A summary of 24 observations of x gave the following information:
∑(x − a) = −73.2 and ∑(x − a)2 = 2115.
The mean of these values of x is 8.95.
(i) Find the value of the constant a.
(ii) Find the standard deviation of these values of x
(i) ∑x - 24a = 24*8.95 - 24a = -73.2
Now use this value of a to get
∑(x-a)^2 = ∑x - 2a∑x + 24a^2 = (24a-73.2) - 2a(24a-73.2) + 24a^2 = 2115
Now you can find a
(ii) review the formula for std and plug in your numbers.
(i) Well, in order to find the value of the constant a, we can start by using the formula for the sum of deviations:
∑(x − a) = −73.2
Since we have 24 observations of x and the mean is given as 8.95, we can rewrite the equation as:
24(8.95 − a) = −73.2
Simplifying this equation, we get:
215.28 - 24a = -73.2
Now, let's isolate the constant a:
24a = 288.48
a = 12.02
So, the value of the constant a is 12.02.
(ii) To find the standard deviation, we need to first find the variance, which is the mean of the squared deviations. We already have the sum of squared deviations:
∑(x − a)2 = 2115
To find the variance, we divide this sum by the total number of observations (24 in this case):
Variance = 2115/24 = 88.125
Finally, to find the standard deviation, we take the square root of the variance:
Standard deviation = √(88.125) = 9.38
Therefore, the standard deviation of the values of x is approximately 9.38.
To find the value of the constant "a", we can use the formula:
∑(x - a) = -73.2
We know that there are 24 observations, so the sum can be written as:
24 * a - 24 * 8.95 = -73.2
Simplifying the equation, we get:
24a - 215 = -73.2
Now, let's isolate the "a" term:
24a = -73.2 + 215
24a = 141.8
Divide both sides by 24:
a = 141.8 / 24
a ≈ 5.91
Therefore, the value of the constant "a" is approximately 5.91.
To find the standard deviation of these values of "x", we can use the formula:
∑(x - a)² = 2115
Again, we know that there are 24 observations, so the sum can be written as:
24 * (x - a)² = 2115
Divide both sides by 24:
(x - a)² = 2115 / 24
(x - a)² ≈ 88.125
Now, let's take the square root of both sides to find the standard deviation:
x - a ≈ √88.125
x - a ≈ 9.37
Add "a" to both sides:
x ≈ 9.37 + a
Substitute the value of "a" we previously found:
x ≈ 9.37 + 5.91
x ≈ 15.28
Therefore, the standard deviation of these values of "x" is approximately 15.28.
To find the value of the constant a, we can use the formula for the sum of deviations from the mean:
∑(x − x̄) = n(x̄ − a),
where ∑ represents the sum notation, n is the number of observations, x̄ is the mean, and a is the constant. Given that ∑(x − a) = -73.2 and the mean x̄ = 8.95, we can substitute these values into the formula:
-73.2 = 24(8.95 - a).
Now, we can solve for a by isolating it:
-73.2 = 215.4 - 24a,
-24a = -288.6,
a = (-288.6) / (-24),
a = 12.
Therefore, the value of the constant a is 12.
To find the standard deviation, we can use the following formula:
standard deviation (σ) = √[∑(x − x̄)² / n],
where ∑ represents the sum notation, n is the number of observations, x̄ is the mean, and σ is the standard deviation.
Given that ∑(x − a)² = 2115 and the number of observations n = 24, we can substitute these values into the formula:
standard deviation (σ) = √[2115 / 24].
Now, we can calculate the standard deviation:
standard deviation (σ) = √88.125,
standard deviation (σ) ≈ 9.38.
Therefore, the standard deviation of these values of x is approximately 9.38.