Alright, let's analyze the given data step-by-step to match the franchise locations with the correct descriptions.
### Locations and their Statistics
We have three locations with the following mean and standard deviation:
- Location A
- Mean: [tex]$6,127.85
- Standard Deviation: $[/tex]413.51
- Location B
- Mean: [tex]$6,296.07
- Standard Deviation: $[/tex]440.67
- Location C
- Mean: [tex]$6,191.28
- Standard Deviation: $[/tex]330.83
### Question Analysis
#### 1. For which store does [tex]$68 \%$[/tex] of the data lie between [tex]$5,714.34 and \$[/tex]6,541.36 (1 SD from the mean)?
To find this, we need to calculate the range for 68% of the data, which is within 1 standard deviation from the mean for each location:
- Location A:
- Lower Bound: [tex]$6,127.85 - 413.51 = 5,714.34$[/tex]
- Upper Bound: [tex]$6,127.85 + 413.51 = 6,541.36$[/tex]
- Range: [tex]$5,714.34$[/tex] to [tex]$6,541.36$[/tex]
- Location B:
- Lower Bound: [tex]$6,296.07 - 440.67 = 5,855.4$[/tex]
- Upper Bound: [tex]$6,296.07 + 440.67 = 6,736.74$[/tex]
- Range: [tex]$5,855.4$[/tex] to [tex]$6,736.74$[/tex]
- Location C:
- Lower Bound: [tex]$6,191.28 - 330.83 = 5,860.45$[/tex]
- Upper Bound: [tex]$6,191.28 + 330.83 = 6,522.11$[/tex]
- Range: [tex]$5,860.45$[/tex] to [tex]$6,522.11$[/tex]
The range [tex]$5,714.34 to $[/tex]6,541.36[tex]$ matches the range for Location A.
#### 2. Which location had the highest average daily profit during the past month?
Here, we compare the means of the three locations:
- Location A: $[/tex]6,127.85
- Location B: [tex]$6,296.07
- Location C: $[/tex]6,191.28[tex]$
The highest mean is $[/tex]6,296.07[tex]$, which corresponds to Location B.
#### 3. For which location does $[/tex]95 \%[tex]$ of the data lie between $[/tex]5,529.62 and [tex]$6,852.94 (2 SD from the mean)?
To find this, we need to calculate the range for 95% of the data, which is within 2 standard deviations from the mean for each location:
- Location A:
- Lower Bound: $[/tex]6,127.85 - 2 \times 413.51 = 5,300.83[tex]$
- Upper Bound: $[/tex]6,127.85 + 2 \times 413.51 = 6,954.87[tex]$
- Range: $[/tex]5,300.83[tex]$ to $[/tex]6,954.87[tex]$
- Location B:
- Lower Bound: $[/tex]6,296.07 - 2 \times 440.67 = 5,414.73[tex]$
- Upper Bound: $[/tex]6,296.07 + 2 \times 440.67 = 7,177.41[tex]$
- Range: $[/tex]5,414.73[tex]$ to $[/tex]7,177.41[tex]$
- Location C:
- Lower Bound: $[/tex]6,191.28 - 2 \times 330.83 = 5,529.62[tex]$
- Upper Bound: $[/tex]6,191.28 + 2 \times 330.83 = 6,852.94[tex]$
- Range: $[/tex]5,529.62[tex]$ to $[/tex]6,852.94[tex]$
The range $[/tex]5,529.62 to [tex]$6,852.94$[/tex] matches the range for Location C.
### Summary:
- Location A:
- [tex]$68 \%$[/tex] of the data lies between [tex]$5,714.34 and $[/tex]6,541.36[tex]$.
- Location B:
- Had the highest average daily profit during the past month ($[/tex]6,296.07).
- Location C:
- [tex]$95 \%$[/tex] of the data lies between [tex]$5,529.62 and $[/tex]6,852.94[tex]$.
Thus, the correct matches are:
- Location A: 68% range matches with $[/tex]5,714.34 to [tex]$6,541.36$[/tex]
- Location B: Highest average daily profit
- Location C: 95% range matches with [tex]$5,529.62 to $[/tex]6,852.94$
