single sample inferences

target parameter : the unknown population parameter that we’re interested in estimating

point estimator : a single variable calculated from a sample that estimates a target population parameter

confidence interval : range of numbers which contain the target param with a high degree of confidence

interval estimator : see confidence interval

confidence coefficient : the probability that the confidence interval contains the population parameter.

confidence level : the confidence coefficient, but as a percentage.

z-statistic : An approximation of a mean using the sample’s mean and a known standard deviation. $z=\frac{\bar{x}-\mu }{\sigma /\sqrt{n}}$

t-statistic : An approximation for mean that takes into account both the sample’s mean $\bar{x}$ as well as the sample’s standard deviation $s$. $t=\frac{\bar{x}-\mu }{s/\sqrt{n}}$

degrees of freedom : the number of things that vary in an equation. It’s the number of entries minus the number of parameters used to calculate the resulting parameter. So for calculating variance ($x=E[(X-\mu {)}^2]$), it’s the number of things in X minus 1 b/c we’re using the sample mean as a parameter.

Confidence intervals based on a normal $z$ statistic

The formula for calculating the confidence interval is $\bar{x}\pm 1.96{\sigma }_{\bar{x}}=\bar{x}\pm \frac{1.96\sigma }{\sqrt{n}}$.

The 1.96 works out to contain the mean with a 95% chance.

confidence level	$\alpha$	$z_{\alpha /2}$
90%	.10	1.645
95%	.05	1.960
98%	.02	2.326
99%	.01	2.575

Notation: $z_{\alpha }$ is the value on a normal distribution ($z$) such that the area ($\alpha$) will be on the right. In practice, that means $z_{\alpha }$ is the symbol for the right-most tail when you calculate a confidence interval.

The generalized formula for large-sample (i.e. $n\ge 30$) is this:

If $\sigma$ is known, $\bar{x}\pm z_{\frac{\alpha }{2}}{\sigma }_{\bar{x}}=\bar{x}\pm z_{\frac{\alpha }{2}}\frac{\sigma }{\sqrt{n}}$ If $\sigma$ is unknown, $\bar{x}\pm z_{\frac{\alpha }{2}}{\sigma }_{\bar{x}}\approx \bar{x}\pm z_{\frac{\alpha }{2}}\frac{s}{\sqrt{n}}$ (where $s$ is the sample’s Standard Deviation)

but to be a valid large-sample confidence interval, we need:

1. A random sample

1. The sample size must be large enough so the Central Limit Theorem applies.

Confidence intervals for a population mean: student’s t-statistic

If we’re operating on a small sample and don’t know $\sigma$, then we can use t-statistics. T-statistics are more variable than z-statistics b/c they have an additional variable (sample standard deviation, rather z-statistics’ population standard deviation).

They use the same formula, but sub $s$ (sample stddev) instead of $\sigma$ (population stddev) and uses the “inverse t” to find the t-statistic (rather than “inverse normal” to find the z-statistic).