# Formula Review

### 8.2A Confidence Interval for a Population Standard Deviation Unknown, Small Sample Case

s = the standard deviation of sample values.

is the formula for the t-score which measures how far away a measure is from the population mean in the Student’s t-distribution

df = n - 1; the degrees of freedom for a Student’s t-distribution where n represents the size of the sample

T~tdf the random variable, T, has a Student’s t-distribution with df degrees of freedom

The general form for a confidence interval for a single mean, population standard deviation unknown, and sample size less than 30 Student's t is given by: $x- - t v,α ( s n ) ≤ μ ≤ x- + t v,α ( s n ) x-- t v,α ( s n )≤μ≤x-+ t v,α ( s n )$

### 8.3A Confidence Interval for A Population Proportion

p′= $xnxn$ where x represents the number of successes in a sample and n represents the sample size. The variable p′ is the sample proportion and serves as the point estimate for the true population proportion.

q′ = 1 – p

The variable p′ has a binomial distribution that can be approximated with the normal distribution shown here. The confidence interval for the true population proportion is given by the formula:

$p' - Zα p'q' n ≤ p ≤ p' + Zα p'q' n p'-Zαp'q' n ≤ p ≤ p'+Zαp'q' n$

provides the number of observations needed to sample to estimate the population proportion, p, with confidence 1 - α and margin of error e. Where e = the acceptable difference between the actual population proportion and the sample proportion.

### 8.4Calculating the Sample Size n: Continuous and Binary Random Variables

n = $Z 2 σ 2 (x--μ)2 Z 2 σ 2 (x--μ)2$ = the formula used to determine the sample size (n) needed to achieve a desired margin of error at a given level of confidence for a continuous random variable

$n = Zα2pq e2 n= Zα2pq e2$ = the formula used to determine the sample size if the random variable is binary