Question

1. (a) The sign test and the Wilcoxon signed-rank test can both be used to test hypotheses about the location of a population. i. When is it more appropriate to use the sign test over the Wilcoxon signed- rank test? Refer to the shape of the population distribution in your an- swer. ii. What (if any) limitations does the sign test have? iii. Let $x_1, x_2,..., x_n$ be a sample taken at random from a population. Describe the steps in computing the Wilcoxon signed-rank test statistic for these data. (15 marks) (b) Ten cows were used in a study to examine methane emissions under two dietary regimes. Five cows were randomly assigned to each regime and the response (CH$_4$ flux in grams per day) was recorded. The dataset is given in the following table. Regime 1 210 220 254 261 280 Regime 2 275 310 330 475 520 A Kolmogorov-Smirnov test was used to compare the two regimes. The test was implemented using R software, with output: Two-sample Kolmogorov-Smirnov test data: r1 and r2 D = 0.8, p-value = 0.07937 alternative hypothesis: two-sided i. State the null and alternative hypothesis. ii. Show how the test statistic was computed. iii. State the conclusion, using $\alpha$ = 0.05. (18 marks)

          1. (a) The sign test and the Wilcoxon signed-rank test can both be used to test
hypotheses about the location of a population.
i. When is it more appropriate to use the sign test over the Wilcoxon signed-
rank test? Refer to the shape of the population distribution in your an-
swer.
ii. What (if any) limitations does the sign test have?
iii. Let $x_1, x_2,..., x_n$ be a sample taken at random from a population. Describe
the steps in computing the Wilcoxon signed-rank test statistic for these
data.
(15 marks)
(b) Ten cows were used in a study to examine methane emissions under two dietary
regimes. Five cows were randomly assigned to each regime and the response
(CH$_4$ flux in grams per day) was recorded. The dataset is given in the following
table.
Regime 1 210 220 254 261 280
Regime 2 275 310 330 475 520
A Kolmogorov-Smirnov test was used to compare the two regimes. The test
was implemented using R software, with output:
Two-sample Kolmogorov-Smirnov test
data: r1 and r2
D = 0.8, p-value = 0.07937
alternative hypothesis: two-sided
i. State the null and alternative hypothesis.
ii. Show how the test statistic was computed.
iii. State the conclusion, using $\alpha$ = 0.05.
(18 marks)

1. (a) The sign test and the Wilcoxon signed-rank test can both be used to test
hypotheses about the location of a population.
i. When is it more appropriate to use the sign test over the Wilcoxon signed-
rank test? Refer to the shape of the population distribution in your an-
swer.
ii. What (if any) limitations does the sign test have?
iii. Let x1, x2,..., xn be a sample taken at random from a population. Describe
the steps in computing the Wilcoxon signed-rank test statistic for these
data.
(15 marks)
(b) Ten cows were used in a study to examine methane emissions under two dietary
regimes. Five cows were randomly assigned to each regime and the response
(CH4 flux in grams per day) was recorded. The dataset is given in the following
table.
Regime 1 210 220 254 261 280
Regime 2 275 310 330 475 520
A Kolmogorov-Smirnov test was used to compare the two regimes. The test
was implemented using R software, with output:
Two-sample Kolmogorov-Smirnov test
data: r1 and r2
D = 0.8, p-value = 0.07937
alternative hypothesis: two-sided
i. State the null and alternative hypothesis.
ii. Show how the test statistic was computed.
iii. State the conclusion, using α = 0.05.
(18 marks)

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