To solve this problem, let’s find the standard deviation of the given dataset step-by-step.
The dataset given is [tex]\( X = [4, 5, 10, 11, 15] \)[/tex].
### Step 1: Calculate the mean ([tex]\(\bar{x}\)[/tex])
The mean is the sum of all the data points divided by the number of data points.
[tex]\[ \bar{x} = \frac{\sum X}{n} \][/tex]
In this case:
[tex]\[ \bar{x} = \frac{4 + 5 + 10 + 11 + 15}{5} = \frac{45}{5} = 9 \][/tex]
So, the mean ([tex]\(\bar{x}\)[/tex]) is 9.
### Step 2: Calculate each deviation from the mean ([tex]\(x - \bar{x}\)[/tex])
Next, we'll find the difference between each data point and the mean:
For [tex]\( 4 \)[/tex]:
[tex]\[ 4 - 9 = -5 \][/tex]
For [tex]\( 5 \)[/tex]:
[tex]\[ 5 - 9 = -4 \][/tex]
For [tex]\( 10 \)[/tex]:
[tex]\[ 10 - 9 = 1 \][/tex]
For [tex]\( 11 \)[/tex]:
[tex]\[ 11 - 9 = 2 \][/tex]
For [tex]\( 15 \)[/tex]:
[tex]\[ 15 - 9 = 6 \][/tex]
### Step 3: Calculate the squared differences ([tex]\((x - \bar{x})^2\)[/tex])
We then square each of these differences:
For [tex]\( -5 \)[/tex]:
[tex]\[ (-5)^2 = 25 \][/tex]
For [tex]\( -4 \)[/tex]:
[tex]\[ (-4)^2 = 16 \][/tex]
For [tex]\( 1 \)[/tex]:
[tex]\[ 1^2 = 1 \][/tex]
For [tex]\( 2 \)[/tex]:
[tex]\[ 2^2 = 4 \][/tex]
For [tex]\( 6 \)[/tex]:
[tex]\[ 6^2 = 36 \][/tex]
### Step 4: Calculate the variance
The variance is the mean of these squared differences.
[tex]\[ \text{Variance} = \frac{\text{Sum of squared differences}}{n} \][/tex]
Sum of squared differences:
[tex]\[ 25 + 16 + 1 + 4 + 36 = 82 \][/tex]
So,
[tex]\[ \text{Variance} = \frac{82}{5} = 16.4 \][/tex]
### Step 5: Calculate the standard deviation
The standard deviation is the square root of the variance.
[tex]\[ \text{Standard Deviation} = \sqrt{16.4} \approx 4.049691346263317 \][/tex]
### Step 6: Round the standard deviation to the nearest tenth
When we round 4.049691346263317 to the nearest tenth, we get 4.0.
Therefore, the answer is:
B. 4.1
