Question

In various places in this module, data on the silver content of coins minted in the reign of the twelfth-century Byzantine king Manuel I Comnenus have been considered. The full dataset is in the Minitab file coins.mwx. The dataset includes, among others, the values of the silver content of nine coins from the first coinage (variable Coin1) and seven from the fourth coinage (variable Coin4) which was produced a number of years later. (For the purposes of this question, you can ignore the variables Coin2 and Coin3.) In particular, in Activity 8 and Exercise 2 of Computer Book B, it was argued that the silver contents in both the first and the fourth coinages can be assumed to be normally distributed. The question of interest is whether there were differences in the silver content of coins minted early and late in Manuel's reign. You are about to investigate this question using a two-sample t-interval. (i) Using Minitab, find either the sample standard deviations of the two variables Coin1 and Coin4, or their sample variances. Hence check for equality of variances using the rule of thumb given in Subsection 4.4 of Unit 8. (ii) Whatever the outcome of part (i), use Minitab to obtain a 90% two-sample t-interval for the difference E(X1) - E(X4), where X1 denotes the silver content in coins of the first coinage, and X4 denotes the silver content in coins of the fourth coinage. State that interval and comment briefly on what it tells us about the silver content of coins in the earlier and later coinages. (iii) Name the distribution used in constructing the confidence interval in part (ii), state the value of its parameter, and show why the parameter takes the value that it does. (iv) What would have been the outcome if you had obtained a 90% two-sample t-interval for E(X4) - E(X1) instead of for E(X1) - E(X4)? Justify your conclusion in terms of the derivative of the parameter transformation involved. coins.mwx Coin1 Coin2 Coin3 Coin4 5.9 6.9 4.9 5.3 6.8 9.0 5.5 5.6 6.4 6.6 4.6 5.5 7.0 8.1 4.5 5.1 6.6 9.3 6.2 7.7 9.2 5.8 7.2 8.6 5.8 6.9 6.2

In various places in this module, data on the silver content of coins minted in the reign of the twelfth-century Byzantine king Manuel I Comnenus have been considered. The full dataset is in the Minitab file coins.mwx. The dataset includes, among others, the values of the silver content of nine coins from the first coinage (variable Coin1) and seven from the fourth coinage (variable Coin4) which was produced a number of years later. (For the purposes of this question, you can ignore the variables Coin2 and Coin3.) In particular, in Activity 8 and Exercise 2 of Computer Book B, it was argued that the silver contents in both the first and the fourth coinages can be assumed to be normally distributed. The question of interest is whether there were differences in the silver content of coins minted early and late in Manuel's reign. You are about to investigate this question using a two-sample t-interval.

(i) Using Minitab, find either the sample standard deviations of the two variables Coin1 and Coin4, or their sample variances. Hence check for equality of variances using the rule of thumb given in Subsection 4.4 of Unit 8.

(ii) Whatever the outcome of part (i), use Minitab to obtain a 90% two-sample t-interval for the difference E(X1) - E(X4), where X1 denotes the silver content in coins of the first coinage, and X4 denotes the silver content in coins of the fourth coinage. State that interval and comment briefly on what it tells us about the silver content of coins in the earlier and later coinages.

(iii) Name the distribution used in constructing the confidence interval in part (ii), state the value of its parameter, and show why the parameter takes the value that it does.

(iv) What would have been the outcome if you had obtained a 90% two-sample t-interval for E(X4) - E(X1) instead of for E(X1) - E(X4)? Justify your conclusion in terms of the derivative of the parameter transformation involved.

coins.mwx
Coin1
Coin2
Coin3
Coin4
5.9
6.9
4.9
5.3
6.8
9.0
5.5
5.6
6.4
6.6
4.6
5.5
7.0
8.1
4.5
5.1
6.6
9.3
6.2
7.7
9.2
5.8
7.2
8.6
5.8
6.9
6.2

Added by Gary Q.