# 8.9 Confidence intervals: summary of formulas

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## General form of a confidence interval

$\left(\text{lower value},\text{upper value}\right)=\left(\text{point estimate}-\text{margin of error},\text{point estimate}+\text{margin of error}\right)$

## To find the error bound when you know the confidence interval

$\text{margin of error}=\text{upper value}-\text{point estimate}\phantom{\rule{20pt}{0ex}}$ OR $\phantom{\rule{20pt}{0ex}}\text{margin of error}=\frac{\text{upper value}-\text{lower value}}{2}$

## Single population mean, known standard deviation, normal distribution

Use the Normal Distribution for Means $\phantom{\rule{20pt}{0ex}}\text{ME}={z}_{\frac{\alpha }{2}}\cdot \frac{\sigma }{\sqrt{n}}$

The confidence interval has the format $\left(\overline{x}-\text{ME},\overline{x}+\text{ME}\right)$ .

## Single population mean, unknown standard deviation, student's-t distribution

Use the Student's-t Distribution with degrees of freedom $\text{df}=n-1$ . $\text{ME}={t}_{\frac{\alpha }{2}}\cdot \frac{s}{\sqrt{n}}$

## Single population proportion, normal distribution

Use the Normal Distribution for a single population proportion $\stackrel{̂}{p}=\frac{x}{n}$

$\text{ME}={z}_{\frac{\alpha }{2}}\cdot \sqrt{\frac{\stackrel{̂}{p}\cdot \stackrel{̂}{q}}{n}}\phantom{\rule{20pt}{0ex}}\stackrel{̂}{p}+\stackrel{̂}{q}=1$

The confidence interval has the format $\left(\stackrel{̂}{p}-\text{ME},\stackrel{̂}{p}+\text{ME}\right)$ .

## Point estimates

$\overline{x}$ is a point estimate for $\mu$

$\stackrel{̂}{p}$ is a point estimate for $\rho$

$s$ is a point estimate for $\sigma$

## Critical values

$\mathrm{z*}$ is a critical value, based on the Normal Curve for an exact confidence interval such as for a 95% confidence interval. ${\mathrm{z* = 1.96}}^{}$

$\mathrm{t*}$ is a critical value, based on the Student's-t for an exact confidence interval.

## Binomial distribution

A discrete random variable (RV) which arises from Bernoulli trials. There are a fixed number, $n$ , of independent trials. “Independent” means that the result of any trial (for example, trial 1) does not affect the results of the following trials, and all trials are conducted under the same conditions. Under these circumstances the binomial RV $X$ is defined as the number of successes in $n$ trials. The notation is: $X$ ~ $B\left(n,p\right)$ . The mean is $\mu =\mathrm{np}$ and the standard deviation is $\sigma =\sqrt{\mathrm{npq}}$ . The probability of exactly $x$ successes in $n$ trials is $P\left(X=x\right)=\left(\genfrac{}{}{0}{}{n}{x}\right){p}^{x}{q}^{n-x}$ .

## Confidence interval (ci)

An interval estimate for an unknown population parameter. This depends on:
(1). The desired confidence level.
(2). Information that is known about the distribution (for example, known standard deviation).
(3). The sample and its size.

## Confidence level

The percent expression for the probability that the confidence interval contains the true population parameter. For example, if the CL=90%, then in 90 out of 100 samples the interval estimate will enclose the true population parameter.

## Degrees of freedom (df)

The number of objects in a sample that are free to vary.

## Margin of error for a population mean (ebm)

The margin of error. Depends on the confidence level, sample size, and known or estimated population standard deviation.

## Margin of error for a population proportion (ebp)

The margin of error. Depends on the confidence level, sample size, and the estimated (from the sample) proportion of successes.

## Inferential statistics

Also called statistical inference or inductive statistics. This facet of statistics deals with estimating a population parameter based on a sample statistic. For example, if 4 out of the 100 calculators sampled are defective we might infer that 4 percent of the production is defective

## Normal distribution

A continuous random variable (RV) with pdf $\text{f(x)}=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\left(x-\mu {\right)}^{2}/{2\sigma }^{2}}$ , where $\mu$ is the mean of the distribution and $\sigma$ is the standard deviation. Notation: $X$ ~ $N\left(\mu ,{\sigma }^{}\right)$ . If $\mu =0$ and $\sigma =1$ , the RV is called the standard normal distribution .

## Parameter

A numerical characteristic of the population.

## Point estimate

A single number computed from a sample and used to estimate a population parameter.

## Student-t distribution

Investigated and reported by William S. Gossett in 1908 and published under the pseudonym Student. The major characteristics of the random variable (RV) are:
(1). It is continuous and assumes any real values.
(2). The pdf is symmetrical about its mean of zero. However, it is more spread out and flatter at the apex than the normal distribution.
(3). It approaches the standard normal distribution as n gets larger.
(4). There is a "family" of t distributions: every representative of the family is completely defined by the number of degrees of freedom which is one less than the number of data

## Standard deviation

A number that is equal to the square root of the variance and measures how far data values are from their mean. Notation: s for sample standard deviation and σ for population standard deviation.

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