Question

Suppose that we have collected a sample of 8 observations with values 12.6, 12.9, 13.4, 12.3, 13.6, 13.5, 12.6, and 13.1. What are the observed sample mean, observed sample variance, and observed sample standard deviation

Answer 1

Answer:

The sample mean is 13.

The sample variance is 0.2286.

The sample standard deviation is 0.4781.

Step-by-step explanation:

The sample is:

S = {12.6, 12.9, 13.4, 12.3, 13.6, 13.5, 12.6, 13.1}

The sample is of size n = 8.

The formula to compute the sample mean, sample variance and sample standard deviation are:

$\bar x=\frac{1}{n} \sum x$

$s^{2}=\frac{1}{n}\sum (x-\bar x)^{2} \\s=\sqrt{\frac{1}{n}\sum (x-\bar x)^{2} }$

Compute the sample mean as follows:

$\bar x=\frac{1}{n} \sum x\\=\frac{1}{8}(12.6+ 12.9+ 13.4+ 12.3+ 13.6+ 13.5+ 12.6+ 13.1)\\=\frac{104}{8}\\ =13$

The sample mean is 13.

Compute the sample variance as follows:

$s^{2}=\frac{1}{n-1}\sum (x-\bar x)^{2} \\=\frac{1}{8-1}[(12.6-13)^{2}+(12.9-13)^{2}+(13.4-13)^{2}+...+(13.1-13)^{2}] \\=\frac{1}{7}\times1.6\\=0.2286$

The sample variance is 0.2286.

Compute the sample standard deviation as follows:

$s=\sqrt{s^{2}}\\=\sqrt{0.2286}\\=0.4781$

The sample standard deviation is 0.4781.