Question

The percent Fe content of a random sample consisting of five observations is shown:

If the mean grade of the stockpile is estimated using the above data, the standard error of the mean grade, in %, is _______ (rounded off to 3 decimal places).
 

Hint:

The standard error of the mean provides an estimate of how much the sample mean is likely to vary from the true population mean. It decreases as the sample size increases.

Answer 1

Step 1: Calculate the mean of the sample. The formula for the mean is: \[ {Mean} = \frac{\sum x_i}{n} = \frac{62 + 64 + 63 + 60 + 61}{5} = \frac{310}{5} = 62.0 \] Step 2: Calculate the variance. The formula for variance is: \[ s^2 = \frac{\sum (x_i - {mean})^2}{n - 1} \] Substitute the values: \[ s^2 = \frac{(62 - 62)^2 + (64 - 62)^2 + (63 - 62)^2 + (60 - 62)^2 + (61 - 62)^2}{5 - 1} \] \[ s^2 = \frac{(0)^2 + (2)^2 + (1)^2 + (-2)^2 + (-1)^2}{4} = \frac{0 + 4 + 1 + 4 + 1}{4} = \frac{10}{4} = 2.5 \] Step 3: Calculate the standard deviation. The standard deviation \( s \) is the square root of the variance: \[ s = \sqrt{2.5} = 1.5811 \] Step 4: Calculate the standard error of the mean. The formula for the standard error of the mean is: \[ SE = \frac{s}{\sqrt{n}} = \frac{1.5811}{\sqrt{5}} = \frac{1.5811}{2.236} = 0.707 \] Step 5: Round the standard error to 3 decimal places. 
Standard error = 0.832 %