Question

Measurements of a sample of 6 weights were found to be 14⋅ 16,3 ⋅ 15,6 ⋅ 14,7 ⋅ 16,8 ⋅ 2

and 15 ⋅ 4 kilogram respectively. (i) Determine an unbiased estimate of the population 

mean. (ii) Compare the sample standard deviation with the estimated population 

standard deviation. 

Answer 1

$$14, 14.7, 15, 15.6, 16.3, 16.8$$

(i)

Arithmetic mean $\bar{x}=\dfrac{\displaystyle\sum_{i=1}^{6}x_i}{6}=\dfrac{92.4}{6}=15.4$

Median = $\dfrac{15+15.6}{2}=15.3$

Mode: All values appeared just once.

The sample mean $\bar{x}=15.4,$ is an unbiased estimator of the population mean.

(i)  Variance

The variance of a sample is:

$$s^2=\dfrac{1}{6-1}\displaystyle\sum_{i=1}^{6}(x_i-\bar{x})^2=\dfrac{5.42}{5}=1.084$$

 Standard Deviation

$$s=\sqrt{s^2}=\sqrt{1.084}\approx1.041153$$

$$\sigma^2=\dfrac{1}{6}\displaystyle\sum_{i=1}^{6}(x_i-\mu)^2=\dfrac{5.42}{6}=\dfrac{2.71}{3}$$

$$\sigma=\sqrt{\sigma^2}=\sqrt{\dfrac{2.71}{3}}\approx0.950438<s$$