Certainly! Let's analyze the given data step-by-step:
### Store 1 Hours: [tex]\(16, 28, 36, 40, 78\)[/tex]
### Store 2 Hours: [tex]\(22, 27, 33, 36, 40\)[/tex]
First, let's compute the relevant statistical measures for each store.
#### Mean (Average):
- Store 1 Mean:
[tex]\[
\text{Mean} = \frac{16 + 28 + 36 + 40 + 78}{5} = \frac{198}{5} = 39.6
\][/tex]
- Store 2 Mean:
[tex]\[
\text{Mean} = \frac{22 + 27 + 33 + 36 + 40}{5} = \frac{158}{5} = 31.6
\][/tex]
#### Median (Middle Value):
- Store 1 Median:
[tex]\[
\text{Median} = 36 \quad \text{(middle value in sorted list)}
\][/tex]
- Store 2 Median:
[tex]\[
\text{Median} = 33 \quad \text{(middle value in sorted list)}
\][/tex]
#### Range (Difference between max and min):
- Store 1 Range:
[tex]\[
\text{Range} = 78 - 16 = 62
\][/tex]
- Store 2 Range:
[tex]\[
\text{Range} = 40 - 22 = 18
\][/tex]
#### Interquartile Range (IQR):
- Store 1 IQR:
[tex]\[
\text{IQR} = Q3 - Q1 = 46 - 34 = 12
\][/tex]
(where [tex]\(Q1\)[/tex] is the 25th percentile and [tex]\(Q3\)[/tex] is the 75th percentile)
- Store 2 IQR:
[tex]\[
\text{IQR} = Q3 - Q1 = 36 - 27 = 9
\][/tex]
(where [tex]\(Q1\)[/tex] is the 25th percentile and [tex]\(Q3\)[/tex] is the 75th percentile)
#### Variance (measure of variability):
- Store 1 Variance:
[tex]\[
\text{Variance} = 435.84
\][/tex]
- Store 2 Variance:
[tex]\[
\text{Variance} = 41.04
\][/tex]
### Conclusions Based on the Measurements:
1. The data for store 1 shows greater variability.
- True, because the variance for Store 1 (435.84) is much higher than for Store 2 (41.04).
2. The median number of hours worked for the two stores is the same.
- False, because the median for Store 1 is 36 hours, while the median for Store 2 is 33 hours.
3. The mean of the data for store 1 is greater than the mean of the data for store 2.
- True, because the mean for Store 1 (39.6) is greater than the mean for Store 2 (31.6).
4. The interquartile range of both data sets is the same.
- False, because the IQR for Store 1 is 12, while the IQR for Store 2 is 9.
5. The range of the data for store 1 is less than the range of the data for store 2.
- False, because the range for Store 1 (62) is greater than the range for Store 2 (18).
### Final Supported Statements:
- The data for store 1 shows greater variability.
- The mean of the data for store 1 is greater than the mean of the data for store 2.
So, the final true statements supported by the data are:
1. The data for store 1 shows greater variability.
2. The mean of the data for store 1 is greater than the mean of the data for store 2.
