Question
Compute the mean, range, and standard deviation for the data items
in each of the three samples. Then describe one way in which the samples are alike and one way in which they are different.
Sample A: 33, 37, 41, 45, 49, 53, 57
Sample B: 33, 35, 37, 45, 53, 56, 57
Sample C: 33, 33, 33, 45, 57, 57,57
in each of the three samples. Then describe one way in which the samples are alike and one way in which they are different.
Sample A: 33, 37, 41, 45, 49, 53, 57
Sample B: 33, 35, 37, 45, 53, 56, 57
Sample C: 33, 33, 33, 45, 57, 57,57
Answers
GPT-4o mini
Let's compute the mean, range, and standard deviation for each of the three samples (A, B, and C) one by one.
### Sample A: 33, 37, 41, 45, 49, 53, 57
1. **Mean:**
\[
\text{Mean} = \frac{33 + 37 + 41 + 45 + 49 + 53 + 57}{7} = \frac{315}{7} = 45
\]
2. **Range:**
\[
\text{Range} = \text{Max} - \text{Min} = 57 - 33 = 24
\]
3. **Standard Deviation:**
\[
\text{Variance} = \frac{(33-45)^2 + (37-45)^2 + (41-45)^2 + (45-45)^2 + (49-45)^2 + (53-45)^2 + (57-45)^2}{7}
\]
\[
= \frac{(144) + (64) + (16) + (0) + (16) + (64) + (144)}{7} = \frac{448}{7} = 64
\]
\[
\text{Standard Deviation} = \sqrt{64} \approx 8
\]
### Sample B: 33, 35, 37, 45, 53, 56, 57
1. **Mean:**
\[
\text{Mean} = \frac{33 + 35 + 37 + 45 + 53 + 56 + 57}{7} = \frac{316}{7} = 45.14 \text{ (approx.)}
\]
2. **Range:**
\[
\text{Range} = 57 - 33 = 24
\]
3. **Standard Deviation:**
\[
\text{Variance} = \frac{(33-45.14)^2 + (35-45.14)^2 + (37-45.14)^2 + (45-45.14)^2 + (53-45.14)^2 + (56-45.14)^2 + (57-45.14)^2}{7}
\]
\[
= \frac{(147.62) + (102.33) + (67.66) + (0.02) + (61.51) + (117.58) + (140.74)}{7} \approx \frac{536.66}{7} \approx 76.66
\]
\[
\text{Standard Deviation} \approx \sqrt{76.66} \approx 8.76 \text{ (approx.)}
\]
### Sample C: 33, 33, 33, 45, 57, 57, 57
1. **Mean:**
\[
\text{Mean} = \frac{33 + 33 + 33 + 45 + 57 + 57 + 57}{7} = \frac{315}{7} = 45
\]
2. **Range:**
\[
\text{Range} = 57 - 33 = 24
\]
3. **Standard Deviation:**
\[
\text{Variance} = \frac{(33-45)^2 + (33-45)^2 + (33-45)^2 + (45-45)^2 + (57-45)^2 + (57-45)^2 + (57-45)^2}{7}
\]
\[
= \frac{(144) + (144) + (144) + (0) + (144) + (144) + (144)}{7} = \frac{864}{7} \approx 123.43
\]
\[
\text{Standard Deviation} = \sqrt{123.43} \approx 11.1 \text{ (approx.)}
\]
### Summary of Results:
- **Sample A:**
- Mean: 45
- Range: 24
- Standard Deviation: 8
- **Sample B:**
- Mean: approximately 45.14
- Range: 24
- Standard Deviation: approximately 8.76
- **Sample C:**
- Mean: 45
- Range: 24
- Standard Deviation: approximately 11.1
### Similarities and Differences:
**Similarity:**
- All three samples have the same range of 24.
**Difference:**
- The standard deviations vary, indicating that the samples have different levels of dispersion around their means. Sample C has the highest standard deviation, showing greater variability in its data.
Let me know if you need further assistance!
### Sample A: 33, 37, 41, 45, 49, 53, 57
1. **Mean:**
\[
\text{Mean} = \frac{33 + 37 + 41 + 45 + 49 + 53 + 57}{7} = \frac{315}{7} = 45
\]
2. **Range:**
\[
\text{Range} = \text{Max} - \text{Min} = 57 - 33 = 24
\]
3. **Standard Deviation:**
\[
\text{Variance} = \frac{(33-45)^2 + (37-45)^2 + (41-45)^2 + (45-45)^2 + (49-45)^2 + (53-45)^2 + (57-45)^2}{7}
\]
\[
= \frac{(144) + (64) + (16) + (0) + (16) + (64) + (144)}{7} = \frac{448}{7} = 64
\]
\[
\text{Standard Deviation} = \sqrt{64} \approx 8
\]
### Sample B: 33, 35, 37, 45, 53, 56, 57
1. **Mean:**
\[
\text{Mean} = \frac{33 + 35 + 37 + 45 + 53 + 56 + 57}{7} = \frac{316}{7} = 45.14 \text{ (approx.)}
\]
2. **Range:**
\[
\text{Range} = 57 - 33 = 24
\]
3. **Standard Deviation:**
\[
\text{Variance} = \frac{(33-45.14)^2 + (35-45.14)^2 + (37-45.14)^2 + (45-45.14)^2 + (53-45.14)^2 + (56-45.14)^2 + (57-45.14)^2}{7}
\]
\[
= \frac{(147.62) + (102.33) + (67.66) + (0.02) + (61.51) + (117.58) + (140.74)}{7} \approx \frac{536.66}{7} \approx 76.66
\]
\[
\text{Standard Deviation} \approx \sqrt{76.66} \approx 8.76 \text{ (approx.)}
\]
### Sample C: 33, 33, 33, 45, 57, 57, 57
1. **Mean:**
\[
\text{Mean} = \frac{33 + 33 + 33 + 45 + 57 + 57 + 57}{7} = \frac{315}{7} = 45
\]
2. **Range:**
\[
\text{Range} = 57 - 33 = 24
\]
3. **Standard Deviation:**
\[
\text{Variance} = \frac{(33-45)^2 + (33-45)^2 + (33-45)^2 + (45-45)^2 + (57-45)^2 + (57-45)^2 + (57-45)^2}{7}
\]
\[
= \frac{(144) + (144) + (144) + (0) + (144) + (144) + (144)}{7} = \frac{864}{7} \approx 123.43
\]
\[
\text{Standard Deviation} = \sqrt{123.43} \approx 11.1 \text{ (approx.)}
\]
### Summary of Results:
- **Sample A:**
- Mean: 45
- Range: 24
- Standard Deviation: 8
- **Sample B:**
- Mean: approximately 45.14
- Range: 24
- Standard Deviation: approximately 8.76
- **Sample C:**
- Mean: 45
- Range: 24
- Standard Deviation: approximately 11.1
### Similarities and Differences:
**Similarity:**
- All three samples have the same range of 24.
**Difference:**
- The standard deviations vary, indicating that the samples have different levels of dispersion around their means. Sample C has the highest standard deviation, showing greater variability in its data.
Let me know if you need further assistance!