Chapter 13, Problem 2CTC

a.

To determine

To rank: The countries based on the total number of medals won.

Answer to Problem 2CTC

The ranking of the countries is,

Country	Total medals	Rank
United States	104	1
Russia	82	2
Germany	44	3
Italy	28	4
Canada	18	5
Argentina	4	7
Norway	4	7
Switzerland	4	7

Explanation of Solution

Given info:

The data represents the gold medals, silver medals and bronze medals obtained by the random sample of 8 countries.

Calculation:

The ranking of the countries should done based on the total number of medals won.

The total number of medals is obtained as follows,

$\text{Total}\text{medals}=\text{Number}\text{of}\left(\text{Gold medals}+\text{Silver medals}+\text{Bronze medals}\right)$

The total number of medals is obtained as follows:

Country	Gold	Silver	Bronze	Total medals
Argentina	1	1	2	$1+1+2=4$
Canada	1	5	12	$1+5+12=18$
Germany	11	19	14	$11+19+14=44$
Italy	8	9	11	$8+9+11=28$
Norway	2	1	1	$2+1+1=4$
Russia	24	26	32	$24+26+32=82$
Switzerland	2	2	0	$2+2+0=4$
United States	46	29	29	$46+29+29=104$

And ranking of the countries based on the total number of medals is as follows,

Country	Total medals	Rank
United States	104	1
Russia	82	2
Germany	44	3
Italy	28	4
Canada	18	5
Argentina	4	7
Norway	4	7
Switzerland	4	7

b.

To determine

To state: The advantages and disadvantages on the ranking method done in part(a).

Explanation of Solution

Advantages:

Some of the advantage of the method used in part (a) is,

• This method of ranking is not time consuming.
• This is an easy method.

Some of the disadvantage of the method used in part (a) is,

• This method of ranking will not give the exact information.
• Sometimes this may be misleading due to highest rank with bronze method and 0 gold medals.

c.

To determine

To rank: The countries based on the Gold medals, silver medals, Bronze medals and total medals won.

Explanation of Solution

Calculation:

Ranking of the countries based on the Gold medals:

The ranking of the countries based on gold medals won is,

Country	Gold	Rank
United States	46	1
Russia	24	2
Germany	11	3
Italy	8	4
Norway	2	5.5
Switzerland	2	5.5
Argentina	1	7.5
Canada	1	7.5

Ranking of the countries based on the silver medals:

The ranking of the countries based on silver medals won is,

Country	Silver	Rank
United States	29	1
Russia	26	2
Germany	19	3
Italy	9	4
Canada	5	5
Switzerland	2	6
Argentina	1	7.5
Norway	1	7.5

Ranking of the countries based on the bronze medals:

The ranking of the countries based on bronze medals won is,

Country	Bronze	Rank
Russia	32	1
United States	29	2
Germany	14	3
Canada	12	4
Italy	11	5
Argentina	2	6
Norway	1	7
Switzerland	0	8

Ranking of the countries based on the sum of the ranks of all the 3 categories:

The sum of ranks of the countries for the 3 categories is,

Country	Sum of ranks of 3 categories
Argentina	$6+7.5+7.5=21$
Canada	$6+7.5+7.5=21$
Germany	$6+7.5+7.5=21$
Italy	$6+7.5+7.5=21$
Norway	$6+7.5+7.5=21$
Russia	$6+7.5+7.5=21$
Switzerland	$6+7.5+7.5=21$
United States	$6+7.5+7.5=21$

The ranking of the countries based on total medals won is,

Country	Sum of the ranks	Rank
Argentina	21	1
Norway	20	2
Switzerland	19.5	3
Canada	16.5	4
Italy	13	5
Germany	9	6
Russia	5	7.5
United States	5	7.5

d.

To determine

To check: Whether the ranks obtained in part (a) and part(c) are same.

Answer to Problem 2CTC

No, the ranks obtained in part (a) and part(c) are not same.

Explanation of Solution

In part (a) the ranks of the countries are,

Country	Total medals	Rank
United States	104	1
Russia	82	2
Germany	44	3
Italy	28	4
Canada	18	5
Argentina	4	7
Norway	4	7
Switzerland	4	7

The ranks of the countries obtained in part(c) are,

Country	Sum of the ranks	Rank
Argentina	21	1
Norway	20	2
Switzerland	19.5	3
Canada	16.5	4
Italy	13	5
Germany	9	6
Russia	5	7.5
United States	5	7.5

Here, both the ranks for the countries are not same.

Because the ranking done is part (a) is based on the total number of medals and ranking done in part(c) are based on the sum of the ranks of the gold, silver and bronze medals.

e.

To determine

To state: The advantages and disadvantages on the ranking method done in part(a).

Explanation of Solution

Advantages:

Some of the advantage of the method used in part (c) is,

• This method gives the accurate information than the previous method.
• This is an easy method.

Some of the disadvantage of the method used in part (a) is,

• This method of is time consuming.
• Sometimes this may be misleading due to highest rank with bronze method and 0 gold medals.

f.

To determine

To rank: The countries with the weighted ranking method.

Answer to Problem 2CTC

The weighted ranks of the countries are,

Explanation of Solution

Calculation:

The given weights are as follows,

The weight for 1 gold medal =3

The weight for 1 silver medal =2

The weight for 1 bronze medal =1

The weighted medals are obtained as follows:

Country	Weighted Gold	Weighted Silver	Weighted Bronze	Total
Argentina	3	2	2	7
Canada	3	10	12	25
Germany	33	38	14	85
Italy	24	18	11	53
Norway	6	2	1	9
Russia	72	52	32	156
Switzerland	6	4	0	10
United States	138	58	29	225

The ranking for the countries based on weights is,

Country	Total	Rank
United States	225	1
Russia	156	2
Germany	85	3
Italy	53	4
Canada	25	5
Switzerland	10	6
Norway	9	7
Argentina	7	8

g.

To determine

To check: Whether the ranks obtained using the 3 methods are same.

Answer to Problem 2CTC

No, the ranks obtained using the 3 methods are not same.

Explanation of Solution

In part(a) the ranks of the countries are,

Country	Total medals	Rank
United States	104	1
Russia	82	2
Germany	44	3
Italy	28	4
Canada	18	5
Argentina	4	7
Norway	4	7
Switzerland	4	7

The ranks of the countries obtained in part(c) are,

Country	Sum of the ranks	Rank
Argentina	21	1
Norway	20	2
Switzerland	19.5	3
Canada	16.5	4
Italy	13	5
Germany	9	6
Russia	5	7.5
United States	5	7.5

The ranks obtained in part(f) are,

Country	Total	Rank
United States	225	1
Russia	156	2
Germany	85	3
Italy	53	4
Canada	25	5
Switzerland	10	6
Norway	9	7
Argentina	7	8

Here, the ranks obtained in part (a) and part (c) is not same.

Because the ranking done is part (a) is based on the total number of medals and ranking done in part (c) are based on the sum of the ranks of the gold, silver and bronze medals.

But, the ranks obtained in part (a) and part (f) is maximum same.

Because the weights given to the 3 categories are linear, and the ranks does not change much due to linear weights.

h.

To determine

To state: The advantages and disadvantages on the ranking method done in part(f).

Explanation of Solution

Advantages:

Some of the advantage of the method used in part (c) is,

• This method gives the accurate information than the previous method.
• This is an easy method.

Some of the disadvantage of the method used in part (a) is,

• This method of is time consuming.
• Sometimes this may be misleading due to highest rank with bronze method and 0 gold medals.

i.

To determine

To test: Whether there is any significant difference between any two ranking methods using the spearman rank correlation test.

Answer to Problem 2CTC

There is sufficient evidence to infer that the ranks obtained in part(a) and part(f) differ significantly.

Explanation of Solution

Calculation:

The testing hypotheses are given below:

Null hypothesis $H_0$:

$H_0:\rho =0$

Alternative hypothesis:

$H_1:\rho \ne 0$

Critical value:

Here, the level of significance is not given.

So, the prior level of significance $\alpha =0.05$ can be used.

And the alternative hypothesis is a two sided test.

From Table L of the rank correlation coefficient in Appendix A, the critical value that corresponds to level of significance $\alpha =0.05$ and sample size $n=8$ is 0.738.

Thus, the critical value for the rank correlation coefficient is 0.738.

Test value:

Software procedure:

Step by step procedure to obtain the spearman ranks correlation coefficient using the MINITAB software:

• Choose Stat > Basis statistics > Correlation.
• In Variables, enter the columns Ranks in part(a), and Ranks in part(f).
• In method select Spearman rho.
• Select display p-value.
• Click OK.

Output using the MINITAB is given below:

From the MINITAB output, the Spearman rank correlation coefficient is 0.976.

Hence, the Spearman rank correlation coefficient is $r_s=0.976$.

Decision rule:

• If the test statistic is greater than or equal to the critical value, then reject the null hypothesis $H_0$.
• If the test statistic is less than the critical value, then fail to reject the null hypothesis $H_0$.

Conclusion:

The test value is, $r_s=976$ and the critical value is 0.738.

Here, the test statistic value is greater than the critical value.

That is, $0.976\left(=r_s\right)>0.738\left(=\text{criticalvalue}\right)$

By the rejection rule “reject the null hypotheses”.

The conclusion is that, sufficient evidence to infer that the ranks obtained in part(a) and part(f) differ significantly.