Alright, let's address the given problem step by step.
(a) Calculating the value of [tex]\( k \)[/tex]:
We are given that the mean of the numbers [tex]\(1, 4, k, (k+4)\)[/tex], and [tex]\(11\)[/tex] is [tex]\((k+1)\)[/tex]. We need to set up this condition in a mathematical equation.
The mean of a set of numbers is calculated by dividing the sum of the numbers by the total count of numbers. So, we first express this formally.
[tex]\[ \text{Mean} = \frac{(1 + 4 + k + (k + 4) + 11)}{5} \][/tex]
Given that this mean equals [tex]\((k+1)\)[/tex], we can write:
[tex]\[ \frac{(1 + 4 + k + (k + 4) + 11)}{5} = k + 1 \][/tex]
Now simplify the numerator:
[tex]\[ 1 + 4 + k + k + 4 + 11 = 20 + 2k \][/tex]
So the equation becomes:
[tex]\[ \frac{(20 + 2k)}{5} = k + 1 \][/tex]
To eliminate the fraction, multiply both sides by 5:
[tex]\[ 20 + 2k = 5(k + 1) \][/tex]
Now expand the right side:
[tex]\[ 20 + 2k = 5k + 5 \][/tex]
To isolate [tex]\( k \)[/tex], first subtract 2k from both sides:
[tex]\[ 20 = 3k + 5 \][/tex]
Then subtract 5 from both sides:
[tex]\[ 15 = 3k \][/tex]
Finally, divide by 3:
[tex]\[ k = 5 \][/tex]
So, the value of [tex]\( k \)[/tex] is:
[tex]\[ \boxed{5} \][/tex]
(b) Calculating the standard deviation:
To find the standard deviation, we first need to identify the mean of the numbers using the value of [tex]\( k \)[/tex]:
The numbers are [tex]\( 1, 4, 5, 9, \)[/tex] and [tex]\( 11 \)[/tex]. Let's first calculate the actual mean [tex]\(\mu\)[/tex] of these numbers:
[tex]\[ \mu = \frac{(1 + 4 + 5 + 9 + 11)}{5} = \frac{30}{5} = 6 \][/tex]
The variance [tex]\( \sigma^2 \)[/tex] is calculated by finding the average of the squared differences from the Mean.
[tex]\[ \sigma^2 = \frac{(1-6)^2 + (4-6)^2 + (5-6)^2 + (9-6)^2 + (11-6)^2}{5} \][/tex]
Calculating each of the squared differences:
[tex]\[ (1-6)^2 = 25 \][/tex]
[tex]\[ (4-6)^2 = 4 \][/tex]
[tex]\[ (5-6)^2 = 1 \][/tex]
[tex]\[ (9-6)^2 = 9 \][/tex]
[tex]\[ (11-6)^2 = 25 \][/tex]
Sum these squared differences:
[tex]\[ 25 + 4 + 1 + 9 + 25 = 64 \][/tex]
So, the variance [tex]\( \sigma^2 \)[/tex] is:
[tex]\[ \sigma^2 = \frac{64}{5} = 12.8 \][/tex]
The standard deviation [tex]\( \sigma \)[/tex] is the square root of the variance:
[tex]\[ \sigma = \sqrt{12.8} \approx 3.58 \][/tex]
So, the standard deviation is approximately:
[tex]\[ \boxed{3.58} \][/tex]
This completes the solution for both parts of the problem.
