Test of Two Variances · Exercise 24

King Manuel I Komnenos ruled the Byzantine Empire from Constantinople (Istanbul) during the years A.D. 1145–1170. The empire was very powerful during his reign but declined significantly afterward. Coins minted during his era were found in Cyprus, an island in the eastern Mediterranean Sea. Nine coins were from his first coinage, seven from the second, four from the third, and seven from the fourth. These spanned most of his reign. We have data on the silver content of the coins:

$$\begin{array}{cccc}\text{First Coinage}& \text{Second Coinage}& \text{Third Coinage}& \text{Fourth Coinage}\\ \text{5.9}& \text{6.9}& \text{4.9}& \text{5.3}\\ \text{6.8}& \text{9.0}& \text{5.5}& \text{5.6}\\ \text{6.4}& \text{6.6}& \text{4.6}& \text{5.5}\\ \text{7.0}& \text{8.1}& \text{4.5}& \text{5.1}\\ \text{6.6}& \text{9.3}& & \text{6.2}\\ \text{7.7}& \text{9.2}& & \text{5.8}\\ \text{7.2}& \text{8.6}& & \text{5.8}\\ \text{6.9}& & & \\ \text{6.2}& & & \end{array}$$
Did the silver content of the coins change over the course of Manuel’s reign?
Here are the means and variances of each coinage. The data are unbalanced.

$$\begin{array}{ccccc}& \text{First}& \text{Second}& \text{Third}& \text{Fourth}\\ \text{Mean}& \text{6.7444}& \text{8.2429}& \text{4.875}& \text{5.6143}\\ \text{Variance}& \text{0.2953}& \text{1.2095}& \text{0.2025}& \text{0.1314}\end{array}$$

Its source prints no answer for this problem, so there's nothing to check against.

OpenStax, Statistics · §13.4 Test of Two Variances · Exercise 24