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Deal with one and two (normal) samples

one_two_sample.Rd

Deal with one and two (normal) samples. For one normal sample x, the function reports descriptive statistics, plot, interval estimation and test of hypothesis of x. For two normal samples x and y, the function reports descriptive statistics, plot, interval estimation and test of hypothesis of x and y, respectively. It also reports interval estimation and test of hypothesis of mu1-mu2 (the difference of the means of x and y) and sigma1^2/sigma2^2 (the ratio of the variances of x and y), tests whether x and y are from the same population, finds the correlation coefficient of x and y if x and y have the same length.

Usage

one_two_sample(x, y = NULL, mu = c(Inf, Inf), sigma = c(-1, -1), 
               var.equal = FALSE, ratio = 1, side = 0, alpha = 0.05)

Arguments

x

A numeric vector.

y

A numeric vector.

mu

If y = NULL, i.e., there is only one sample. See the argument mu in one_sample. For two normal samples x and y, mu plays one role: the population means. However, mu is used in two places: one is the two sided or one sided interval estimation of sigma1^2 / sigma2^2 of two normal samples, another is the two sided or one sided test of hypothesis of sigma1^2 and sigma2^2 of two normal samples. When mu is known, input it, and the function computes the interval endpoints (or the F value) using an F distribution with degree of freedom (n1, n2). When it is unknown, ignore it, and the function computes the interval endpoints (or the F value) using an F distribution with degree of freedom (n1-1, n2-1).

sigma

If y = NULL, i.e., there is only one sample. See the argument sigma in one_sample. For two normal samples x and y, sigma plays one role: the population standard deviations. However, sigma is used in two places: one is the two sided or one sided interval estimation of mu1-mu2 of two normal samples, another is the two sided or one sided test of hypothesis of mu1 and mu2 of two normal samples. When the standard deviations are known, input it, then the function computes the interval endpoints using normal population; when the standard deviations are unknown, ignore it, now we need to consider whether the two populations have equal variances. See var.equal below.

var.equal

A logical variable indicating whether to treat the two variances as being equal. If TRUE then the pooled variance is used to estimate the variance otherwise the Welch (or Satterthwaite) approximation to the degrees of freedom is used.

ratio

The hypothesized ratio of the population variances of x and y. It is used in var.test(x, y, ratio = ratio, ...), i.e., when computing the interval estimation and test of hypothesis of sigma1^2 / sigma2^2 when mu1 or mu2 is unknown.

side

If y = NULL, i.e., there is only one sample. See the argument side in one_sample. For two normal samples x and y, sigma is used in four places: interval estimation of mu1-mu2, test of hypothesis of mu1 and mu2, interval estimation of sigma1^2 / sigma2^2, test of hypothesis of sigma1^2 and sigma2^2. In interval estimation of mu1-mu2 or sigma1^2 / sigma2^2, side is a parameter used to control whether to compute two sided or one sided interval estimation. When computing the one sided upper limit, input side = -1 (or a number < 0); when computing the one sided lower limit, input side = 1 (or a number > 0); when computing the two sided limits, input side = 0 (default). In test of hypothesis of mu1 and mu2 or sigma1^2 and sigma2^2, side is a parameter used to control two sided or one sided test of hypothesis. When inputting side = 0 (default), the function computes two sided test of hypothesis, and H1: mu1 != mu2 or H1: sigma1^2 != sigma2^2; when inputting side = -1 (or a number < 0), the function computes one sided test of hypothesis, and H1: mu1 < mu2 or H1: sigma1^2 < sigma2^2; when inputting side = 1 (or a number > 0), the function computes one sided test of hypothesis, and H1: mu1 > mu2 or H1: sigma1^2 > sigma2^2.

alpha

The significance level, a real number in [0, 1]. Default to 0.05. 1-alpha is the degree of confidence.

Value

A list with the following components:

one_sample_x

It contains the results by one_sample(x, ...).

one_sample_y

It contains the results by one_sample(y, ...).

mu1_mu2_interval

It contains the results of interval estimation of mu1-mu2.

mu1_mu2_hypothesis

It contains the results of test of hypothesis of mu1-mu2.

sigma_ratio_interval

It contains the results of interval estimation of sigma1^2 / sigma2^2.

sigma_ratio_hypothesis

It contains the results of test of hypothesis of sigma1^2 / sigma2^2.

res.ks

It contains the results of ks.test(x,y).

res.binom

It contains the results of binom.test(sum(x<y), length(x)).

res.wilcox

It contains the results of wilcox.test(x, y, ...).

cor.pearson

It contains the results of cor.test(x, y, method = "pearson", ...).

cor.kendall

It contains the results of cor.test(x, y, method = "kendall", ...).

cor.spearman

It contains the results of cor.test(x, y, method = "spearman", ...).

References

Zhang, Y. Y., Wei, Y. (2013), One and two samples using only an R funtion, doi:10.2991/asshm-13.2013.29 .

Author

Ying-Ying Zhang (Robert) robertzhangyying@qq.com

Examples

## One sample
x=rnorm(10, mean = 1, sd = 0.2); x
#>  [1] 1.0298811 0.9227715 0.9539012 1.0784212 1.0939866 1.1977686 0.8572544
#>  [8] 1.2539380 1.1418132 1.2093845

## one_sample(x, ...) == one_two_sample(x, ...)
one_sample(x, mu = 1, sigma = 0.2, side = 1)
#> quantile of x
#>        0%       25%       50%       75%      100% 
#> 0.8572544 0.9728962 1.0862039 1.1837797 1.2539380 
#> data_outline of x
#>    N     Mean        Var   std_dev   Median   std_mean       CV       CSS
#> 1 10 1.073912 0.01747354 0.1321875 1.086204 0.04180136 12.30897 0.1572618
#>        USS         R        R1   Skewness  Kurtosis
#> 1 11.69013 0.3966836 0.2108836 -0.2943003 -1.065115
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  x
#> W = 0.95981, p-value = 0.7837
#> 
#> 
#> The data is from the normal population.
#> 
#> The data is from the normal population.
#> 
#> Interval estimation and test of hypothesis of mu
#> Interval estimation: interval_estimate4()
#>       mean df         a   b
#> 1 1.073912 10 0.9698824 Inf
#> Test of hypothesis: mean_test1()
#> H0: mu <= 1     H1: mu > 1 
#>       mean df        Z  p_value
#> 1 1.073912 10 1.168652 0.121272
#> 
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#>          var df          a   b
#> 1 0.02118917 10 0.01157433 Inf
#> Test of hypothesis: var_test1()
#> H0: sigma2 <= 0.04     H1: sigma2 > 0.04 
#>          var df   chisq2   P_value
#> 1 0.02118917 10 5.297293 0.8704546
one_two_sample(x, mu = 1, sigma = 0.2, side = 1)
#> quantile of x
#>        0%       25%       50%       75%      100% 
#> 0.8572544 0.9728962 1.0862039 1.1837797 1.2539380 
#> data_outline of x
#>    N     Mean        Var   std_dev   Median   std_mean       CV       CSS
#> 1 10 1.073912 0.01747354 0.1321875 1.086204 0.04180136 12.30897 0.1572618
#>        USS         R        R1   Skewness  Kurtosis
#> 1 11.69013 0.3966836 0.2108836 -0.2943003 -1.065115
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  x
#> W = 0.95981, p-value = 0.7837
#> 
#> 
#> The data is from the normal population.
#> 
#> The data is from the normal population.
#> 
#> Interval estimation and test of hypothesis of mu
#> Interval estimation: interval_estimate4()
#>       mean df         a   b
#> 1 1.073912 10 0.9698824 Inf
#> Test of hypothesis: mean_test1()
#> H0: mu <= 1     H1: mu > 1 
#>       mean df        Z  p_value
#> 1 1.073912 10 1.168652 0.121272
#> 
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#>          var df          a   b
#> 1 0.02118917 10 0.01157433 Inf
#> Test of hypothesis: var_test1()
#> H0: sigma2 <= 0.04     H1: sigma2 > 0.04 
#>          var df   chisq2   P_value
#> 1 0.02118917 10 5.297293 0.8704546

one_sample(x, sigma = 0.2, side = 1)
#> quantile of x
#>        0%       25%       50%       75%      100% 
#> 0.8572544 0.9728962 1.0862039 1.1837797 1.2539380 
#> data_outline of x
#>    N     Mean        Var   std_dev   Median   std_mean       CV       CSS
#> 1 10 1.073912 0.01747354 0.1321875 1.086204 0.04180136 12.30897 0.1572618
#>        USS         R        R1   Skewness  Kurtosis
#> 1 11.69013 0.3966836 0.2108836 -0.2943003 -1.065115
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  x
#> W = 0.95981, p-value = 0.7837
#> 
#> 
#> The data is from the normal population.
#> 
#> The data is from the normal population.
#> 
#> Interval estimation and test of hypothesis of mu
#> Interval estimation: interval_estimate4()
#>       mean df         a   b
#> 1 1.073912 10 0.9698824 Inf
#> Test of hypothesis: mean_test1()
#> H0: mu <= 0     H1: mu > 0 
#>       mean df        Z p_value
#> 1 1.073912 10 16.98004       0
#> 
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#>          var df           a   b
#> 1 0.01747354  9 0.009294995 Inf
#> Test of hypothesis: var_test1()
#> H0: sigma2 <= 0.04     H1: sigma2 > 0.04 
#>          var df   chisq2   P_value
#> 1 0.01747354  9 3.931546 0.9158599
one_two_sample(x, sigma = 0.2, side = 1)
#> quantile of x
#>        0%       25%       50%       75%      100% 
#> 0.8572544 0.9728962 1.0862039 1.1837797 1.2539380 
#> data_outline of x
#>    N     Mean        Var   std_dev   Median   std_mean       CV       CSS
#> 1 10 1.073912 0.01747354 0.1321875 1.086204 0.04180136 12.30897 0.1572618
#>        USS         R        R1   Skewness  Kurtosis
#> 1 11.69013 0.3966836 0.2108836 -0.2943003 -1.065115
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  x
#> W = 0.95981, p-value = 0.7837
#> 
#> 
#> The data is from the normal population.
#> 
#> The data is from the normal population.
#> 
#> Interval estimation and test of hypothesis of mu
#> Interval estimation: interval_estimate4()
#>       mean df         a   b
#> 1 1.073912 10 0.9698824 Inf
#> Test of hypothesis: mean_test1()
#> H0: mu <= 0     H1: mu > 0 
#>       mean df        Z p_value
#> 1 1.073912 10 16.98004       0
#> 
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#>          var df           a   b
#> 1 0.01747354  9 0.009294995 Inf
#> Test of hypothesis: var_test1()
#> H0: sigma2 <= 0.04     H1: sigma2 > 0.04 
#>          var df   chisq2   P_value
#> 1 0.01747354  9 3.931546 0.9158599

one_sample(x, mu = 1, side = 1)
#> quantile of x
#>        0%       25%       50%       75%      100% 
#> 0.8572544 0.9728962 1.0862039 1.1837797 1.2539380 
#> data_outline of x
#>    N     Mean        Var   std_dev   Median   std_mean       CV       CSS
#> 1 10 1.073912 0.01747354 0.1321875 1.086204 0.04180136 12.30897 0.1572618
#>        USS         R        R1   Skewness  Kurtosis
#> 1 11.69013 0.3966836 0.2108836 -0.2943003 -1.065115
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  x
#> W = 0.95981, p-value = 0.7837
#> 
#> 
#> The data is from the normal population.
#> 
#> The data is from the normal population.
#> 
#> Interval estimation and test of hypothesis of mu
#> Interval estimation and test of hypothesis: t.test()
#> H0: mu <= 1     H1: mu > 1 
#> 
#> 	One Sample t-test
#> 
#> data:  x
#> t = 1.7682, df = 9, p-value = 0.05541
#> alternative hypothesis: true mean is greater than 1
#> 95 percent confidence interval:
#>  0.9972854       Inf
#> sample estimates:
#> mean of x 
#>  1.073912 
#> 
#> 
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#>          var df          a   b
#> 1 0.02118917 10 0.01157433 Inf
#> Test of hypothesis: var_test1()
#> H0: sigma2 <= 1     H1: sigma2 > 1 
#>          var df    chisq2   P_value
#> 1 0.02118917 10 0.2118917 0.9999999
one_two_sample(x, mu = 1, side = 1)
#> quantile of x
#>        0%       25%       50%       75%      100% 
#> 0.8572544 0.9728962 1.0862039 1.1837797 1.2539380 
#> data_outline of x
#>    N     Mean        Var   std_dev   Median   std_mean       CV       CSS
#> 1 10 1.073912 0.01747354 0.1321875 1.086204 0.04180136 12.30897 0.1572618
#>        USS         R        R1   Skewness  Kurtosis
#> 1 11.69013 0.3966836 0.2108836 -0.2943003 -1.065115
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  x
#> W = 0.95981, p-value = 0.7837
#> 
#> 
#> The data is from the normal population.
#> 
#> The data is from the normal population.
#> 
#> Interval estimation and test of hypothesis of mu
#> Interval estimation and test of hypothesis: t.test()
#> H0: mu <= 1     H1: mu > 1 
#> 
#> 	One Sample t-test
#> 
#> data:  x
#> t = 1.7682, df = 9, p-value = 0.05541
#> alternative hypothesis: true mean is greater than 1
#> 95 percent confidence interval:
#>  0.9972854       Inf
#> sample estimates:
#> mean of x 
#>  1.073912 
#> 
#> 
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#>          var df          a   b
#> 1 0.02118917 10 0.01157433 Inf
#> Test of hypothesis: var_test1()
#> H0: sigma2 <= 1     H1: sigma2 > 1 
#>          var df    chisq2   P_value
#> 1 0.02118917 10 0.2118917 0.9999999

one_sample(x)
#> quantile of x
#>        0%       25%       50%       75%      100% 
#> 0.8572544 0.9728962 1.0862039 1.1837797 1.2539380 
#> data_outline of x
#>    N     Mean        Var   std_dev   Median   std_mean       CV       CSS
#> 1 10 1.073912 0.01747354 0.1321875 1.086204 0.04180136 12.30897 0.1572618
#>        USS         R        R1   Skewness  Kurtosis
#> 1 11.69013 0.3966836 0.2108836 -0.2943003 -1.065115
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  x
#> W = 0.95981, p-value = 0.7837
#> 
#> 
#> The data is from the normal population.
#> 
#> The data is from the normal population.
#> 
#> Interval estimation and test of hypothesis of mu
#> Interval estimation and test of hypothesis: t.test()
#> H0: mu = 0     H1: mu != 0 
#> 
#> 	One Sample t-test
#> 
#> data:  x
#> t = 25.691, df = 9, p-value = 9.881e-10
#> alternative hypothesis: true mean is not equal to 0
#> 95 percent confidence interval:
#>  0.9793508 1.1684733
#> sample estimates:
#> mean of x 
#>  1.073912 
#> 
#> 
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#>          var df           a          b
#> 1 0.01747354  9 0.008267031 0.05823672
#> Test of hypothesis: var_test1()
#> H0: sigma2 = 1     H1: sigma2 != 1 
#>          var df    chisq2      P_value
#> 1 0.01747354  9 0.1572618 3.840908e-07
one_two_sample(x)
#> quantile of x
#>        0%       25%       50%       75%      100% 
#> 0.8572544 0.9728962 1.0862039 1.1837797 1.2539380 
#> data_outline of x
#>    N     Mean        Var   std_dev   Median   std_mean       CV       CSS
#> 1 10 1.073912 0.01747354 0.1321875 1.086204 0.04180136 12.30897 0.1572618
#>        USS         R        R1   Skewness  Kurtosis
#> 1 11.69013 0.3966836 0.2108836 -0.2943003 -1.065115
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  x
#> W = 0.95981, p-value = 0.7837
#> 
#> 
#> The data is from the normal population.
#> 
#> The data is from the normal population.
#> 
#> Interval estimation and test of hypothesis of mu
#> Interval estimation and test of hypothesis: t.test()
#> H0: mu = 0     H1: mu != 0 
#> 
#> 	One Sample t-test
#> 
#> data:  x
#> t = 25.691, df = 9, p-value = 9.881e-10
#> alternative hypothesis: true mean is not equal to 0
#> 95 percent confidence interval:
#>  0.9793508 1.1684733
#> sample estimates:
#> mean of x 
#>  1.073912 
#> 
#> 
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#>          var df           a          b
#> 1 0.01747354  9 0.008267031 0.05823672
#> Test of hypothesis: var_test1()
#> H0: sigma2 = 1     H1: sigma2 != 1 
#>          var df    chisq2      P_value
#> 1 0.01747354  9 0.1572618 3.840908e-07

## Two samples
set.seed(1)
x=rnorm(10, mean = 1, sd = 0.2); x
#>  [1] 0.8747092 1.0367287 0.8328743 1.3190562 1.0659016 0.8359063 1.0974858
#>  [8] 1.1476649 1.1151563 0.9389223
y=rnorm(20, mean = 2, sd = 0.3); y
#>  [1] 2.453534 2.116953 1.813628 1.335590 2.337479 1.986520 1.995143 2.283151
#>  [9] 2.246366 2.178170 2.275693 2.234641 2.022369 1.403194 2.185948 1.983161
#> [17] 1.953261 1.558774 1.856555 2.125382
y2=rnorm(20, mean = 2, sd = 0.2); y2
#>  [1] 2.271736 1.979442 2.077534 1.989239 1.724588 1.917001 1.921142 1.988137
#>  [9] 2.220005 2.152635 1.967095 1.949328 2.139393 2.111333 1.862249 1.858501
#> [17] 2.072916 2.153707 1.977531 2.176222

## sigma1, sigma2 known; mu1, mu2 known
one_two_sample(x, y, sigma = c(0.2, 0.3), mu = c(1, 2))
#> Interval estimation and test of hypothesis
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  x
#> W = 0.93828, p-value = 0.534
#> 
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  y
#> W = 0.91028, p-value = 0.06452
#> 
#> 
#> x and y are both from the normal populations.
#> 
#> x: descriptive statistics, plot, interval estimation and test of hypothesis
#> quantile of x
#>        0%       25%       50%       75%      100% 
#> 0.8328743 0.8907625 1.0513151 1.1107387 1.3190562 
#> data_outline of x
#>    N     Mean        Var   std_dev   Median   std_mean       CV       CSS
#> 1 10 1.026441 0.02437258 0.1561172 1.051315 0.04936859 15.20957 0.2193532
#>        USS         R        R1  Skewness   Kurtosis
#> 1 10.75516 0.4861819 0.2199761 0.3512426 -0.3169031
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  x
#> W = 0.93828, p-value = 0.534
#> 
#> 
#> The data is from the normal population.
#> 
#> The data is from the normal population.
#> 
#> Interval estimation and test of hypothesis of mu
#> Interval estimation: interval_estimate4()
#>       mean df         a      b
#> 1 1.026441 10 0.9024815 1.1504
#> Test of hypothesis: mean_test1()
#> H0: mu = 1     H1: mu != 1 
#>       mean df         Z   p_value
#> 1 1.026441 10 0.4180619 0.6759019
#> 
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#>          var df          a          b
#> 1 0.02263442 10 0.01105025 0.06970931
#> Test of hypothesis: var_test1()
#> H0: sigma2 = 0.04     H1: sigma2 != 0.04 
#>          var df   chisq2   P_value
#> 1 0.02263442 10 5.658606 0.3138319
#> 
#> y: descriptive statistics, plot, interval estimation and test of hypothesis
#> quantile of y
#>       0%      25%      50%      75%     100% 
#> 1.335590 1.929085 2.069661 2.237572 2.453534 
#> data_outline of y
#>    N     Mean        Var   std_dev   Median   std_mean       CV      CSS
#> 1 20 2.017276 0.09116068 0.3019283 2.069661 0.06751321 14.96713 1.732053
#>        USS        R        R1  Skewness  Kurtosis
#> 1 83.12008 1.117944 0.3084875 -1.003374 0.5258495
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  y
#> W = 0.91028, p-value = 0.06452
#> 
#> 
#> The data is from the normal population.
#> 
#> The data is from the normal population.
#> 
#> Interval estimation and test of hypothesis of mu
#> Interval estimation: interval_estimate4()
#>       mean df        a        b
#> 1 2.017276 20 1.885797 2.148754
#> Test of hypothesis: mean_test1()
#> H0: mu = 2     H1: mu != 2 
#>       mean df         Z   p_value
#> 1 2.017276 20 0.2575318 0.7967683
#> 
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#>         var df          a        b
#> 1 0.0869011 20 0.05086456 0.181218
#> Test of hypothesis: var_test1()
#> H0: sigma2 = 0.09     H1: sigma2 != 0.09 
#>         var df   chisq2   P_value
#> 1 0.0869011 20 19.31135 0.9966434
#> 
#> Interval estimation and test of hypothesis of mu1-mu2
#> 
#> Interval estimation: interval_estimate5()
#>         mean df         a          b
#> 1 -0.9908352 30 -1.171535 -0.8101355
#> 
#> Test of hypothesis: mean_test2()
#>         mean df         Z      p_value
#> 1 -0.9908352 30 -10.74712 6.114309e-27
#> 
#> Interval estimation and test of hypothesis of sigma1^2/sigma2^2
#> Interval estimation: interval_var4()
#>        rate df1 df2         a         b
#> 1 0.2604619  10  20 0.0939051 0.8904003
#> Test of hypothesis: var_test2()
#>        rate df1 df2         F    P_value
#> 1 0.2604619  10  20 0.2604619 0.03318465
#> n1 != n2
#> 
#> Test whether x and y are from the same population
#> H0: x and y are from the same population (without significant difference)
#> ks.test(x,y)
#> 
#> 	Exact two-sample Kolmogorov-Smirnov test
#> 
#> data:  x and y
#> D = 1, p-value = 6.657e-08
#> alternative hypothesis: two-sided
#> 
#> wilcox.test(x, y, alternative = alternative)
#> 
#> 	Wilcoxon rank sum exact test
#> 
#> data:  x and y
#> W = 0, p-value = 6.657e-08
#> alternative hypothesis: true location shift is not equal to 0
#> 

## sigma1 = sigma2 unknown; mu1, mu2 known
one_two_sample(x, y2, var.equal = TRUE, mu = c(1, 2))
#> Interval estimation and test of hypothesis
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  x
#> W = 0.93828, p-value = 0.534
#> 
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  y
#> W = 0.975, p-value = 0.8548
#> 
#> 
#> x and y are both from the normal populations.
#> 
#> x: descriptive statistics, plot, interval estimation and test of hypothesis
#> quantile of x
#>        0%       25%       50%       75%      100% 
#> 0.8328743 0.8907625 1.0513151 1.1107387 1.3190562 
#> data_outline of x
#>    N     Mean        Var   std_dev   Median   std_mean       CV       CSS
#> 1 10 1.026441 0.02437258 0.1561172 1.051315 0.04936859 15.20957 0.2193532
#>        USS         R        R1  Skewness   Kurtosis
#> 1 10.75516 0.4861819 0.2199761 0.3512426 -0.3169031
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  x
#> W = 0.93828, p-value = 0.534
#> 
#> 
#> The data is from the normal population.
#> 
#> The data is from the normal population.
#> 
#> Interval estimation and test of hypothesis of mu
#> Interval estimation and test of hypothesis: t.test()
#> H0: mu = 1     H1: mu != 1 
#> 
#> 	One Sample t-test
#> 
#> data:  x
#> t = 0.53557, df = 9, p-value = 0.6052
#> alternative hypothesis: true mean is not equal to 1
#> 95 percent confidence interval:
#>  0.914761 1.138120
#> sample estimates:
#> mean of x 
#>  1.026441 
#> 
#> 
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#>          var df          a          b
#> 1 0.02263442 10 0.01105025 0.06970931
#> Test of hypothesis: var_test1()
#> H0: sigma2 = 1     H1: sigma2 != 1 
#>          var df    chisq2      P_value
#> 1 0.02263442 10 0.2263442 2.816068e-07
#> 
#> y: descriptive statistics, plot, interval estimation and test of hypothesis
#> quantile of y
#>       0%      25%      50%      75%     100% 
#> 1.724588 1.942281 1.988688 2.142703 2.271736 
#> data_outline of y
#>    N     Mean        Var   std_dev   Median   std_mean       CV      CSS
#> 1 20 2.025487 0.01911437 0.1382547 1.988688 0.03091469 6.825753 0.363173
#>       USS         R       R1   Skewness  Kurtosis
#> 1 82.4151 0.5471478 0.200422 -0.1631305 -0.298063
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  y
#> W = 0.975, p-value = 0.8548
#> 
#> 
#> The data is from the normal population.
#> 
#> The data is from the normal population.
#> 
#> Interval estimation and test of hypothesis of mu
#> Interval estimation and test of hypothesis: t.test()
#> H0: mu = 2     H1: mu != 2 
#> 
#> 	One Sample t-test
#> 
#> data:  y
#> t = 0.82442, df = 19, p-value = 0.4199
#> alternative hypothesis: true mean is not equal to 2
#> 95 percent confidence interval:
#>  1.960781 2.090192
#> sample estimates:
#> mean of x 
#>  2.025487 
#> 
#> 
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#>          var df          a          b
#> 1 0.01880822 20 0.01100874 0.03922147
#> Test of hypothesis: var_test1()
#> H0: sigma2 = 1     H1: sigma2 != 1 
#>          var df    chisq2      P_value
#> 1 0.01880822 20 0.3761644 2.573594e-14
#> 
#> Interval estimation and test of hypothesis of mu1-mu2
#> 
#> Interval estimation and test of hypothesis: t.test()
#> 
#> 	Two Sample t-test
#> 
#> data:  x and y
#> t = -17.884, df = 28, p-value < 2.2e-16
#> alternative hypothesis: true difference in means is not equal to 0
#> 95 percent confidence interval:
#>  -1.1134763 -0.8846159
#> sample estimates:
#> mean of x mean of y 
#>  1.026441  2.025487 
#> 
#> 
#> Interval estimation and test of hypothesis of sigma1^2/sigma2^2
#> Interval estimation: interval_var4()
#>       rate df1 df2         a        b
#> 1 1.203432  10  20 0.4338771 4.113986
#> Test of hypothesis: var_test2()
#>       rate df1 df2        F   P_value
#> 1 1.203432  10  20 1.203432 0.6914616
#> n1 != n2
#> 
#> Test whether x and y are from the same population
#> H0: x and y are from the same population (without significant difference)
#> ks.test(x,y)
#> 
#> 	Exact two-sample Kolmogorov-Smirnov test
#> 
#> data:  x and y
#> D = 1, p-value = 6.657e-08
#> alternative hypothesis: two-sided
#> 
#> wilcox.test(x, y, alternative = alternative)
#> 
#> 	Wilcoxon rank sum exact test
#> 
#> data:  x and y
#> W = 0, p-value = 6.657e-08
#> alternative hypothesis: true location shift is not equal to 0
#> 

## sigma1 != sigma2 unknown; mu1, mu2 known
one_two_sample(x, y, mu = c(1, 2))
#> Interval estimation and test of hypothesis
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  x
#> W = 0.93828, p-value = 0.534
#> 
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  y
#> W = 0.91028, p-value = 0.06452
#> 
#> 
#> x and y are both from the normal populations.
#> 
#> x: descriptive statistics, plot, interval estimation and test of hypothesis
#> quantile of x
#>        0%       25%       50%       75%      100% 
#> 0.8328743 0.8907625 1.0513151 1.1107387 1.3190562 
#> data_outline of x
#>    N     Mean        Var   std_dev   Median   std_mean       CV       CSS
#> 1 10 1.026441 0.02437258 0.1561172 1.051315 0.04936859 15.20957 0.2193532
#>        USS         R        R1  Skewness   Kurtosis
#> 1 10.75516 0.4861819 0.2199761 0.3512426 -0.3169031
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  x
#> W = 0.93828, p-value = 0.534
#> 
#> 
#> The data is from the normal population.
#> 
#> The data is from the normal population.
#> 
#> Interval estimation and test of hypothesis of mu
#> Interval estimation and test of hypothesis: t.test()
#> H0: mu = 1     H1: mu != 1 
#> 
#> 	One Sample t-test
#> 
#> data:  x
#> t = 0.53557, df = 9, p-value = 0.6052
#> alternative hypothesis: true mean is not equal to 1
#> 95 percent confidence interval:
#>  0.914761 1.138120
#> sample estimates:
#> mean of x 
#>  1.026441 
#> 
#> 
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#>          var df          a          b
#> 1 0.02263442 10 0.01105025 0.06970931
#> Test of hypothesis: var_test1()
#> H0: sigma2 = 1     H1: sigma2 != 1 
#>          var df    chisq2      P_value
#> 1 0.02263442 10 0.2263442 2.816068e-07
#> 
#> y: descriptive statistics, plot, interval estimation and test of hypothesis
#> quantile of y
#>       0%      25%      50%      75%     100% 
#> 1.335590 1.929085 2.069661 2.237572 2.453534 
#> data_outline of y
#>    N     Mean        Var   std_dev   Median   std_mean       CV      CSS
#> 1 20 2.017276 0.09116068 0.3019283 2.069661 0.06751321 14.96713 1.732053
#>        USS        R        R1  Skewness  Kurtosis
#> 1 83.12008 1.117944 0.3084875 -1.003374 0.5258495
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  y
#> W = 0.91028, p-value = 0.06452
#> 
#> 
#> The data is from the normal population.
#> 
#> The data is from the normal population.
#> 
#> Interval estimation and test of hypothesis of mu
#> Interval estimation and test of hypothesis: t.test()
#> H0: mu = 2     H1: mu != 2 
#> 
#> 	One Sample t-test
#> 
#> data:  y
#> t = 0.25589, df = 19, p-value = 0.8008
#> alternative hypothesis: true mean is not equal to 2
#> 95 percent confidence interval:
#>  1.875969 2.158583
#> sample estimates:
#> mean of x 
#>  2.017276 
#> 
#> 
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#>         var df          a        b
#> 1 0.0869011 20 0.05086456 0.181218
#> Test of hypothesis: var_test1()
#> H0: sigma2 = 1     H1: sigma2 != 1 
#>         var df   chisq2      P_value
#> 1 0.0869011 20 1.738022 6.160195e-08
#> 
#> Interval estimation and test of hypothesis of mu1-mu2
#> 
#> Interval estimation and test of hypothesis: t.test()
#> 
#> 	Welch Two Sample t-test
#> 
#> data:  x and y
#> t = -11.847, df = 27.907, p-value = 2.111e-12
#> alternative hypothesis: true difference in means is not equal to 0
#> 95 percent confidence interval:
#>  -1.162185 -0.819485
#> sample estimates:
#> mean of x mean of y 
#>  1.026441  2.017276 
#> 
#> 
#> Interval estimation and test of hypothesis of sigma1^2/sigma2^2
#> Interval estimation: interval_var4()
#>        rate df1 df2         a         b
#> 1 0.2604619  10  20 0.0939051 0.8904003
#> Test of hypothesis: var_test2()
#>        rate df1 df2         F    P_value
#> 1 0.2604619  10  20 0.2604619 0.03318465
#> n1 != n2
#> 
#> Test whether x and y are from the same population
#> H0: x and y are from the same population (without significant difference)
#> ks.test(x,y)
#> 
#> 	Exact two-sample Kolmogorov-Smirnov test
#> 
#> data:  x and y
#> D = 1, p-value = 6.657e-08
#> alternative hypothesis: two-sided
#> 
#> wilcox.test(x, y, alternative = alternative)
#> 
#> 	Wilcoxon rank sum exact test
#> 
#> data:  x and y
#> W = 0, p-value = 6.657e-08
#> alternative hypothesis: true location shift is not equal to 0
#> 

## sigma1, sigma2 known; mu1, mu2 unknown
one_two_sample(x, y, sigma = c(0.2, 0.3))
#> Interval estimation and test of hypothesis
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  x
#> W = 0.93828, p-value = 0.534
#> 
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  y
#> W = 0.91028, p-value = 0.06452
#> 
#> 
#> x and y are both from the normal populations.
#> 
#> x: descriptive statistics, plot, interval estimation and test of hypothesis
#> quantile of x
#>        0%       25%       50%       75%      100% 
#> 0.8328743 0.8907625 1.0513151 1.1107387 1.3190562 
#> data_outline of x
#>    N     Mean        Var   std_dev   Median   std_mean       CV       CSS
#> 1 10 1.026441 0.02437258 0.1561172 1.051315 0.04936859 15.20957 0.2193532
#>        USS         R        R1  Skewness   Kurtosis
#> 1 10.75516 0.4861819 0.2199761 0.3512426 -0.3169031
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  x
#> W = 0.93828, p-value = 0.534
#> 
#> 
#> The data is from the normal population.
#> 
#> The data is from the normal population.
#> 
#> Interval estimation and test of hypothesis of mu
#> Interval estimation: interval_estimate4()
#>       mean df         a      b
#> 1 1.026441 10 0.9024815 1.1504
#> Test of hypothesis: mean_test1()
#> H0: mu = 0     H1: mu != 0 
#>       mean df        Z p_value
#> 1 1.026441 10 16.22945       0
#> 
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#>          var df          a          b
#> 1 0.02437258  9 0.01153109 0.08123021
#> Test of hypothesis: var_test1()
#> H0: sigma2 = 0.04     H1: sigma2 != 0.04 
#>          var df  chisq2   P_value
#> 1 0.02437258  9 5.48383 0.4194824
#> 
#> y: descriptive statistics, plot, interval estimation and test of hypothesis
#> quantile of y
#>       0%      25%      50%      75%     100% 
#> 1.335590 1.929085 2.069661 2.237572 2.453534 
#> data_outline of y
#>    N     Mean        Var   std_dev   Median   std_mean       CV      CSS
#> 1 20 2.017276 0.09116068 0.3019283 2.069661 0.06751321 14.96713 1.732053
#>        USS        R        R1  Skewness  Kurtosis
#> 1 83.12008 1.117944 0.3084875 -1.003374 0.5258495
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  y
#> W = 0.91028, p-value = 0.06452
#> 
#> 
#> The data is from the normal population.
#> 
#> The data is from the normal population.
#> 
#> Interval estimation and test of hypothesis of mu
#> Interval estimation: interval_estimate4()
#>       mean df        a        b
#> 1 2.017276 20 1.885797 2.148754
#> Test of hypothesis: mean_test1()
#> H0: mu = 0     H1: mu != 0 
#>       mean df        Z p_value
#> 1 2.017276 20 30.07177       0
#> 
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#>          var df          a         b
#> 1 0.09116068 19 0.05272238 0.1944703
#> Test of hypothesis: var_test1()
#> H0: sigma2 = 0.09     H1: sigma2 != 0.09 
#>          var df   chisq2   P_value
#> 1 0.09116068 19 19.24503 0.8824428
#> 
#> Interval estimation and test of hypothesis of mu1-mu2
#> 
#> Interval estimation: interval_estimate5()
#>         mean df         a          b
#> 1 -0.9908352 30 -1.171535 -0.8101355
#> 
#> Test of hypothesis: mean_test2()
#>         mean df         Z      p_value
#> 1 -0.9908352 30 -10.74712 6.114309e-27
#> 
#> Interval estimation and test of hypothesis of sigma1^2/sigma2^2
#> Interval estimation and test of hypothesis: var.test()
#> 
#> 	F test to compare two variances
#> 
#> data:  x and y
#> F = 0.26736, num df = 9, denom df = 19, p-value = 0.04757
#> alternative hypothesis: true ratio of variances is not equal to 1
#> 95 percent confidence interval:
#>  0.09283112 0.98477156
#> sample estimates:
#> ratio of variances 
#>          0.2673585 
#> 
#> n1 != n2
#> 
#> Test whether x and y are from the same population
#> H0: x and y are from the same population (without significant difference)
#> ks.test(x,y)
#> 
#> 	Exact two-sample Kolmogorov-Smirnov test
#> 
#> data:  x and y
#> D = 1, p-value = 6.657e-08
#> alternative hypothesis: two-sided
#> 
#> wilcox.test(x, y, alternative = alternative)
#> 
#> 	Wilcoxon rank sum exact test
#> 
#> data:  x and y
#> W = 0, p-value = 6.657e-08
#> alternative hypothesis: true location shift is not equal to 0
#> 

## sigma1 = sigma2 unknown; mu1, mu2 unknown
one_two_sample(x, y2, var.equal = TRUE)
#> Interval estimation and test of hypothesis
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  x
#> W = 0.93828, p-value = 0.534
#> 
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  y
#> W = 0.975, p-value = 0.8548
#> 
#> 
#> x and y are both from the normal populations.
#> 
#> x: descriptive statistics, plot, interval estimation and test of hypothesis
#> quantile of x
#>        0%       25%       50%       75%      100% 
#> 0.8328743 0.8907625 1.0513151 1.1107387 1.3190562 
#> data_outline of x
#>    N     Mean        Var   std_dev   Median   std_mean       CV       CSS
#> 1 10 1.026441 0.02437258 0.1561172 1.051315 0.04936859 15.20957 0.2193532
#>        USS         R        R1  Skewness   Kurtosis
#> 1 10.75516 0.4861819 0.2199761 0.3512426 -0.3169031
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  x
#> W = 0.93828, p-value = 0.534
#> 
#> 
#> The data is from the normal population.
#> 
#> The data is from the normal population.
#> 
#> Interval estimation and test of hypothesis of mu
#> Interval estimation and test of hypothesis: t.test()
#> H0: mu = 0     H1: mu != 0 
#> 
#> 	One Sample t-test
#> 
#> data:  x
#> t = 20.791, df = 9, p-value = 6.446e-09
#> alternative hypothesis: true mean is not equal to 0
#> 95 percent confidence interval:
#>  0.914761 1.138120
#> sample estimates:
#> mean of x 
#>  1.026441 
#> 
#> 
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#>          var df          a          b
#> 1 0.02437258  9 0.01153109 0.08123021
#> Test of hypothesis: var_test1()
#> H0: sigma2 = 1     H1: sigma2 != 1 
#>          var df    chisq2      P_value
#> 1 0.02437258  9 0.2193532 1.674074e-06
#> 
#> y: descriptive statistics, plot, interval estimation and test of hypothesis
#> quantile of y
#>       0%      25%      50%      75%     100% 
#> 1.724588 1.942281 1.988688 2.142703 2.271736 
#> data_outline of y
#>    N     Mean        Var   std_dev   Median   std_mean       CV      CSS
#> 1 20 2.025487 0.01911437 0.1382547 1.988688 0.03091469 6.825753 0.363173
#>       USS         R       R1   Skewness  Kurtosis
#> 1 82.4151 0.5471478 0.200422 -0.1631305 -0.298063
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  y
#> W = 0.975, p-value = 0.8548
#> 
#> 
#> The data is from the normal population.
#> 
#> The data is from the normal population.
#> 
#> Interval estimation and test of hypothesis of mu
#> Interval estimation and test of hypothesis: t.test()
#> H0: mu = 0     H1: mu != 0 
#> 
#> 	One Sample t-test
#> 
#> data:  y
#> t = 65.519, df = 19, p-value < 2.2e-16
#> alternative hypothesis: true mean is not equal to 0
#> 95 percent confidence interval:
#>  1.960781 2.090192
#> sample estimates:
#> mean of x 
#>  2.025487 
#> 
#> 
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#>          var df          a         b
#> 1 0.01911437 19 0.01105471 0.0407761
#> Test of hypothesis: var_test1()
#> H0: sigma2 = 1     H1: sigma2 != 1 
#>          var df   chisq2      P_value
#> 1 0.01911437 19 0.363173 1.369903e-13
#> 
#> Interval estimation and test of hypothesis of mu1-mu2
#> 
#> Interval estimation and test of hypothesis: t.test()
#> 
#> 	Two Sample t-test
#> 
#> data:  x and y
#> t = -17.884, df = 28, p-value < 2.2e-16
#> alternative hypothesis: true difference in means is not equal to 0
#> 95 percent confidence interval:
#>  -1.1134763 -0.8846159
#> sample estimates:
#> mean of x mean of y 
#>  1.026441  2.025487 
#> 
#> 
#> Interval estimation and test of hypothesis of sigma1^2/sigma2^2
#> Interval estimation and test of hypothesis: var.test()
#> 
#> 	F test to compare two variances
#> 
#> data:  x and y
#> F = 1.2751, num df = 9, denom df = 19, p-value = 0.6233
#> alternative hypothesis: true ratio of variances is not equal to 1
#> 95 percent confidence interval:
#>  0.4427323 4.6965951
#> sample estimates:
#> ratio of variances 
#>           1.275092 
#> 
#> n1 != n2
#> 
#> Test whether x and y are from the same population
#> H0: x and y are from the same population (without significant difference)
#> ks.test(x,y)
#> 
#> 	Exact two-sample Kolmogorov-Smirnov test
#> 
#> data:  x and y
#> D = 1, p-value = 6.657e-08
#> alternative hypothesis: two-sided
#> 
#> wilcox.test(x, y, alternative = alternative)
#> 
#> 	Wilcoxon rank sum exact test
#> 
#> data:  x and y
#> W = 0, p-value = 6.657e-08
#> alternative hypothesis: true location shift is not equal to 0
#> 

## sigma1 != sigma2 unknown; mu1, mu2 unknown
one_two_sample(x, y)
#> Interval estimation and test of hypothesis
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  x
#> W = 0.93828, p-value = 0.534
#> 
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  y
#> W = 0.91028, p-value = 0.06452
#> 
#> 
#> x and y are both from the normal populations.
#> 
#> x: descriptive statistics, plot, interval estimation and test of hypothesis
#> quantile of x
#>        0%       25%       50%       75%      100% 
#> 0.8328743 0.8907625 1.0513151 1.1107387 1.3190562 
#> data_outline of x
#>    N     Mean        Var   std_dev   Median   std_mean       CV       CSS
#> 1 10 1.026441 0.02437258 0.1561172 1.051315 0.04936859 15.20957 0.2193532
#>        USS         R        R1  Skewness   Kurtosis
#> 1 10.75516 0.4861819 0.2199761 0.3512426 -0.3169031
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  x
#> W = 0.93828, p-value = 0.534
#> 
#> 
#> The data is from the normal population.
#> 
#> The data is from the normal population.
#> 
#> Interval estimation and test of hypothesis of mu
#> Interval estimation and test of hypothesis: t.test()
#> H0: mu = 0     H1: mu != 0 
#> 
#> 	One Sample t-test
#> 
#> data:  x
#> t = 20.791, df = 9, p-value = 6.446e-09
#> alternative hypothesis: true mean is not equal to 0
#> 95 percent confidence interval:
#>  0.914761 1.138120
#> sample estimates:
#> mean of x 
#>  1.026441 
#> 
#> 
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#>          var df          a          b
#> 1 0.02437258  9 0.01153109 0.08123021
#> Test of hypothesis: var_test1()
#> H0: sigma2 = 1     H1: sigma2 != 1 
#>          var df    chisq2      P_value
#> 1 0.02437258  9 0.2193532 1.674074e-06
#> 
#> y: descriptive statistics, plot, interval estimation and test of hypothesis
#> quantile of y
#>       0%      25%      50%      75%     100% 
#> 1.335590 1.929085 2.069661 2.237572 2.453534 
#> data_outline of y
#>    N     Mean        Var   std_dev   Median   std_mean       CV      CSS
#> 1 20 2.017276 0.09116068 0.3019283 2.069661 0.06751321 14.96713 1.732053
#>        USS        R        R1  Skewness  Kurtosis
#> 1 83.12008 1.117944 0.3084875 -1.003374 0.5258495
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  y
#> W = 0.91028, p-value = 0.06452
#> 
#> 
#> The data is from the normal population.
#> 
#> The data is from the normal population.
#> 
#> Interval estimation and test of hypothesis of mu
#> Interval estimation and test of hypothesis: t.test()
#> H0: mu = 0     H1: mu != 0 
#> 
#> 	One Sample t-test
#> 
#> data:  y
#> t = 29.88, df = 19, p-value < 2.2e-16
#> alternative hypothesis: true mean is not equal to 0
#> 95 percent confidence interval:
#>  1.875969 2.158583
#> sample estimates:
#> mean of x 
#>  2.017276 
#> 
#> 
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#>          var df          a         b
#> 1 0.09116068 19 0.05272238 0.1944703
#> Test of hypothesis: var_test1()
#> H0: sigma2 = 1     H1: sigma2 != 1 
#>          var df   chisq2      P_value
#> 1 0.09116068 19 1.732053 2.061657e-07
#> 
#> Interval estimation and test of hypothesis of mu1-mu2
#> 
#> Interval estimation and test of hypothesis: t.test()
#> 
#> 	Welch Two Sample t-test
#> 
#> data:  x and y
#> t = -11.847, df = 27.907, p-value = 2.111e-12
#> alternative hypothesis: true difference in means is not equal to 0
#> 95 percent confidence interval:
#>  -1.162185 -0.819485
#> sample estimates:
#> mean of x mean of y 
#>  1.026441  2.017276 
#> 
#> 
#> Interval estimation and test of hypothesis of sigma1^2/sigma2^2
#> Interval estimation and test of hypothesis: var.test()
#> 
#> 	F test to compare two variances
#> 
#> data:  x and y
#> F = 0.26736, num df = 9, denom df = 19, p-value = 0.04757
#> alternative hypothesis: true ratio of variances is not equal to 1
#> 95 percent confidence interval:
#>  0.09283112 0.98477156
#> sample estimates:
#> ratio of variances 
#>          0.2673585 
#> 
#> n1 != n2
#> 
#> Test whether x and y are from the same population
#> H0: x and y are from the same population (without significant difference)
#> ks.test(x,y)
#> 
#> 	Exact two-sample Kolmogorov-Smirnov test
#> 
#> data:  x and y
#> D = 1, p-value = 6.657e-08
#> alternative hypothesis: two-sided
#> 
#> wilcox.test(x, y, alternative = alternative)
#> 
#> 	Wilcoxon rank sum exact test
#> 
#> data:  x and y
#> W = 0, p-value = 6.657e-08
#> alternative hypothesis: true location shift is not equal to 0
#>