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:

Question

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.

Answer 1

To help Alejandro find the sample mean and the mean absolute deviation (MAD) of the ages, we first need the list of ages you refer to in the problem. However, since the ages are not provided, I will guide you through the calculations using a hypothetical list of ages.

Sample Mean Calculation

The sample mean can be calculated using the formula:

\[ \text{Sample Mean} = \frac{\sum_{i=1}^{n} x_i}{n} \]

where \( x_i \) are the ages and \( n \) is the number of samples (in this case, 10).

Mean Absolute Deviation (MAD) Calculation

The mean absolute deviation can be calculated using the formula:

\[ \text{MAD} = \frac{\sum_{i=1}^{n} |x_i - \text{Sample Mean}|}{n} \]

where \( |x_i - \text{Sample Mean}| \) is the absolute deviation of each observation from the sample mean.

Example Calculation

Let's consider a hypothetical set of ages:

Ages = [25, 30, 35, 40, 45, 50, 55, 60, 65, 70]

1. Calculate the Sample Mean:

\[ \text{Sample Mean} = \frac{25 + 30 + 35 + 40 + 45 + 50 + 55 + 60 + 65 + 70}{10} = \frac{ 450 }{10} = 45 \]

2. Calculate the Absolute Deviations:

\[ |25 - 45| = 20, \quad |30 - 45| = 15, \quad |35 - 45| = 10, \quad |40 - 45| = 5 \] \[ |45 - 45| = 0, \quad |50 - 45| = 5, \quad |55 - 45| = 10, \quad |60 - 45| = 15 \] \[ |65 - 45| = 20, \quad |70 - 45| = 25 \]

3. Sum the Absolute Deviations:

\[ 20 + 15 + 10 + 5 + 0 + 5 + 10 + 15 + 20 + 25 = 120 \]

4. Calculate the MAD:

\[ \text{MAD} = \frac{120}{10} = 12 \]

Interpretation

Now, based on the MAD value obtained (12 in this case), you would evaluate how far off the sample mean (45 in this case) is from the actual mean. The correct response would depend on whether 12 is considered relatively large or small in the context provided.

Given the options:

1. If we assume the actual mean is around the sample mean, we would consider 12 small.
2. If we assume the actual mean is far off from the sample mean, we might consider 12 large.

Without the actual ages and contextual information, we cannot confirm the correct answer. Please provide the actual ages or adjust the given options based on your values if my hypothetical ages do not apply.