To determine the mean weekly grocery bill for the given data, we need to use the method of midpoints for each bill range multiplied by their respective frequencies. Let's break down the process:
1. Identify the midpoints of each bill range:
- For the range [tex]$135-139$[/tex], the midpoint is [tex]\((135 + 139) / 2 = 137\)[/tex].
- For the range [tex]$140-144$[/tex], the midpoint is [tex]\((140 + 144) / 2 = 142\)[/tex].
- For the range [tex]$145-149$[/tex], the midpoint is [tex]\((145 + 149) / 2 = 147\)[/tex].
- For the range [tex]$150-154$[/tex], the midpoint is [tex]\((150 + 154) / 2 = 152\)[/tex].
- For the range [tex]$155-159$[/tex], the midpoint is [tex]\((155 + 159) / 2 = 157\)[/tex].
2. Multiply each midpoint by their respective frequencies:
- [tex]$137 \times 20 = 2740$[/tex]
- [tex]$142 \times 17 = 2414$[/tex]
- [tex]$147 \times 16 = 2352$[/tex]
- [tex]$152 \times 14 = 2128$[/tex]
- [tex]$157 \times 19 = 2983$[/tex]
3. Calculate the total sum of the products obtained above:
- [tex]\( 2740 + 2414 + 2352 + 2128 + 2983 = 12617 \)[/tex]
4. Find the total frequency:
- [tex]\( 20 + 17 + 16 + 14 + 19 = 86 \)[/tex]
5. Calculate the approximate mean weekly grocery bill:
- Mean = Total sum of midpoints' products / Total frequency
- Mean = [tex]\( \frac{12617}{86} \approx 146.71 \)[/tex]
Therefore, the approximate mean weekly grocery bill is \[tex]$146.71. Given the options provided in the question, this matches none of the exact choices directly, but the closest value is more accurately represented. The mean weekly grocery bill calculated is approximately \$[/tex]146.71.
