Answer :
To find the sample mean and sample standard deviation of the given data set, let's go through the steps together:
### Step 1: Calculate the Sample Mean ([tex]\(\bar{x}\)[/tex])
The sample mean is the average of all the data points. You can calculate it by summing up all the data points and then dividing by the number of data points.
Given the data set:
[tex]\[
\begin{align*}
&0.50, 3.50, 4.75, 6.25, 8.00, 1.50, 3.75, 5.00, 6.25, 8.00, \\
&1.50, 4.00, 5.25, 6.25, 8.00, 1.75, 4.00, 5.50, 6.25, 8.00, \\
&1.75, 4.00, 5.50, 6.50, 8.75, 1.75, 4.25, 5.50, 6.50, 8.75, \\
&2.25, 4.25, 5.50, 6.75, 9.00, 3.00, 4.25, 5.75, 7.00, 9.25, \\
&3.25, 4.25, 5.75, 7.25, 9.75, 3.25, 4.50, 6.00, 7.75, 10.75
\end{align*}
\][/tex]
First, find the total sum of the data:
[tex]\[
\text{Sum} = 0.50 + 3.50 + 4.75 + \ldots + 10.75 = 270.75
\][/tex]
Next, count the total number of data points:
[tex]\[
\text{Number of data points} = 50
\][/tex]
Now, divide the sum by the number of data points to get the sample mean:
[tex]\[
\bar{x} = \frac{270.75}{50} = 5.415
\][/tex]
Rounding this to two decimal places, we get:
[tex]\[
\bar{x} = 5.42
\][/tex]
### Step 2: Calculate the Sample Standard Deviation ([tex]\(s\)[/tex])
The sample standard deviation measures the spread of the data points. Here's how to calculate it:
1. Subtract the mean from each data point and square the result.
2. Sum these squared differences.
3. Divide by the number of data points minus one.
4. Take the square root of the result.
Without listing each calculation step:
- Calculate the squared differences and their sum.
- Divide the total by 49 (since there are 50 data points, use [tex]\(n-1\)[/tex] for sample standard deviation).
- Square root the result.
The calculated value is:
[tex]\[
s \approx 2.400366681172198
\][/tex]
Rounding this to two decimal places, we get:
[tex]\[
s = 2.40
\][/tex]
In conclusion, the sample mean is [tex]\( \bar{x} = 5.42 \)[/tex] minutes and the sample standard deviation is [tex]\( s = 2.40 \)[/tex] minutes.
### Step 1: Calculate the Sample Mean ([tex]\(\bar{x}\)[/tex])
The sample mean is the average of all the data points. You can calculate it by summing up all the data points and then dividing by the number of data points.
Given the data set:
[tex]\[
\begin{align*}
&0.50, 3.50, 4.75, 6.25, 8.00, 1.50, 3.75, 5.00, 6.25, 8.00, \\
&1.50, 4.00, 5.25, 6.25, 8.00, 1.75, 4.00, 5.50, 6.25, 8.00, \\
&1.75, 4.00, 5.50, 6.50, 8.75, 1.75, 4.25, 5.50, 6.50, 8.75, \\
&2.25, 4.25, 5.50, 6.75, 9.00, 3.00, 4.25, 5.75, 7.00, 9.25, \\
&3.25, 4.25, 5.75, 7.25, 9.75, 3.25, 4.50, 6.00, 7.75, 10.75
\end{align*}
\][/tex]
First, find the total sum of the data:
[tex]\[
\text{Sum} = 0.50 + 3.50 + 4.75 + \ldots + 10.75 = 270.75
\][/tex]
Next, count the total number of data points:
[tex]\[
\text{Number of data points} = 50
\][/tex]
Now, divide the sum by the number of data points to get the sample mean:
[tex]\[
\bar{x} = \frac{270.75}{50} = 5.415
\][/tex]
Rounding this to two decimal places, we get:
[tex]\[
\bar{x} = 5.42
\][/tex]
### Step 2: Calculate the Sample Standard Deviation ([tex]\(s\)[/tex])
The sample standard deviation measures the spread of the data points. Here's how to calculate it:
1. Subtract the mean from each data point and square the result.
2. Sum these squared differences.
3. Divide by the number of data points minus one.
4. Take the square root of the result.
Without listing each calculation step:
- Calculate the squared differences and their sum.
- Divide the total by 49 (since there are 50 data points, use [tex]\(n-1\)[/tex] for sample standard deviation).
- Square root the result.
The calculated value is:
[tex]\[
s \approx 2.400366681172198
\][/tex]
Rounding this to two decimal places, we get:
[tex]\[
s = 2.40
\][/tex]
In conclusion, the sample mean is [tex]\( \bar{x} = 5.42 \)[/tex] minutes and the sample standard deviation is [tex]\( s = 2.40 \)[/tex] minutes.
