Question
36. Sylvia and Alberto go to the same school. Each of them randomly sampled 20 students in the school,
asking, “How many people are in your household, including yourself?”. The results are shown below.
Sylvia’s Sample
4 5 4 5 3
5 2 4 6 3
7 6 3 8 4
2 5 6 4 6
Alberto’s Sample
6 8 4 9 4
5 5 3 3 5
4 6 3 5 4
8 4 4 2 4
What is the mean of Sylvia’s sample?
Enter your answer and your work in the box provided.
Downloaded by S. FARSAKH at SOUTHEAST MIDDLE SCHOOL. This resource expires on 6/30/2025.
sylvla sample
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Assessment 1
25
GO ON
Part B
What is the mean of Alberto’s sample?
Enter your answer and your work in the box provided.
Part C
Are the means of the two samples the same? If not, would that show that somebody made
an error? Explain your reasoning.
Enter your answer and your work or explanation in the box provided.
Downloaded by S. FARSAKH at SOUTHEAST MIDDLE SCHOOL. This resource expires on 6/30/2025.
©Curriculum Associates, LLC Copying is not permitted.
Assessment 1
26
Part D
What would be a good estimate for the average number of people per household for all of the students
in the school? Explain how you got your estimate.
Enter your answer and your work or explanation in the box provided.
asking, “How many people are in your household, including yourself?”. The results are shown below.
Sylvia’s Sample
4 5 4 5 3
5 2 4 6 3
7 6 3 8 4
2 5 6 4 6
Alberto’s Sample
6 8 4 9 4
5 5 3 3 5
4 6 3 5 4
8 4 4 2 4
What is the mean of Sylvia’s sample?
Enter your answer and your work in the box provided.
Downloaded by S. FARSAKH at SOUTHEAST MIDDLE SCHOOL. This resource expires on 6/30/2025.
sylvla sample
©Curriculum Associates, LLC Copying is not permitted.
Assessment 1
25
GO ON
Part B
What is the mean of Alberto’s sample?
Enter your answer and your work in the box provided.
Part C
Are the means of the two samples the same? If not, would that show that somebody made
an error? Explain your reasoning.
Enter your answer and your work or explanation in the box provided.
Downloaded by S. FARSAKH at SOUTHEAST MIDDLE SCHOOL. This resource expires on 6/30/2025.
©Curriculum Associates, LLC Copying is not permitted.
Assessment 1
26
Part D
What would be a good estimate for the average number of people per household for all of the students
in the school? Explain how you got your estimate.
Enter your answer and your work or explanation in the box provided.
Answers
GPT-4o mini
### Part A: Mean of Sylvia's Sample
To calculate the mean of Sylvia's sample, we first need to add all her responses and then divide by the number of students in her sample.
**Sylvia’s Sample Responses:**
```
4, 5, 4, 5, 3,
5, 2, 4, 6, 3,
7, 6, 3, 8, 4,
2, 5, 6, 4, 6
```
**Sum of Sylvia's Responses:**
\[
4 + 5 + 4 + 5 + 3 + 5 + 2 + 4 + 6 + 3 + 7 + 6 + 3 + 8 + 4 + 2 + 5 + 6 + 4 + 6 = 5(6) + 4(6) + 3(5) + 2(3) + 8 = 114
\]
**Mean Calculation:**
\[
\text{Mean} = \frac{\text{Total Sum}}{\text{Number of Responses}} = \frac{114}{20} = 5.7
\]
### Part B: Mean of Alberto's Sample
Next, we calculate the mean of Alberto's sample.
**Alberto’s Sample Responses:**
```
6, 8, 4, 9, 4,
5, 5, 3, 3, 5,
4, 6, 3, 5, 4,
8, 4, 4, 2, 4
```
**Sum of Alberto's Responses:**
\[
6 + 8 + 4 + 9 + 4 + 5 + 5 + 3 + 3 + 5 + 4 + 6 + 3 + 5 + 4 + 8 + 4 + 4 + 2 + 4 = 8(5) + 4(6) + 9 + 3(4) + 3 + 5 + 6 + 2 = 102
\]
**Mean Calculation:**
\[
\text{Mean} = \frac{\text{Total Sum}}{\text{Number of Responses}} = \frac{102}{20} = 5.1
\]
### Part C: Are the Means of the Two Samples the Same?
**Comparing Means:**
- Mean of Sylvia's Sample = 5.7
- Mean of Alberto's Sample = 5.1
**Conclusion:**
The means are not the same. A difference in means does not necessarily indicate that someone made an error. It is possible that their samples represent different groups of students or that the individual responses varied widely. Differences can arise naturally due to sampling variability.
### Part D: Estimate for the Average Number of People Per Household
A reasonable estimate for the average number of people per household for all students in the school could be the average of the two sample means, assuming both samples are representative of the overall student population.
**Estimate Calculation:**
\[
\text{Estimate} = \frac{\text{Mean of Sylvia} + \text{Mean of Alberto}}{2} = \frac{5.7 + 5.1}{2} = \frac{10.8}{2} = 5.4
\]
**Explanation:**
Using the average of the two sample means provides a better overall estimate as it accounts for variability between the two independent samples. Thus, a good estimate for the average number of people per household for all students in the school would be 5.4.
To calculate the mean of Sylvia's sample, we first need to add all her responses and then divide by the number of students in her sample.
**Sylvia’s Sample Responses:**
```
4, 5, 4, 5, 3,
5, 2, 4, 6, 3,
7, 6, 3, 8, 4,
2, 5, 6, 4, 6
```
**Sum of Sylvia's Responses:**
\[
4 + 5 + 4 + 5 + 3 + 5 + 2 + 4 + 6 + 3 + 7 + 6 + 3 + 8 + 4 + 2 + 5 + 6 + 4 + 6 = 5(6) + 4(6) + 3(5) + 2(3) + 8 = 114
\]
**Mean Calculation:**
\[
\text{Mean} = \frac{\text{Total Sum}}{\text{Number of Responses}} = \frac{114}{20} = 5.7
\]
### Part B: Mean of Alberto's Sample
Next, we calculate the mean of Alberto's sample.
**Alberto’s Sample Responses:**
```
6, 8, 4, 9, 4,
5, 5, 3, 3, 5,
4, 6, 3, 5, 4,
8, 4, 4, 2, 4
```
**Sum of Alberto's Responses:**
\[
6 + 8 + 4 + 9 + 4 + 5 + 5 + 3 + 3 + 5 + 4 + 6 + 3 + 5 + 4 + 8 + 4 + 4 + 2 + 4 = 8(5) + 4(6) + 9 + 3(4) + 3 + 5 + 6 + 2 = 102
\]
**Mean Calculation:**
\[
\text{Mean} = \frac{\text{Total Sum}}{\text{Number of Responses}} = \frac{102}{20} = 5.1
\]
### Part C: Are the Means of the Two Samples the Same?
**Comparing Means:**
- Mean of Sylvia's Sample = 5.7
- Mean of Alberto's Sample = 5.1
**Conclusion:**
The means are not the same. A difference in means does not necessarily indicate that someone made an error. It is possible that their samples represent different groups of students or that the individual responses varied widely. Differences can arise naturally due to sampling variability.
### Part D: Estimate for the Average Number of People Per Household
A reasonable estimate for the average number of people per household for all students in the school could be the average of the two sample means, assuming both samples are representative of the overall student population.
**Estimate Calculation:**
\[
\text{Estimate} = \frac{\text{Mean of Sylvia} + \text{Mean of Alberto}}{2} = \frac{5.7 + 5.1}{2} = \frac{10.8}{2} = 5.4
\]
**Explanation:**
Using the average of the two sample means provides a better overall estimate as it accounts for variability between the two independent samples. Thus, a good estimate for the average number of people per household for all students in the school would be 5.4.