Skip to contents

Deal with one (normal) sample

one_sample.Rd

Deal with one sample x, especially normal. Report descriptive statistics, plot, interval estimation and test of hypothesis of x.

Usage

one_sample(x, mu = Inf, sigma = -1, side = 0, alpha = 0.05)

Arguments

x

A numeric vector.

mu

mu plays two roles.

In two sided or one sided interval estimation (or test of hypothesis) of sigma^2 of one normal sample, mu is the population mean. When it is known, input it, and the function computes the interval endpoints (or chi-square statistic) using a chi-square distribution with degree of freedom n. When it is unknown, ignore it (the default), and the function computes the interval endpoints (or chi-square statistic) using a chi-square distribution with degree of freedom n-1.

In two sided or one sided test of hypothesis of mu of one normal sample, mu is mu0 in the null hypothesis, and mu0 = if (mu < Inf) mu else 0.

sigma

sigma plays two roles.

In two sided or one sided interval estimation (or test of hypothesis) of mu of one normal sample, sigma is the standard deviation of the population. sigma>=0 indicates it is known, and the function computes the interval endpoints (or Z statistic) using a standard normal distribution. sigma<0 indicates it is unknown, and the function computes the interval endpoints (or T statistic) using a t distribution with degree of freedom n-1. Default to unknown standard deviation.

In two sided or one sided test of hypothesis of sigma^2 of one normal sample, sigma is sigma0 in the null hypothesis. Default is 1, i.e., H0: sigma^2 = 1.

side

side plays two roles and is used in four places.

In two sided or one sided interval estimation of mu of one normal sample, 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; when computing the one sided lower limit, input side = 1; when computing the two sided limits, input side = 0 (default).

In two sided or one sided interval estimation of sigma^2 of one normal sample, 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; when computing the one sided lower limit, input side = 1; when computing the two sided limits, input side = 0 (default).

In two sided or one sided test of hypothesis of mu of one normal sample, 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: mu != mu0; when inputting side = -1 (or a number < 0), the function computes one sided test of hypothesis, and H1: mu < mu0; when inputting side = 1 (or a number > 0), the function computes one sided test of hypothesis, and H1: mu > mu0.

In two sided or one sided test of hypothesis of sigma^2 of one normal sample, 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: sigma^2 != sigma0^2; when inputting side = -1 (or a number < 0), the function computes one sided test of hypothesis, and H1: sigma^2 < sigma0^2; when inputting side = 1 (or a number > 0), the function computes one sided test of hypothesis, and H1: sigma^2 > sigma0^2.

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:

mu_interval

It contains the results of interval estimation of mu.

mu_hypothesis

It contains the results of test of hypothesis of mu.

sigma_interval

It contains the results of interval estimation of sigma.

sigma_hypothesis

It contains the results of test of hypothesis of sigma.

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

x=rnorm(10, mean = 1, sd = 0.2); x
#>  [1] 0.7543430 1.2079064 1.0911204 0.5240150 1.1195131 1.0255681 0.6566256
#>  [8] 1.0072219 1.2014536 1.1426218
one_sample(x, mu = 1, sigma = 0.2, side = 1)
#> quantile of x
#>        0%       25%       50%       75%      100% 
#> 0.5240150 0.8175627 1.0583442 1.1368447 1.2079064 
#> data_outline of x
#>    N      Mean        Var   std_dev   Median   std_mean       CV       CSS
#> 1 10 0.9730389 0.05833198 0.2415202 1.058344 0.07637538 24.82122 0.5249879
#>        USS         R        R1   Skewness   Kurtosis
#> 1 9.993035 0.6838915 0.3192819 -0.9656092 -0.4767062
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  x
#> W = 0.85822, p-value = 0.07271
#> 
#> 
#> 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 0.9730389 10 0.8690092 Inf
#> Test of hypothesis: mean_test1()
#> H0: mu <= 1     H1: mu > 1 
#>        mean df          Z   p_value
#> 1 0.9730389 10 -0.4262925 0.6650526
#> 
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#>          var df          a   b
#> 1 0.05322569 10 0.02907389 Inf
#> Test of hypothesis: var_test1()
#> H0: sigma2 <= 0.04     H1: sigma2 > 0.04 
#>          var df   chisq2   P_value
#> 1 0.05322569 10 13.30642 0.2070405
one_sample(x, sigma = 0.2, side = 1)
#> quantile of x
#>        0%       25%       50%       75%      100% 
#> 0.5240150 0.8175627 1.0583442 1.1368447 1.2079064 
#> data_outline of x
#>    N      Mean        Var   std_dev   Median   std_mean       CV       CSS
#> 1 10 0.9730389 0.05833198 0.2415202 1.058344 0.07637538 24.82122 0.5249879
#>        USS         R        R1   Skewness   Kurtosis
#> 1 9.993035 0.6838915 0.3192819 -0.9656092 -0.4767062
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  x
#> W = 0.85822, p-value = 0.07271
#> 
#> 
#> 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 0.9730389 10 0.8690092 Inf
#> Test of hypothesis: mean_test1()
#> H0: mu <= 0     H1: mu > 0 
#>        mean df       Z p_value
#> 1 0.9730389 10 15.3851       0
#> 
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#>          var df          a   b
#> 1 0.05833198  9 0.03102953 Inf
#> Test of hypothesis: var_test1()
#> H0: sigma2 <= 0.04     H1: sigma2 > 0.04 
#>          var df  chisq2   P_value
#> 1 0.05833198  9 13.1247 0.1570446
one_sample(x, mu = 1, side = 1)
#> quantile of x
#>        0%       25%       50%       75%      100% 
#> 0.5240150 0.8175627 1.0583442 1.1368447 1.2079064 
#> data_outline of x
#>    N      Mean        Var   std_dev   Median   std_mean       CV       CSS
#> 1 10 0.9730389 0.05833198 0.2415202 1.058344 0.07637538 24.82122 0.5249879
#>        USS         R        R1   Skewness   Kurtosis
#> 1 9.993035 0.6838915 0.3192819 -0.9656092 -0.4767062
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  x
#> W = 0.85822, p-value = 0.07271
#> 
#> 
#> 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 = -0.35301, df = 9, p-value = 0.6339
#> alternative hypothesis: true mean is greater than 1
#> 95 percent confidence interval:
#>  0.8330342       Inf
#> sample estimates:
#> mean of x 
#> 0.9730389 
#> 
#> 
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#>          var df          a   b
#> 1 0.05322569 10 0.02907389 Inf
#> Test of hypothesis: var_test1()
#> H0: sigma2 <= 1     H1: sigma2 > 1 
#>          var df    chisq2   P_value
#> 1 0.05322569 10 0.5322569 0.9999911
one_sample(x)
#> quantile of x
#>        0%       25%       50%       75%      100% 
#> 0.5240150 0.8175627 1.0583442 1.1368447 1.2079064 
#> data_outline of x
#>    N      Mean        Var   std_dev   Median   std_mean       CV       CSS
#> 1 10 0.9730389 0.05833198 0.2415202 1.058344 0.07637538 24.82122 0.5249879
#>        USS         R        R1   Skewness   Kurtosis
#> 1 9.993035 0.6838915 0.3192819 -0.9656092 -0.4767062
#> 
#> 	Shapiro-Wilk normality test
#> 
#> data:  x
#> W = 0.85822, p-value = 0.07271
#> 
#> 
#> 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 = 12.74, df = 9, p-value = 4.617e-07
#> alternative hypothesis: true mean is not equal to 0
#> 95 percent confidence interval:
#>  0.8002658 1.1458120
#> sample estimates:
#> mean of x 
#> 0.9730389 
#> 
#> 
#> Interval estimation and test of hypothesis of sigma
#> Interval estimation: interval_var3()
#>          var df          a         b
#> 1 0.05833198  9 0.02759787 0.1944119
#> Test of hypothesis: var_test1()
#> H0: sigma2 = 1     H1: sigma2 != 1 
#>          var df    chisq2      P_value
#> 1 0.05833198  9 0.5249879 7.503865e-05