Question

9. An archaeologist discovers only seven fossil skeletons from a previously unknown species of miniature horse. Reconstruction of the skeletons of these seven miniature horses show the shoulder height in centimeters to be 45.3 47.1 44.2 46.8 46.5 45.5 47.6 Find a 99% confidence interval for u, the mean shoulder height of the entire population of such horses.

Answer 1

From the given sample data : n=7 , $\sum X=323$ , $\sum X^2=14912.64$

The sample mean is ,

$\bar{X}=\frac{\sum X}{n} =\frac{323}{7}=46.143$

The sample standard deviation is ,

$s=\sqrt{\frac{\sum X^2-\frac{(\sum X)^2}{n}}{n-1}}=\sqrt{\frac{14912.64 -\frac{323^2}{7}}{7-1}}=1.19$

The critical value is ,

$t_{n-1,\alpha/2}=t_{7-1,0.01/2}=3.707$ ; From t-table

Therefore , the 99% confidence interval for $\mu$ is ,

$\bar{X}\pm t_{n-1,\alpha/2}\frac{s}{\sqrt{n}}$

$46.143\pm 3.707*\frac{1.19}{\sqrt{7}}$

$46.143\pm 1.6673$

$44.4757<\mu<47.8103$