Find solutions to your questions with the help of IDNLearn.com's expert community. Discover reliable answers to your questions with our extensive database of expert knowledge.
Sagot :
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.
(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.
We are happy to have you as part of our community. Keep asking, answering, and sharing your insights. Together, we can create a valuable knowledge resource. IDNLearn.com is your go-to source for accurate answers. Thanks for stopping by, and come back for more helpful information.
