Let’s analyze the given table and find the range of values around the sample mean step-by-step.
### Step 1: Gather all the data points
First, we need to collect all the data points from the table into a single list:
[tex]\[
[4, 2, 3, 0, 0, 1, 1, 2, 0, 4, 1, 4, 4, 2, 3, 2, 0, 1, 4]
\][/tex]
### Step 2: Calculate the Sample Mean
The sample mean ([tex]\(\mu\)[/tex]) is the average of all the data points. It is computed as follows:
[tex]\[
\mu = \frac{\text{sum of all data points}}{\text{number of data points}}
\][/tex]
Given the data points, the sample mean is:
[tex]\[
\mu = 2.0
\][/tex]
### Step 3: Calculate the Sample Standard Deviation
The sample standard deviation ([tex]\(\sigma\)[/tex]) gives us an indication of the spread of the data points around the mean. For the given data, the sample standard deviation is:
[tex]\[
\sigma = 1.4867838833500564
\][/tex]
### Step 4: Determine the Range of Values
To find the range of values where most of the data points lie, we will calculate the range as one standard deviation below and above the mean. This is often referred to as the 68% interval in a normal distribution.
The lower bound of the range is:
[tex]\[
\mu - \sigma = 2.0 - 1.4867838833500564 = 0.5132161166499436
\][/tex]
The upper bound of the range is:
[tex]\[
\mu + \sigma = 2.0 + 1.4867838833500564 = 3.486783883350056
\][/tex]
### Conclusion
The range of the values for the sample means, inclusive of one standard deviation, is from approximately [tex]\(0.513\)[/tex] to [tex]\(3.487\)[/tex]. This means that most data points lie within this range on either side of the mean.
