2. Comparing two unrelated samples

ONE

The data in Table 4.8 were obtained from a reading-level test for 1st grade children. Compare the performance gains of the two different methods for teaching reading.

Table 4.8
Method Gain score Method Gain score
One on one 16 Small group 11
One on one 13 Small group 2
One on one 16 Small group 10
One on one 16 Small group 4
One on one 13 Small group 9
One on one 9 Small group 8
One on one 12 Small group 5
One on one 12 Small group 6
One on one 20 Small group 4
One on one 17 Small group 16

Use two-tailed Mann-Whitney U and Kolmogorov-Smirnov two-sample tests to determine which method was better for teaching reading. Set \(\alpha = 0.05\). Report your findings

\[{H}_{0}:\text{There is no difference in reading gain scores between one-on-one and small group instruction}\\ \\ {H}_{1}:\text{There is a difference in reading gain scores between the two methods}\]

## Two-tailed Mann-Whitney U test
## ---------------------------------------
## p-value: 0.0021
## Decision: Reject the null hypothesis
## 
## Kolmogorov-Smirnov two-sample test
## ------------------------------------
## p-value: 0.0021
## Decision: Reject the null hypothesis

Two-tailed Mann-Whitney U and Kolmogorov-Smirnov two-sample tests were conducted to compare reading gain scores between one-on-one and small group instruction among \({n}_{1}={n}_{2}=10\) first-grade children, with \(\alpha = 0.05\). Median gain scores were \(14.5\text{ (one-on-one) and }7.0\text{ (small group). }\) Both tests were significant — the MWU test \((W = 91, p = 0.0021)\) and the KS test \((D = 0.8, p = 0.0021)\). We reject the null hypothesis under both tests. There is strong evidence that one-on-one instruction produces higher reading gain scores than small group instruction.

TWO

Table 4.10
Method 1 Method 2
53 91
41 18
17 14
45 21
44 23
12 99
49 16
50 10

Table 4.10 shows assessment scores of two different classes who are being taught computer skills using two different methods

Use two-tailed Mann-Whitney U and Kolmogorov-Smirnov two-sample tests to determine which method was better for teaching compute skills. Set \(\alpha = 0.05\). Report your findings

\[{H}_{0}:\text{There is no difference in computer skills assessment scoresbetween method 1 and method 2}\\ \\ {H}_{1}:\text{There is a difference in assessment scores between the two methods}\]

## Two-tailed Mann-Whitney U test
## ---------------------------------------
## p-value: 0.4418
## Decision: Fail to reject the null hypothesis
## 
## Kolmogorov-Smirnov two-sample test
## ------------------------------------
## p-value: 0.2827
## Decision: Fail to reject the null hypothesis

Two-tailed Mann-Whitney U and Kolmogorov-Smirnov tests were applied to compare computer skills assessment scores between two teaching methods \(({n}_{1}={n}_{2}=8)\), with \(\alpha = 0.05\) . Median scores were \(44.5\text{ (method 1) and }19.5\text{ (method 2) }\). Despite the descriptive difference, neither test was significant — MWU \((W = 40, p = 0.4418)\) and KS \((D = 0.5, p = 0.2827)\). Both tests fail to reject \({H}_{0}\). The high variability within each group, particularly the extreme scores of 91 and 99 in method 2, inflates within-group spread and reduces power. The data does not provide sufficient evidence to conclude that one teaching method is superior.

THREE

Two methods were used to provide instruction in Science for 7th Grade. Method 1 included a laboratory each week and method 2 has only classroom work with lecture and worksheets. Table 4.12 shows end-of-term test performance for the two methods. Construct a 95% median confidence interval based on the difference between two independent samples to compare the two methods.

Table 4.12
Method 1 Method 2
15 8
23 15
9 10
12 13
18 17
22 5
17 18
20 7

\[{H}_{0}:\text{There is no difference in end-of-term test performance between laboratory and classroom instruction}\\ \\ {H}_{1}:\text{There is a difference in end-of-term test performance between the two methods}\]

## 95% confidence interval between the scores of the two methods:
## [0, 12]

A two-tailed Mann-Whitney U test with a 95% confidence interval was conducted to compare end-of-term science scores between laboratory-based (method 1) and classroom-based (method 2) instruction, with \({n}_{1}={n}_{2}=8\) and \(\alpha = 0.05\). Median scores were \(17.5\text{ (method 1) and }11.5\text{ (method 2) }\). The test was marginally non-significant \((W = 50.5, p = 0.0582)\). We fail to reject \({H}_{0}\) at the \(5\%\) level. The Hodges-Lehmann estimate of the median difference was \(5.0\) marks in favour of method 1, with a \(95\%\) CI of \([0, 12]\). The lower bound of zero indicates that the interval just touches the null value, consistent with the borderline p-value. While the evidence leans toward a method 1 advantage, the small sample size limits power and no definitive conclusion can be drawn.

FOUR

An alloy is composed of zinc, copper and tin. It may be made at one of two temperatures H (higher) or L (lower). We wish to know if one temperature produces a harder alloy. A sample is taken from each of 9 batches at L and 7 at H. To arrange them in ascending order of hardness, all specimens are scraped against one another to see which makes a deeper scratch (a deeper scratch indicates a softer specimen). On this basis the specimens are ranked 1 (softest) to 16 (hardest) with the results given below. Should we reject the hypothesis that hardness is unaffected by temperature?

rank temperature
1 H
2 L
3 H
4 H
5 H
6 L
7 H
8 L
9 L
10 H
11 H
12 L
13 L
14 L
15 L
16 L

\[{H}_{0}:\text{Alloy hardness is unaffected by production temperature}\\ \\ {H}_{1}:\text{Alloy hardness differs between the production temperature}\]

## Mann-Whitney U test
## ---------------------
## p-value: 0.0549
## Decision: Fail to reject the null hypothesis

A two-tailed Mann-Whitney U test with exact p-value computation was used to determine whether production temperature affected alloy hardness, with \({n}_{H}=7\) batches at higher temperature and \({n}_{L}=9\) at lower temperature, and \(\alpha = 0.05\). Specimens were ranked 1 (softest) to 16 (hardest). Rank sums were \(41\text{ (H) and }95\text{ (L), with median ranks of }5\text{ (H) and }12\text{ (L), }\)suggesting that lower temperature batches tended to produce harder alloys. However, the test was marginally non-significant \((W = 13, p = 0.0549)\). We fail to reject \({H}_{0}\) — there is insufficient evidence at the \(5\%\) level to conclude that temperature affects hardness, though the result is borderline and warrants caution. A larger sample would help clarify whether the observed ranking pattern reflects a true population difference.

FIVE

A psychologist notes total time (in seconds) needed to perform a series of simple manual tasks for each of eight children with learning difficulties and seven children without learning difficulties. The times are:

without difficulties with difficulties
204 243
218 228
197 261
183 202
227 343
233 242
191 220
239

Use a Smirnov test to find whether the psychologist is justified in asserting these samples are likely to be from different populations.

\[{H}_{0}:\text{Children with and without learning difficulties are drawn from the same population}\\ \\ {H}_{1}:\text{The two groups are drawn from the same population}\]

## Mann-Whitney U test
## ---------------------
## p-value: 0.0559
## Decision: Fail to reject the null hypothesis

A two-sample Kolmogorov-Smirnov (Smirnov) test with exact p-value computation was conducted to determine whether task completion times differed between \({n}_{1}=8\) children with learning difficulties and \({n}_{2}=7\) children without, with \(\alpha = 0.05\). Median times were \(240.5\) seconds (with difficulties) and \(204.0\) seconds (without difficulties), suggesting the group with learning difficulties tended to take longer. The test was marginally non-significant \((D = 0.625, p = 0.0559)\). We fail to reject \({H}_{0}\) at the \(5\%\) level — the data does not provide sufficient evidence to justify the psychologist’s assertion that the two samples are from different populations, though the result is borderline and the descriptive pattern is consistent with the hypothesis

SIX

The following data are DMF scores for 34 male and 54 female first-year dental students. The DMF score is the total of the numbers of decayed + missing + filled teeth.

gender DMF
male 8
male 6
male 4
male 2
male 10
male 5
male 6
male 6
male 19
male 4
male 10
male 4
male 10
male 12
male 7
male 2
male 5
male 1
male 8
male 2
male 0
male 7
male 6
male 4
male 4
male 11
male 2
male 16
male 8
male 7
male 8
male 4
male 0
male 2
female 4
female 7
female 13
female 8
female 8
female 4
female 14
female 5
female 6
female 4
female 12
female 9
female 9
female 9
female 8
female 12
female 4
female 8
female 8
female 4
female 11
female 6
female 15
female 9
female 8
female 14
female 9
female 8
female 9
female 7
female 12
female 11
female 7
female 4
female 10
female 7
female 8
female 8
female 7
female 9
female 10
female 16
female 14
female 15
female 10
female 4
female 6
female 3
female 9
female 3
female 10
female 3
female 8

Use an asymptotic WMW test to determine whether the DMF score differs significantly between males and females.

\[{H}_{0}:\text{DMF scores are identically distributed between male and female first year dental students}\\ \\ {H}_{1}:\text{DMF scores differ between male and female first-year students}\]

## Mann-Whitney U test
## ---------------------
## p-value: 0.0045
## Decision: Reject the null hypothesis

An asymptotic Wilcoxon-Mann-Whitney test (normal approximation) was applied to compare DMF scores between \({n}_{male}=34 \text{ and } {n}_{f}=54\) female first-year dental students, \(\alpha = 0.05\). The asymptotic form was appropriate here given the large combined sample size \((N=88)\) and the presence of ties, which preclude exact computation. Median DMF scores were \(6\) (male) and \(8\) (female), with means of \(6.18\) and \(8.33\) respectively, suggesting females tended to have higher DMF scores.The test was significant $(W = 587.5, p = 0.0045). We reject the null hypothesis. DMF scores differ significantly between male and female dental students, with females exhibiting a higher burden of decayed, missing and filled teeth

4. Comparing more than two unrelated samples

ONE

A researcher conducted a study with n = 15 participants to investigate strength gains from exercise. The participants were divided into three groups and given one of three treatments. Participants’ strength gains were measured and ranked. The rankings are presented in Table 6.8 below

Table 6.8
I II III
7 13 12
2 1 5
4 7 16
11 8 9
15 3 14

Use a Kruskal-Wallis H test with \(\alpha = 0.05\) to determine if one or more of the groups were significantly different. If a significant difference exists, use a two-tailed Mann-Whitney U test or a two-sample Kolmogorov-Smirnov test to identify which groups are significantly different. Use the Bonferroni procedure to limit the type I error rate. Report your findings

\[ {H}_{0}: \text{The perceived effectiveness rankings are identical across all three attractiveness groups}\\ \\ {H}_{1}:\text{At least one attractiveness group differs in perceived effectiveness rankings} \]

## Kruskal-Wallis Test
## ------------------------------------------------
## p-value: 0.2567
## Decision: Fail to reject the null hypothesis

A Kruskal-Wallis H test was conducted to compare strength gains across three treatment groups (I, II, and III) among \(\mathit{n=15}\) participants (5 per group), with \(\mathit{\alpha = 0.05}\). Strength gains were ranked globally, yielding rank sums of \({R}_{I}=36.5, {R}_{II}=30.5 \text{ and } {R}_{III}=53\). The test was \(\mathit{non-significant } ({\chi^2}_{(2)}=2.72, p=0.257)\). We therefore fail to reject \({H}_{0}\) — there is insufficient evidence to conclude that any treatment group produced significantly different strength gains. Since the omnibus test was non-significant, post-hoc pairwise comparisons were not warranted; no further inference is made about individual group differences.

TWO

A researcher investigated how physical attraction influences the perception among others of a person’s effectiveness with difficult tasks. The photographs of 24 people were shown to a focus group. The group was asked to classify the photos into three groups: very attractive, average and very unattractive. Then, the group ranked the photographs according to their impressions of how capable they were of solving difficult problems. Table 6.9 shows the classification and rankings of the people in the photos(1 = most effective, 24 = least effective)

Table 6.9
very attractive average very unattractive
1 3 11
2 4 15
5 8 16
6 9 18
7 13 20
10 14 21
12 19 23
17 22 24

Use a Kruskal-Wallis H test with \(\alpha = 0.05\) to determine if one or more of the groups are significantly different. If a significant difference exists, use two-tailed Mann-Whitney U tests to identify which groups are significantly different. Use the Bonferroni procedure to limit the type I error rate. Report you findings.

\[{H}_{0}: \text{The perceived effectiveness rankings are identical across all three attractiveness groups} \\ \\ {H}_{1}: \text{At least one attractiveness group differs in perceived effectiveness rankings}\]

## Kruskal-Wallis Test
## ------------------------------------------------
## p-value: 0.007
## Decision: Reject the null hypothesis

A Kruskal-Wallis H test was performed to compare perceived effectiveness rankings (1=most effective, 24=least effective) A Kruskal-Wallis H test was performed to compare perceived effectiveness rankings across three attractiveness classifications — very attractive, average, and very unattractive - among \(n=24\) individuals, with \(\alpha=0.05.\) Mean rankings were \(7.5, 11.5 \text{ and }18.5\) respectively, suggesting a systematic gradient. The test was significant \(\mathit{({\chi^2}_{(2)}=9.92, p=0.007) }\) indicating that at least one group differed. We reject the null hypothesis.

Now that we have established the existence of a significance in perceived ranking amongst the three attractiveness groups, we conduct the Mann-Whitney tests to determine its origin. The Bonferroni procedure adjusts our critical p-value to \(\mathit{0.05/3=0.0167}\). Below are the test results;

## Bonferroni-Mann-Whitney U test (very attractive vs average)
## ------------------------------------------------------------
## p-value: 0.2271
## Decision: Fail to reject the null hypothesis
## 
## Bonferroni-Mann-Whitney U test (very attractive vs very unattractive)
## ---------------------------------------------------------------------
## p-value: 0.0039
## Decision: Reject the null hypothesis
## 
## Bonferroni-Mann-Whitney U test (average vs very unattractive)
## -------------------------------------------------------------
## p-value: 0.0406
## Decision: Fail to reject the null hypothesis

The comparison between very attractive and average groups was \(\mathit{non-significant } \mathit{(W = 20, p = 0.2271)}\). The comparison between very attractive and very unattractive groups was \(\mathit{significant } \mathit{(W = 4, p = 0.0039)}\). The comparison between average and very unattractive groups was \(\mathit{non-significant } \mathit{(W = 12, p = 0.0406)}\). The data therefore provides sufficient evidence that individuals perceived as very attractive were ranked significantly more effective at solving difficult problems than those perceived as very unattractive

THREE

Lubischew (1962) gives measurements of maximum head width in units of 0.01 mm for three species of Chaetocnema. Part of his data is given below. Use a Kruskal–Wallis test to see if there is a species difference in head widths

species 1 species 2 species 3
53 49 58
50 49 51
52 47 45
50 54 53
49 43 49
47 51 51
54 49 50
51 51 51
52 50
57 46
49

\[{H}_{0}:\text{The distribution of maximum head widths is identical across all three \textit{Chaetocnema} species} \\ \\ {H}_{1}:\text{At least one species differs in head width distribution}\]

## Kruskal-Wallis Test
## ------------------------------------------------
## p-value: 0.1088
## Decision: Fail to reject the null hypothesis

A Kruskal-Wallis H test was used to assess species differences in maximum head width (units:0.01mm) across three species of Chaetocnema \({n}_{1}=10,{n}_{2}=11 \text{ and }{n}{3}=8\), with \(\alpha =0.05.\) Median head widths were \(51.5, 49.0, \text{ and } 51.0\) for species 1, 2, and 3 respectively.The test was \(\mathit{non-significant } ({\chi^2}_{2}=4.44 p = 0.1088).\) We fail to reject the null hypothesis since the data does not provide sufficient evidence of a species difference in head widths at the 5% significance level.

5. Tests for Nominal scale data

ONE

A researcher wishes to determine if there is an association between the level of a teacher’s education and his/her job satisfaction. He surveyed 158 teachers. The frequencies of the corresponding results are displayed in Table 8.19

Table 8.19
Bachelor’s degree Master’s degree Post-Master’s degree
satisfied 60 41 19
unsatisfied 10 13 15

First, use a \({\chi}^{2}\)-test for independence with \(\alpha = 0.05\) to determine if there is an association between level of education and job satisfaction. Then, determine the effect size for the association. Report your findings

\[{H}_{0}:\text{Level of education and job satisfaction are independent} \] \[{H}_{1}:\text{Level of education and job satisfaction are associated}\]

## Test of independence (Chi-Square approximation)
## ------------------------------------------------
## p-value: 0.0038
## Decision: Reject the null hypothesis

In conclusion, education level and job satisfaction are statistically associative at the 5% significance level.

TWO

A professor gave her class a 10-item survey to determine the students’ satisfaction with the course. Survey question responses were measured using a five-point Likert scale. The survey had a score range from +20 to −20. Table 8.20 shows the scores of the students in a class of 13 students who rated the professor

gender score satisfaction
male +12 +
male +6 +
male -5 -
male -10 -
male +17 +
male +4 +
female -2 -
female -13 -
female +10 +
female -8 -
female -11 -
female -4 -
female -14 -

Use a Fisher exact test with \(\alpha = 0.05\) to determine if there is an association between gender and course satisfaction of the professor’s class. Then, determine the effect size for the association. Report your findings.

\[{H}_{0}: \text{Gender and course satisfaction are independent}\] \[{H}_{0}: \text{Gender and course satisfaction are associated}\]

## Test of independence (Fisher's Exact Test)
## ------------------------------------------------
## p-value: 0.1026
## Decision: Fail to reject the null hypothesis

There exists insufficient evidence against the null hypothesis of independence between gender and course satisfaction at 5% significance level

THREE

In an English parliamentary electoral constituency a random sample of 400 voters are classified by age and political affiliation as follows

30 or under 31-40 41-55 56 or over
Conservative 31 32 39 34
Liberal Democrat 16 19 25 31
Labour 36 27 58 52

Is there evidence of an association between political affiliation and age? It is generally (though not universally) accepted that the Conservative, Liberal Democrat and Labour parties represent an ordering of right, middle and left in the political spectrum

## Test of independence (Chi-Square approximation)
## ------------------------------------------------
## p-value: 0.4435
## Decision: Fail to reject the null hypothesis

Political affiliation and age are independent of each other at the 5% level

FOUR

Agresti (1984) quotes the following data on cross-classification of attitudes towards abortion and amounts of schooling based on the US General Social Survey, 1972. Test these data for evidence of association between attitudes and educational background

Disapprove Neutral Approve
Less than high school 209 101 237
High school 151 126 426
More than high school 16 21 138 
## Test of independence (Chi-Square approximation)
## ------------------------------------------------
## p-value: 0
## Decision: Reject the null hypothesis

At the 5% significance level, there exists sufficient evidence in favor of the claim of present association between subjects’ attitudes towards abortion and educational background

6. Variable comparison

ONE

a

A china manufacturer is investigating market response to seven designs of dinner set. The main markets are the British and American. To get some idea of preferences in the two markets a survey of 100 British and 100 American women is carried out and each woman is asked to rank the designs in order of preference from 1 for favorite to 7 for least acceptable. For each country the 100 rank scores for each design is totalled. The design with the lowest total is assigned rank 1, that with the next lowest total rank 2, and so on. Overall rankings for each country are

Design British rank American rank
A 1 3
B 2 4
C 3 1
D 4 5
E 5 2
F 6 7
G 7 6

Calculate the Spearman and Kendall correlation coefficients. Is there evidence of a positive association between orders of preference

\[\text{Spearman's rank correlation coefficeint}\\ {H}_{0}:{\rho}_{s}=0\\ {H}_{1}:{\rho}_{s}\neq0\\ \\ \text{Kendall's rank correlation coefficeint}\\ {H}_{0}:{\tau}_{s}=0\\ {H}_{1}:{\tau}_{s}\neq0\]

## Spearman's correlation
## --------------------------
## value: 0.5714
## p-value: 0.1
## Decision: Fail to reject the null hypothesis
## 
## ====================================================
## 
## Kendall's correlation
## --------------------------
## value: 0.4286
## p-value: 0.1194
## Decision: Fail to reject the null hypothesis

We fail to reject the null hypotheses for both rank coefficients. The data provides sufficient evidence in favor of the claim that the 7 dinner set designs have an insignificant difference in the order of preference from the British and American markets.

b

The manufacturer above later decides to assess preferences in the Canadian and Australian markets by a similar method. Calculate the Spearman and Kendall correlation and coefficients. Is there evidence of a positive association between orders of preference.

## Spearman's correlation
## --------------------------
## value: 0
## p-value: 0.5183
## Decision: Fail to reject the null hypothesis
## 
## ====================================================
## 
## Kendall's correlation
## --------------------------
## value: 0.0476
## p-value: 0.5
## Decision: Fail to reject the null hypothesis

We fail to reject the null hypotheses for both rank coefficients. The data provides sufficient evidence in favor of the claim that the 7 dinner set designs have an insignificant difference in the order of preference from the Canadian and Australian markets.

c

Perform an appropriate analysis of the ranked data for all four countries in Exercises 7.6 and 7.7 to assess the evidence for any overall concordance

UK-CAN

## Spearman's correlation
## --------------------------
## value: 0.3929
## p-value: 0.1978
## Decision: Fail to reject the null hypothesis
## 
## ====================================================
## 
## Kendall's correlation
## --------------------------
## value: 0.2381
## p-value: 0.281
## Decision: Fail to reject the null hypothesis

We fail to reject the null hypotheses for both rank coefficients. The data provides sufficient evidence in favor of the claim that the 7 dinner set designs have an insignificant difference in the order of preference from the British and Canadian markets.

UK-AUS

## Spearman's correlation
## --------------------------
## value: 0.6786
## p-value: 0.0548
## Decision: Fail to reject the null hypothesis
## 
## ====================================================
## 
## Kendall's correlation
## --------------------------
## value: 0.4286
## p-value: 0.1194
## Decision: Fail to reject the null hypothesis

We fail to reject the null hypotheses for both rank coefficients. The data provides sufficient evidence in favor of the claim that the 7 dinner set designs have an insignificant difference in the order of preference from the British and Australian markets.

USA-CAN

## Spearman's correlation
## --------------------------
## value: 0.8214
## p-value: 0.0171
## Decision: Reject the null hypothesis
## 
## ====================================================
## 
## Kendall's correlation
## --------------------------
## value: 0.619
## p-value: 0.0345
## Decision: Reject the null hypothesis

We reject the null hypotheses for both rank coefficients. The data provides sufficient evidence against the initial claim that the 7 dinner set designs have an insignificant difference in the order of preference from the American and Canadian markets.

USA-AUS

## Spearman's correlation
## --------------------------
## value: 0.0357
## p-value: 0.4817
## Decision: Fail to reject the null hypothesis
## 
## ====================================================
## 
## Kendall's correlation
## --------------------------
## value: 0.0476
## p-value: 0.5
## Decision: Fail to reject the null hypothesis

We fail to reject the null hypotheses for both rank coefficients. The data provides sufficient evidence in favor of the claim that the 7 dinner set designs have an insignificant difference in the order of preference from the British and Australian market.

TWO

In a pharmacological experiment involving \(\beta\) -blocking agents, Sweeting (1982) recorded for a control group of dogs, cardiac oxygen consumption (MVO) and left ventricular pressure (LVP). Calculate the Kendall and Spearman correlation coefficients. Is there evidence of correlation

Dog MVO LVP
A 78 32
B 92 33
C 116 45
D 90 30
E 106 38
F 78 24
G 99 44

\(\text{We test the relevant hypotheses:}\)

\[\text{Spearman's rank correlation coefficeint}\\ {H}_{0}:{\rho}_{s}=0\\ {H}_{1}:{\rho}_{s}\neq0\\ \\ \text{Kendall's rank correlation coefficeint}\\ {H}_{0}:{\tau}_{s}=0\\ {H}_{1}:{\tau}_{s}\neq0\]

## Spearman's correlation
## --------------------------
## value: 0.9009
## p-value: 0.0056
## Decision: Reject the null hypothesis
## 
## ====================================================
## 
## Kendall's correlation
## --------------------------
## value: 0.7807
## p-value: 0.0151
## Decision: Reject the null hypothesis

For this pharmacological experiment involving 7 dogs (n=7) selected from the control group, we collect their data on cardiac oxygen consumption and left ventricular pressure. The paired observations from each of the 7 subjects is ranked for the purpose of the analysis. The Spearman rank correlation coeficient was significant \(\mathit{({r}_{s} = 0.9009, p < 0.05)}.\) Kendall’s coeficient was also significant \(\mathit{({\tau} = 0.7807, p < 0.05)}\).This data shows that there exists a significant monotonic association between cardiac oxygen consumption and left ventricular pressure.

THREE

Bardsley and Chambers (1984) gave numbers of beef cattle and sheep on 19 large farms in a region. Is there evidence of correlation

cattle sheep
41 4716
0 4605
42 4951
15 2745
47 6592
0 8934
0 9165
0 5917
56 2618
67 1105
707 150
368 2005
231 3222
104 7150
132 8658
200 6304
172 1800
146 5270
0 1537

\(\text{We test the relevant hypotheses:}\)

\[\text{Spearman's rank correlation coefficeint}\\ {H}_{0}:{\rho}_{s}=0\\ {H}_{1}:{\rho}_{s}\neq0\\ \\ \text{Kendall's rank correlation coefficeint}\\ {H}_{0}:{\tau}_{s}=0\\ {H}_{1}:{\tau}_{s}\neq0\]

## Spearman's correlation
## --------------------------
## value: -0.331
## p-value: 0.1663
## Decision: Fail to reject the null hypothesis
## 
## ====================================================
## 
## Kendall's correlation
## --------------------------
## value: -0.235
## p-value: 0.168
## Decision: Fail to reject the null hypothesis

The number of beef cattle and sheep was collected from 19 randomly selected farms. These observations are ranked for subsequent analysis. The Spearman rank correlation coefficeint was insignificant \(\mathit{({r}_{s} = -0.331, p > 0.05)}\) .The same applies for the Kendall’s coefficient \(\mathit{({\tau} = -0.235, p > 0.05)}\). There’s an insignificant negative monotonic association between the number of cattle owned and sheep owned by the farms in that region.

7. References

\(\text{- Conover, W. J. (1999). }\textit{Practical Nonparametric Statistics}\text{ (3rd ed.). John Wiley & Sons}\) \(\text{- Corder, G. W., & Foreman, D. I. (2014). }\textit{Nonparametric Statistics: A Step-by-Step Approach}\text{ (2nd ed.). John Wiley & Sons.}\)