Question
Alejandro wants to estimate the average age of people at his gym. He takes a random sample of 10 people at his gym and records the age of each person. His results are summarized below:
Ages =
PART A
Find the sample mean.
Sample mean
Question 2
PART B
Find the mean absolute deviation (MAD) of the sample and use it to evaluate how far off the sample mean is from the actual mean.
Responses
A The MAD is 12.6. Since this number is relatively small, the sample mean is a pretty good estimate for the actual mean.The MAD is 12.6. Since this number is relatively small, the sample mean is a pretty good estimate for the actual mean.
B The MAD is 12.6. Since this number is relatively large, the sample mean is likely far from the actual mean.The MAD is 12.6. Since this number is relatively large, the sample mean is likely far from the actual mean.
C The MAD is 47. Since this number is relatively small, the sample mean is a pretty good estimate for the actual mean.The MAD is 47. Since this number is relatively small, the sample mean is a pretty good estimate for the actual mean.
D The MAD is 47. Since this number is relatively large, the sample mean is likely far from the actual mean.The MAD is 47. Since this number is relatively large, the sample mean is likely far from the actual mean.
Ages =
PART A
Find the sample mean.
Sample mean
Question 2
PART B
Find the mean absolute deviation (MAD) of the sample and use it to evaluate how far off the sample mean is from the actual mean.
Responses
A The MAD is 12.6. Since this number is relatively small, the sample mean is a pretty good estimate for the actual mean.The MAD is 12.6. Since this number is relatively small, the sample mean is a pretty good estimate for the actual mean.
B The MAD is 12.6. Since this number is relatively large, the sample mean is likely far from the actual mean.The MAD is 12.6. Since this number is relatively large, the sample mean is likely far from the actual mean.
C The MAD is 47. Since this number is relatively small, the sample mean is a pretty good estimate for the actual mean.The MAD is 47. Since this number is relatively small, the sample mean is a pretty good estimate for the actual mean.
D The MAD is 47. Since this number is relatively large, the sample mean is likely far from the actual mean.The MAD is 47. Since this number is relatively large, the sample mean is likely far from the actual mean.
Answers
GPT-4o mini
To solve this problem, let's go through the necessary calculations step by step.
### Part A: Find the Sample Mean
1. **List the Ages**: Since you didn't provide specific ages, I can't calculate the exact mean. However, let's denote the ages collected as \( A_1, A_2, A_3, A_4, A_5, A_6, A_7, A_8, A_9, A_{10} \).
2. **Calculate the Sample Mean**:
The sample mean (denoted as \(\bar{x}\)) is given by:
\[
\bar{x} = \frac{A_1 + A_2 + A_3 + A_4 + A_5 + A_6 + A_7 + A_8 + A_9 + A_{10}}{10}
\]
3. **Insert Age Values**: If you provide the ages, I'd be able to give you the exact mean calculation.
### Part B: Mean Absolute Deviation (MAD)
1. **Calculate Absolute Deviations**: The absolute deviation for each age from the sample mean is calculated as:
\[
|A_i - \bar{x}|
\]
for each \( i \) from 1 to 10.
2. **Calculate Mean Absolute Deviation**:
The MAD is given by:
\[
\text{MAD} = \frac{|A_1 - \bar{x}| + |A_2 - \bar{x}| + |A_3 - \bar{x}| + ... + |A_{10} - \bar{x}|}{10}
\]
3. **Interpret the MAD**: After calculating the MAD, you can compare it to the sample mean:
- If the MAD is relatively small, it suggests the sample mean is a good estimate for the actual mean.
- If the MAD is relatively large, it indicates the sample mean might not be a good estimate for the actual mean.
### Final Choice Based on MAD
- If you find \( \text{MAD} = 12.6 \), you would choose **Answer A or B** based on whether you consider 12.6 small or large.
- If you find \( \text{MAD} = 47 \), you would choose **Answer C or D** based on the perception of 47 being small or large.
Feel free to provide the exact ages so that I can assist in calculating the samples.
### Part A: Find the Sample Mean
1. **List the Ages**: Since you didn't provide specific ages, I can't calculate the exact mean. However, let's denote the ages collected as \( A_1, A_2, A_3, A_4, A_5, A_6, A_7, A_8, A_9, A_{10} \).
2. **Calculate the Sample Mean**:
The sample mean (denoted as \(\bar{x}\)) is given by:
\[
\bar{x} = \frac{A_1 + A_2 + A_3 + A_4 + A_5 + A_6 + A_7 + A_8 + A_9 + A_{10}}{10}
\]
3. **Insert Age Values**: If you provide the ages, I'd be able to give you the exact mean calculation.
### Part B: Mean Absolute Deviation (MAD)
1. **Calculate Absolute Deviations**: The absolute deviation for each age from the sample mean is calculated as:
\[
|A_i - \bar{x}|
\]
for each \( i \) from 1 to 10.
2. **Calculate Mean Absolute Deviation**:
The MAD is given by:
\[
\text{MAD} = \frac{|A_1 - \bar{x}| + |A_2 - \bar{x}| + |A_3 - \bar{x}| + ... + |A_{10} - \bar{x}|}{10}
\]
3. **Interpret the MAD**: After calculating the MAD, you can compare it to the sample mean:
- If the MAD is relatively small, it suggests the sample mean is a good estimate for the actual mean.
- If the MAD is relatively large, it indicates the sample mean might not be a good estimate for the actual mean.
### Final Choice Based on MAD
- If you find \( \text{MAD} = 12.6 \), you would choose **Answer A or B** based on whether you consider 12.6 small or large.
- If you find \( \text{MAD} = 47 \), you would choose **Answer C or D** based on the perception of 47 being small or large.
Feel free to provide the exact ages so that I can assist in calculating the samples.