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.
HELP PLZ!
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.
HELP PLZ!
Since no probability is given, we denote the mean of x as $\displaystyle \mu = \Sigma x/n$
We know n is 24, and $\displaystyle \mu = 8.95$. Therefore, $\displaystyle \Sigma x = (24)(8.95) = 214.8$
Your problem gives $\displaystyle \Sigma (x - a) = -73.2$
That can be rewritten as $\displaystyle \Sigma x - a = -73.2$.
$\displaystyle a = 214.8 + 73.2$ <------ $\displaystyle \Sigma x = 214.8$ from above
$\displaystyle a = 288$
I'll take a stab at part (ii) as well.
From above, for your second equation, substitute our value of 288 for a, we get $\displaystyle \Sigma (x - 288)^2 = 2115$
Expanding and simplifying, we get:
$\displaystyle \Sigma x^2 - 576\Sigma x + 82944 = 2115$
$\displaystyle \Sigma x^2 - 576(214.8) + 82944 = 2115$ <------ remember that $\displaystyle \Sigma x = 214.8$ from part i
$\displaystyle \Sigma x^2 = 42895.8$
Now, we know that Variance, denoted as $\displaystyle \sigma ^ 2 = \Sigma (x - \mu)^2$
Plugging in our value for $\displaystyle \mu = 8.95$ and simplifying, we get:
$\displaystyle \sigma^2 = \Sigma x^2 - 17.9\Sigma x + \Sigma 80.1025$
$\displaystyle \sigma^2 = 42895.8 - 17.9(214.8) + 80.1025$ <---- we figured out earlier $\displaystyle \Sigma x$ and $\displaystyle \Sigma x ^2$
$\displaystyle \sigma^2 = 42895.8 - 3844.92 + 80.1025$
$\displaystyle \sigma^2 = 39130.98$
Now calculate standard deviation, denoted as $\displaystyle \sigma$
$\displaystyle \sigma = \sqrt{\sigma^2}$
$\displaystyle \sigma = \sqrt{39130.98}$
$\displaystyle \sigma = 197.8155$
um... which one is right?
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