Test of Two Variances · Exercise 24

King Manuel I, Komnenus ruled the Byzantine Empire from Constantinople (Istanbul) during the years 1145 to 1180 A.D. The empire was very powerful during his reign, but declined significantly afterwards. 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 a 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, Introductory Statistics 2e · §13.4 Test of Two Variances · Exercise 24