Sampling Distribution

Let there be $Y_{1},Y_{2},\ldots ,Y_{n}$, where each $Y_{i}$ is a randomv variable from the population.

Every Y have the same mean and distribution that we don't know.

$E(Y_{i})=\mu ,Var(Y_{i})=\sigma ^{2}$

We then have the sample mean

${\bar {Y}}={\frac {1}{n}}\sum _{i=1}^{n}Y_{i}$

The sample mean is expected to be $\mu$ through a pretty easy direct proof

The variance of the sample mean is ${\frac {\sigma ^{2}}{n}}$, also through a pretty easy direct proof.

Central limit theorem

The central limit theorem states that the distribution of the sample mean follows normal distribution.

${\bar {Y}}\sim N(\mu ,{\frac {\sigma ^{2}}{n}})$

As long as the following two conditions are satisfied, CLT applies, regardless of the population's distribution.

1. The population distribution of $Y$ is normal, or
2. The sample size for each $Y_{i}$ is large $n>30$

By extension, we also have

$S\sim N(\mu _{S}=n\mu ,\sigma _{S}={\sqrt {n\sigma }})$

where $S=\sum Y_{i}$

Confidence Interval

Estimation is the guess for the unknown parameter. A point estimate is a "best guess" of the population parameter, where as the confidence interval is the range of reasonable values that are intended to contain the parameter of interest with a certain degree of confidence, calculated with

(point estimate - margin of error, point estimate + margin of error)

Constructing CIs

By CLT, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{Y} \sim N(\mu, \frac{\sigma^2}{n} )} . The confidence interval is the range of plausible Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{Y}} .

If we define the middle 90% to be plausible, to find the confidence interval, simply find the 5th and 95th percentile.

Generalized, if we want a confidence interval of the middle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1 - \alpha) 100%} , have a confidence interval of

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{y} \pm Z_{\alpha / 2} \frac{\sigma}{ \sqrt{n} } }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{y}} is the sample mean and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_{x}} is the z score of the x-th percentile.

T-Distribution

T distribution table

CLT has several restrictions, the biggest one being a large sample size. T-

Since we don't know the population variance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma^2} , we have to use the sample variance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s} to estimate it. This introduces more uncertainty, accounted for by the t-distribution.

T-distribution is the distribution of sample mean based on population mean, sample variance and degrees of freedom (covered later). It looks very similar to normal distribution.

When the sample size Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is small, there is greater uncertainty in the estimates. T-di

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_{\alpha/2} > Z_{\alpha/2} }

The spread of t-distribution depends on the degrees of freedom, which is based on sample size. When looking up the table, round down df.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \upsilon = n - 1 }

As the sample size increases, degrees of freedom increase, the spread of t-distribution decreases, and t-distribution approaches normal distribution.

Based on CLT and normal distribution, we had the confidence interval

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{Y} \pm Z_{\alpha / 2} \frac{\sigma}{ \sqrt{n} } }

Now, based on T-distribution, we have the CI

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{Y} \pm t_{\alpha / 2} \frac{s}{ \sqrt{n} } }

Find Sample Size

To calculate sample size needed depending on desired error margin and sample variance by assuming that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \upsilon = \infty}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = \frac{Z^2_{\alpha/2} s^2}{E^2} }

We want to always round up to stay within the error margin.

I don't really know why.