To estimate the mean book thickness from the given data, we need to follow these steps:
1. Determine the midpoints for each interval:
- For the interval [tex]\( 0 < x \leq 2 \)[/tex]: The midpoint is [tex]\(\frac{0 + 2}{2} = 1.0\)[/tex] mm.
- For the interval [tex]\( 2 < x \leq 4 \)[/tex]: The midpoint is [tex]\(\frac{2 + 4}{2} = 3.0\)[/tex] mm.
- For the interval [tex]\( 4 < x \leq 6 \)[/tex]: The midpoint is [tex]\(\frac{4 + 6}{2} = 5.0\)[/tex] mm.
2. Multiply each midpoint by its corresponding frequency to find the sum of the products:
- For the interval [tex]\( 0 < x \leq 2 \)[/tex]: [tex]\(1.0 \times 4 = 4.0\)[/tex]
- For the interval [tex]\( 2 < x \leq 4 \)[/tex]: [tex]\(3.0 \times 9 = 27.0\)[/tex]
- For the interval [tex]\( 4 < x \leq 6 \)[/tex]: [tex]\(5.0 \times 7 = 35.0\)[/tex]
3. Calculate the total sum of these products:
[tex]\[ 4.0 + 27.0 + 35.0 = 66.0 \][/tex]
4. Calculate the total frequency (total number of books):
[tex]\[ 4 + 9 + 7 = 20 \][/tex]
5. Divide the sum of the products by the total frequency to get the estimated mean thickness:
[tex]\[ \frac{66.0}{20} = 3.3 \text{ mm} \][/tex]
Thus, the estimate for the mean book thickness is [tex]\( \boxed{3.3 \text{ mm}} \)[/tex].
