Estimating Populations and Samples

Point Estimator

an estimate for the value of a population parameter, derived from sample data
The $\hat{}$ symbol is added to the population parameter when you’re talking about its point estimator. As an example, the point estimator for $μ$ is $\hat{μ}$

Sample Mean

To find the mean of the sample, use the formula

\bar{x} = \frac{\sum x}{n}

where $x$ represents the values in the sample, and $n$ is the sample size.

Point Estimator for Population Mean

The point estimator for the population mean is found by calculating $x$ . In other words,

\hat{μ} = \bar{x}

This means that if you want a good estimate for the true value of the population mean, you can use the mean of the sample.

Point Estimator for Population Variance

\hat{σ^{2}} = s^{2} = \frac{\sum (x - \bar{x})^{2}}{n - 1}

Population Proportion

The population proportion is represented using $p$ . It’s the proportion of successes within the population. $p$ = probability = proportion
The point estimator for $p$ is given by $p_{s}$ , where $p_{s}$ is the proportion of successes in the sample.

\hat{p} = p_{s}

You calculate $p_{s}$ by dividing the number of successes in the sample by the size of the sample.

p_{s} = \frac{number of successes}{number in sample}

Sampling Distribution of Proportions

what you get if you consider all possible samples of size $n$ taken from the same population and form a distribution out of their proportions. We use $P_{s}$ to represent the sample proportion of random variable $X$ :

P_{s} = \frac{X}{n}

The expectation and variance of $P_{s}$ are defined as

E (P_{s}) = p

V a r (P_{s}) = \frac{p q}{n}

where $p$ is the population proportion.

Standard Error of Proportion

the standard deviation of this distribution. It’s given by

\sqrt{V a r (P_{s})}

If $n > 30$ , then $P_{s}$ follows a normal distribution, so

P_{s} \sim N (p, \frac{p q}{n})

for large $n$ . When working with this, you need to apply a continuity correction of

\pm \frac{1}{2 n}

Sampling Distribution of Means

what you get if you consider all possible samples of size $n$ taken from the same population and form a distribution out of their means. We use $\bar{X}$ to represent the sample mean random variable.

E (\bar{X}) = μ

V a r (\bar{X}) = \frac{σ^{2}}{n}

where $μ$ and $σ^{2}$ are the mean and variance of the population.

Standard Error of the Mean