500 beanbag chairs produced each day to be understuffed. Therefore, the correct option is:
They would expect 75 chairs to be understuffed.
