6-1. The Mean and Standard Deviation of the Sample Mean

样本均值

In actual practice we would typically take just one sample.

Here is an example with such a small population and small sample size that we can actually write down every single sample.

[ Solution ]

all possible samples

Mean and Variance of Sample Means

7 possible sample means : {152, 154, 156, 158, 160, 162, 164}
probability distribution of the sample means : {1/16, 2/16, 3/16, 4/16, 3/16, 2/16, 1/16}

Mean and Variance of Population

population : {152, 156, 160, 164}

# 1. Mean and Variance of Sample Means

x <- c(152, 154, 156, 158, 160, 162, 164)
p <- c(1, 2, 3, 4, 3, 2, 1)/16

mu_x <- sum(x*p); mu_x
var_x <- sum(x^2 * p) - mu_x^2; var_x


# 2. Mean and Variance of Population

y <- c(152, 156, 160, 164)

mu_y <- sum(y)/length(y); mu_y
var_y <- sum(y^2)/length(y) - mu_y^2; var_y

> mu_x <- sum(x*p); mu_x
## [1] 158
> var_x <- sum(x^2 * p) - mu_x^2; var_x
## [1] 10

> mu_y <- sum(y)/length(y); mu_y
## [1] 158
> var_y <- sum(y^2)/length(y) - mu_y^2; var_y 
## [1] 20

[ Solution ]

n <- 100
mu <- 13525
si <- 4180

# 1. Mean of Sample Mean
mu_x <- mu; mu

# 2. Standard Deviation of Sample Mean
sig_x <- si / sqrt(n); sig_x

> # 1. Mean of Sample Mean
> mu_x <- mu; mu
## [1] 13525
> 
> # 2. Standard Deviation of Sample Mean
> sig_x <- si / sqrt(n); sig_x
## [1] 418

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# 1. Mean and Variance of Sample Means x <- c(152, 154, 156, 158, 160, 162, 164) p <- c(1, 2, 3, 4, 3, 2, 1)/16 mu_x <- sum(x*p); mu_x var_x <- sum(x^2 * p) - mu_x^2; var_x # 2. Mean and Variance of Population y <- c(152, 156, 160, 164) mu_y <- sum(y)/length(y); mu_y var_y <- sum(y^2)/length(y) - mu_y^2; var_y