To find the value of [tex]\( y \)[/tex] such that the standard deviation of the data set [tex]\[ 5, y, 5, 5, 5 \][/tex] is 5, we can follow these steps:
### Step 1: Calculating the Mean
First, we calculate the mean ([tex]\( \mu \)[/tex]) of the data set. The mean is the sum of all the values divided by the number of values.
[tex]\[ \mu = \frac{5 + y + 5 + 5 + 5}{5} = \frac{20 + y}{5} \][/tex]
### Step 2: Calculating the Variance
Next, we use the mean to calculate the variance ([tex]\( \sigma^2 \)[/tex]). Variance is the average of the squared differences from the mean.
The formula for variance is:
[tex]\[ \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2 \][/tex]
Substitute the values into the formula:
[tex]\[ \sigma^2 = \frac{(5 - \mu)^2 + (y - \mu)^2 + (5 - \mu)^2 + (5 - \mu)^2 + (5 - \mu)^2}{5} \][/tex]
Since [tex]\( \mu = \frac{20 + y}{5} \)[/tex], the variance becomes:
[tex]\[ \sigma^2 = \frac{(5 - \frac{20 + y}{5})^2 + (y - \frac{20 + y}{5})^2 + (5 - \frac{20 + y}{5})^2 + (5 - \frac{20 + y}{5})^2 + (5 - \frac{20 + y}{5})^2}{5} \][/tex]
### Step 3: Substituting the Value of the Mean
Simplify each term inside the summation:
[tex]\[ 5 - \frac{20 + y}{5} = \frac{25 - 20 - y}{5} = \frac{5 - y}{5} \][/tex]
[tex]\[ y - \frac{20 + y}{5} = \frac{5y - 20 - y}{5} = \frac{4y - 20}{5} = \frac{4(y - 5)}{5} \][/tex]
So the variance equation becomes:
[tex]\[ \sigma^2 = \frac{4 \left(\frac{5 - y}{5}\right)^2 + \left(\frac{4(y - 5)}{5}\right)^2}{5} = \frac{4 \cdot \frac{(5 - y)^2}{25} + \frac{16(y - 5)^2}{25}}{5} \][/tex]
Simplify further:
[tex]\[ \sigma^2 = \frac{4 \cdot \frac{(5 - y)^2}{25} + \frac{16(y - 5)^2}{25}}{5} = \frac{4(5 - y)^2 + 16(y - 5)^2}{125} \][/tex]
[tex]\[ \sigma^2 = \frac{(5 - y)^2 + 4(y - 5)^2}{25} \][/tex]
### Step 4: Equate to the Given Standard Deviation
Given that the standard deviation [tex]\( \sigma \)[/tex] is 5, we know:
[tex]\[ \sqrt{\sigma^2} = 5 \][/tex]
Therefore,
[tex]\[ \sigma^2 = 25 \][/tex]
So,
[tex]\[ \frac{(5 - y)^2 + 4(y - 5)^2}{25} = \frac{25}{5} \][/tex]
This simplifies to:
[tex]\[ (5 - y)^2 + 4(y - 5)^2 = 100 \][/tex]
### Step 5: Solving for [tex]\( y \)[/tex]
Now, expand and simplify the expression:
[tex]\[ (5 - y)^2 + 4(y - 5)^2 = 100 \][/tex]
[tex]\[ (5 - y)^2 + 4(y^2 - 10y + 25) = 100 \][/tex]
[tex]\[ 25 - 10y + y^2 + 4y^2 - 40y + 100 = 100 \][/tex]
[tex]\[ 5y^2 - 50y + 125 = 100 \][/tex]
[tex]\[ 5y^2 - 50y + 25 = 0 \][/tex]
[tex]\[ y^2 - 10y + 5 = 0 \][/tex]
Solving this quadratic equation using the quadratic formula [tex]\( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
[tex]\[ y = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 5}}{2 \cdot 1} \][/tex]
[tex]\[ y = \frac{10 \pm \sqrt{100 - 20}}{2} \][/tex]
[tex]\[ y = \frac{10 \pm \sqrt{80}}{2} \][/tex]
[tex]\[ y = \frac{10 \pm 4\sqrt{5}}{2} \][/tex]
[tex]\[ y = 5 \pm 2\sqrt{5} \][/tex]
Thus, the values of [tex]\( y \)[/tex] are:
[tex]\[ y = 5 + 2\sqrt{5} \quad \text{or} \quad y = 5 - 2\sqrt{5} \][/tex]
