Mathematics

Statistical inference

Sampling techniques, distribution of sample means and proportions (CLT), and confidence intervals for parameter estimation.

Concepts

Population: set of elements on which a certain study is carried out. Its number is designated by $\mathbf{N}$ .
Sample: any selected subset of the population. Its number is designated by $\mathbf{n}$ .
Sampling: the method followed to select the sample.

Concepts

Population: set of elements on which a certain study is carried out. Its number is designated by $\mathbf{N}$ .
Sample: any selected subset of the population. Its number is designated by $\mathbf{n}$ .
Sampling: the method followed to select the sample.

Probabilistic sampling

Simple random sampling: the population is numbered from 1 to $N$ , and $n$ numbers are drawn for the sample.
Systematic random sampling: the first number is chosen at random among the first $k$ and the rest every $k$ , where $k = N/n$ .
Stratified random sampling: The total population is divided into $k$ $k$ strata of size $N_1, N_2, \dots, N_k$ $N_{1}, N_{2}, \dots, N_{k}$ . The sample $n$ $n$ is selected following two criteria:
- Uniform allocation: the same number of elements from each stratum is selected, equal to $N/k$ .
- Proportional allocation: the number of elements of each stratum is proportional to its size. It is a problem of proportional distribution.

Example: We want to select a sample of 60 people in 3 populations of size 1600, 1200 and 2400 people. Calculate the sampling with proportional allocation.
The total population size is

N = 5200

	Pop. A	Pop. B	Pop. C
Size $N_i$	1600	1200	2400
Pop. fraction	4/13	3/13	6/13
Sample size	$\frac{4}{13} \cdot 60 = 18$	$\frac{3}{13} \cdot 60 \approx 14$	$\frac{6}{13} \cdot 60 \approx 28$

Distribution of sample means
Let

N

be a population with mean

\mu

and standard deviation

\sigma

. Samples of size

n

and mean

\bar{X}

are taken. Let the variable

\bar{X}

describe the distribution of sample means:
Central Limit Theorem: The mean of the variable

\bar{X}

is equal to the mean of the population

\mu

. The standard deviation of

\bar{X}

\sigma/\sqrt{n}

, then the distribution of sample means follows a normal distribution:

Distribution of sample means
Let

N

be a population with mean

\mu

and standard deviation

\sigma

. Samples of size

n

and mean

\bar{X}

are taken. Let the variable

\bar{X}

describe the distribution of sample means:
Central Limit Theorem: The mean of the variable

\bar{X}

is equal to the mean of the population

\mu

. The standard deviation of

\bar{X}

\sigma/\sqrt{n}

, then the distribution of sample means follows a normal distribution:

N\left(\mu, \frac{\sigma}{\sqrt{n}}\right)

Distribution of sample proportions
Let

N

be a binomial population, with proportion

p

of a certain characteristic. Samples of size

n

with proportion

\hat{p}

of this characteristic are taken. Let

\hat{P}

be the variable that describes the distribution of sample proportions:
Central Limit Theorem:

\hat{P}

approximates a normal distribution defined as:

N\left(p, \sqrt{\frac{p \cdot q}{n}}\right) \quad \text{for } n \ge 30

It is about obtaining an interval that includes the mean

\mu

of a certain variable

X

that we want to investigate in the population, with a confidence level

(1-\alpha)

. The parameter

\alpha

is called the significance level and is equivalent to the probability or risk that the parameter is outside the calculated confidence interval.

It is about obtaining an interval that includes the mean

\mu

of a certain variable

X

that we want to investigate in the population, with a confidence level

(1-\alpha)

. The parameter

\alpha

is called the significance level and is equivalent to the probability or risk that the parameter is outside the calculated confidence interval.

Confidence intervals for the mean
Population confidence interval for $\mu$ :

\left( \bar{x} - Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} , \; \bar{x} + Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} \right)

Where

\bar{x}

is the sample mean,

\sigma

is the population standard deviation,

n

is the sample size and

Z_{\alpha/2}

is the critical value obtained from the table.
Maximum error made in the estimation will be:

E_{\max} = Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}

The minimum sample size based on the maximum error to be assumed is:

n = \left( Z_{\alpha/2} \cdot \frac{\sigma}{E_{\max}} \right)^2

Confidence intervals for the proportion
Population confidence interval for $p$ :

\left( \hat{p} - Z_{\alpha/2} \cdot \sqrt{\frac{\hat{p} \cdot \hat{q}}{n}} , \; \hat{p} + Z_{\alpha/2} \cdot \sqrt{\frac{\hat{p} \cdot \hat{q}}{n}} \right)

Where

\hat{p}

is the sample proportion (probability of success),

\hat{q} = 1 - \hat{p}

is the probability of failure.
Maximum error made in the estimation will be:

E_{\max} = Z_{\alpha/2} \cdot \sqrt{\frac{\hat{p} \cdot \hat{q}}{n}}

The minimum sample size based on the maximum error to be assumed is:

n = \left( Z_{\alpha/2} \cdot \frac{\sqrt{\hat{p} \cdot \hat{q}}}{E_{\max}} \right)^2

Table to obtain the critical value

Z_{\alpha/2}

based on the desired significance level

\alpha

Table to obtain the critical value

Z_{\alpha/2}

based on the desired significance level

\alpha

Statistical inference

Sampling

Sampling distributions

Confidence interval estimation

Z Critical Values Table