https://doi.org/10.32614/CRAN.package.OneTwoSamples
The goal of OneTwoSamples is to introduce an R function one_two_sample() which can deal with one and two (normal) samples, Ying-Ying Zhang, Yi Wei (2012) doi:10.2991/asshm-13.2013.29. 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.
Installation
You can install the released version of OneTwoSamples from CRAN with:
install.packages("OneTwoSamples")You can install the development version of OneTwoSamples from github with:
devtools::install_github("fbertran/OneTwoSamples")One sample
x=rnorm(10, mean = 1, sd = 0.2); x
#> [1] 1.0796212 0.8775947 1.0682239 0.7741274 1.2866047 1.3960800 0.9265557 0.7911731
#> [9] 1.1139439 0.9729891one_sample(x, …) == one_two_sample(x, …)
one_sample(x, mu = 1, sigma = 0.2, side = 1)
#> quantile of x
#> 0% 25% 50% 75% 100%
#> 0.7741274 0.8898350 1.0206065 1.1053632 1.3960800
#> data_outline of x
#> N Mean Var std_dev Median std_mean CV CSS USS
#> 1 10 1.028691 0.04116675 0.2028959 1.020607 0.06416132 19.72369 0.3705008 10.95256
#> R R1 Skewness Kurtosis
#> 1 0.6219526 0.2155283 0.5482993 -0.3821252
#>
#> Shapiro-Wilk normality test
#>
#> data: x
#> W = 0.94937, p-value = 0.6611
#>
#>
#> 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 1.028691 10 0.9246617 Inf
#> Test of hypothesis: mean_test1()
#> H0: mu <= 1 H1: mu > 1
#> mean df Z p_value
#> 1 1.028691 10 0.4536504 0.3250402
#>
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#> var df a b
#> 1 0.03787327 10 0.02068782 Inf
#> Test of hypothesis: var_test1()
#> H0: sigma2 <= 0.04 H1: sigma2 > 0.04
#> var df chisq2 P_value
#> 1 0.03787327 10 9.468318 0.4883082
one_two_sample(x, mu = 1, sigma = 0.2, side = 1)
#> quantile of x
#> 0% 25% 50% 75% 100%
#> 0.7741274 0.8898350 1.0206065 1.1053632 1.3960800
#> data_outline of x
#> N Mean Var std_dev Median std_mean CV CSS USS
#> 1 10 1.028691 0.04116675 0.2028959 1.020607 0.06416132 19.72369 0.3705008 10.95256
#> R R1 Skewness Kurtosis
#> 1 0.6219526 0.2155283 0.5482993 -0.3821252
#>
#> Shapiro-Wilk normality test
#>
#> data: x
#> W = 0.94937, p-value = 0.6611
#>
#>
#> 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 1.028691 10 0.9246617 Inf
#> Test of hypothesis: mean_test1()
#> H0: mu <= 1 H1: mu > 1
#> mean df Z p_value
#> 1 1.028691 10 0.4536504 0.3250402
#>
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#> var df a b
#> 1 0.03787327 10 0.02068782 Inf
#> Test of hypothesis: var_test1()
#> H0: sigma2 <= 0.04 H1: sigma2 > 0.04
#> var df chisq2 P_value
#> 1 0.03787327 10 9.468318 0.4883082
one_sample(x, sigma = 0.2, side = 1)
#> quantile of x
#> 0% 25% 50% 75% 100%
#> 0.7741274 0.8898350 1.0206065 1.1053632 1.3960800
#> data_outline of x
#> N Mean Var std_dev Median std_mean CV CSS USS
#> 1 10 1.028691 0.04116675 0.2028959 1.020607 0.06416132 19.72369 0.3705008 10.95256
#> R R1 Skewness Kurtosis
#> 1 0.6219526 0.2155283 0.5482993 -0.3821252
#>
#> Shapiro-Wilk normality test
#>
#> data: x
#> W = 0.94937, p-value = 0.6611
#>
#>
#> 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 1.028691 10 0.9246617 Inf
#> Test of hypothesis: mean_test1()
#> H0: mu <= 0 H1: mu > 0
#> mean df Z p_value
#> 1 1.028691 10 16.26504 0
#>
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#> var df a b
#> 1 0.04116675 9 0.02189853 Inf
#> Test of hypothesis: var_test1()
#> H0: sigma2 <= 0.04 H1: sigma2 > 0.04
#> var df chisq2 P_value
#> 1 0.04116675 9 9.262519 0.4134029
one_two_sample(x, sigma = 0.2, side = 1)
#> quantile of x
#> 0% 25% 50% 75% 100%
#> 0.7741274 0.8898350 1.0206065 1.1053632 1.3960800
#> data_outline of x
#> N Mean Var std_dev Median std_mean CV CSS USS
#> 1 10 1.028691 0.04116675 0.2028959 1.020607 0.06416132 19.72369 0.3705008 10.95256
#> R R1 Skewness Kurtosis
#> 1 0.6219526 0.2155283 0.5482993 -0.3821252
#>
#> Shapiro-Wilk normality test
#>
#> data: x
#> W = 0.94937, p-value = 0.6611
#>
#>
#> 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 1.028691 10 0.9246617 Inf
#> Test of hypothesis: mean_test1()
#> H0: mu <= 0 H1: mu > 0
#> mean df Z p_value
#> 1 1.028691 10 16.26504 0
#>
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#> var df a b
#> 1 0.04116675 9 0.02189853 Inf
#> Test of hypothesis: var_test1()
#> H0: sigma2 <= 0.04 H1: sigma2 > 0.04
#> var df chisq2 P_value
#> 1 0.04116675 9 9.262519 0.4134029
one_sample(x, mu = 1, side = 1)
#> quantile of x
#> 0% 25% 50% 75% 100%
#> 0.7741274 0.8898350 1.0206065 1.1053632 1.3960800
#> data_outline of x
#> N Mean Var std_dev Median std_mean CV CSS USS
#> 1 10 1.028691 0.04116675 0.2028959 1.020607 0.06416132 19.72369 0.3705008 10.95256
#> R R1 Skewness Kurtosis
#> 1 0.6219526 0.2155283 0.5482993 -0.3821252
#>
#> Shapiro-Wilk normality test
#>
#> data: x
#> W = 0.94937, p-value = 0.6611
#>
#>
#> 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 = 0.44718, df = 9, p-value = 0.3327
#> alternative hypothesis: true mean is greater than 1
#> 95 percent confidence interval:
#> 0.9110764 Inf
#> sample estimates:
#> mean of x
#> 1.028691
#>
#>
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#> var df a b
#> 1 0.03787327 10 0.02068782 Inf
#> Test of hypothesis: var_test1()
#> H0: sigma2 <= 1 H1: sigma2 > 1
#> var df chisq2 P_value
#> 1 0.03787327 10 0.3787327 0.9999983
one_two_sample(x, mu = 1, side = 1)
#> quantile of x
#> 0% 25% 50% 75% 100%
#> 0.7741274 0.8898350 1.0206065 1.1053632 1.3960800
#> data_outline of x
#> N Mean Var std_dev Median std_mean CV CSS USS
#> 1 10 1.028691 0.04116675 0.2028959 1.020607 0.06416132 19.72369 0.3705008 10.95256
#> R R1 Skewness Kurtosis
#> 1 0.6219526 0.2155283 0.5482993 -0.3821252
#>
#> Shapiro-Wilk normality test
#>
#> data: x
#> W = 0.94937, p-value = 0.6611
#>
#>
#> 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 = 0.44718, df = 9, p-value = 0.3327
#> alternative hypothesis: true mean is greater than 1
#> 95 percent confidence interval:
#> 0.9110764 Inf
#> sample estimates:
#> mean of x
#> 1.028691
#>
#>
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#> var df a b
#> 1 0.03787327 10 0.02068782 Inf
#> Test of hypothesis: var_test1()
#> H0: sigma2 <= 1 H1: sigma2 > 1
#> var df chisq2 P_value
#> 1 0.03787327 10 0.3787327 0.9999983
one_sample(x)
#> quantile of x
#> 0% 25% 50% 75% 100%
#> 0.7741274 0.8898350 1.0206065 1.1053632 1.3960800
#> data_outline of x
#> N Mean Var std_dev Median std_mean CV CSS USS
#> 1 10 1.028691 0.04116675 0.2028959 1.020607 0.06416132 19.72369 0.3705008 10.95256
#> R R1 Skewness Kurtosis
#> 1 0.6219526 0.2155283 0.5482993 -0.3821252
#>
#> Shapiro-Wilk normality test
#>
#> data: x
#> W = 0.94937, p-value = 0.6611
#>
#>
#> 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 = 16.033, df = 9, p-value = 6.318e-08
#> alternative hypothesis: true mean is not equal to 0
#> 95 percent confidence interval:
#> 0.8835484 1.1738344
#> sample estimates:
#> mean of x
#> 1.028691
#>
#>
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#> var df a b
#> 1 0.04116675 9 0.0194767 0.1372027
#> Test of hypothesis: var_test1()
#> H0: sigma2 = 1 H1: sigma2 != 1
#> var df chisq2 P_value
#> 1 0.04116675 9 0.3705008 1.665084e-05
one_two_sample(x)
#> quantile of x
#> 0% 25% 50% 75% 100%
#> 0.7741274 0.8898350 1.0206065 1.1053632 1.3960800
#> data_outline of x
#> N Mean Var std_dev Median std_mean CV CSS USS
#> 1 10 1.028691 0.04116675 0.2028959 1.020607 0.06416132 19.72369 0.3705008 10.95256
#> R R1 Skewness Kurtosis
#> 1 0.6219526 0.2155283 0.5482993 -0.3821252
#>
#> Shapiro-Wilk normality test
#>
#> data: x
#> W = 0.94937, p-value = 0.6611
#>
#>
#> 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 = 16.033, df = 9, p-value = 6.318e-08
#> alternative hypothesis: true mean is not equal to 0
#> 95 percent confidence interval:
#> 0.8835484 1.1738344
#> sample estimates:
#> mean of x
#> 1.028691
#>
#>
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#> var df a b
#> 1 0.04116675 9 0.0194767 0.1372027
#> Test of hypothesis: var_test1()
#> H0: sigma2 = 1 H1: sigma2 != 1
#> var df chisq2 P_value
#> 1 0.04116675 9 0.3705008 1.665084e-05Two 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 1.1476649
#> [9] 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 2.246366
#> [10] 2.178170 2.275693 2.234641 2.022369 1.403194 2.185948 1.983161 1.953261 1.558774
#> [19] 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 2.220005
#> [10] 2.152635 1.967095 1.949328 2.139393 2.111333 1.862249 1.858501 2.072916 2.153707
#> [19] 1.977531 2.176222sigma1, 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 USS
#> 1 10 1.026441 0.02437258 0.1561172 1.051315 0.04936859 15.20957 0.2193532 10.75516
#> R R1 Skewness Kurtosis
#> 1 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 USS
#> 1 20 2.017276 0.09116068 0.3019283 2.069661 0.06751321 14.96713 1.732053 83.12008
#> R R1 Skewness Kurtosis
#> 1 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 0sigma1 = 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 USS
#> 1 10 1.026441 0.02437258 0.1561172 1.051315 0.04936859 15.20957 0.2193532 10.75516
#> R R1 Skewness Kurtosis
#> 1 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 USS
#> 1 20 2.025487 0.01911437 0.1382547 1.988688 0.03091469 6.825753 0.363173 82.4151
#> R R1 Skewness Kurtosis
#> 1 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 0sigma1 != 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 USS
#> 1 10 1.026441 0.02437258 0.1561172 1.051315 0.04936859 15.20957 0.2193532 10.75516
#> R R1 Skewness Kurtosis
#> 1 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 USS
#> 1 20 2.017276 0.09116068 0.3019283 2.069661 0.06751321 14.96713 1.732053 83.12008
#> R R1 Skewness Kurtosis
#> 1 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 0sigma1, 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 USS
#> 1 10 1.026441 0.02437258 0.1561172 1.051315 0.04936859 15.20957 0.2193532 10.75516
#> R R1 Skewness Kurtosis
#> 1 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 USS
#> 1 20 2.017276 0.09116068 0.3019283 2.069661 0.06751321 14.96713 1.732053 83.12008
#> R R1 Skewness Kurtosis
#> 1 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 0sigma1 = 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 USS
#> 1 10 1.026441 0.02437258 0.1561172 1.051315 0.04936859 15.20957 0.2193532 10.75516
#> R R1 Skewness Kurtosis
#> 1 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 USS
#> 1 20 2.025487 0.01911437 0.1382547 1.988688 0.03091469 6.825753 0.363173 82.4151
#> R R1 Skewness Kurtosis
#> 1 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 0sigma1 != 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 USS
#> 1 10 1.026441 0.02437258 0.1561172 1.051315 0.04936859 15.20957 0.2193532 10.75516
#> R R1 Skewness Kurtosis
#> 1 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 USS
#> 1 20 2.017276 0.09116068 0.3019283 2.069661 0.06751321 14.96713 1.732053 83.12008
#> R R1 Skewness Kurtosis
#> 1 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