Formula Review

8.1 A Single Population Mean Using the Normal Distribution

X¯~N(μX,σn)X¯~N(μX,σn) The distribution of sample means is normally distributed with mean equal to the population mean and standard deviation given by the population standard deviation divided by the square root of the sample size.

The general form for a confidence interval for a single population mean, known standard deviation, normal distribution is given by


 
(lower bound, upper bound) = (point estimate – EBM, point estimate + EBM)
=(x¯EBM,x¯+EBM)=(x¯EBM,x¯+EBM)
=(x¯zσn,x¯+zσn).=(x¯zσn,x¯+zσn).

EBM = zσnzσn = the error bound for the mean, or the margin of error for a single population mean; this formula is used when the population standard deviation is known.

CL = confidence level, or the proportion of confidence intervals created that is expected to contain the true population parameter

α = 1 – CL = the proportion of confidence intervals that will not contain the population parameter

zα2zα2 = the z-score with the property that the area to the right of the z-score is  2 2; this is the z-score, used in the calculation of EBM, where α = 1 – CL.

n = z2σ2EBM2z2σ2EBM2 = the formula used to determine the sample size (n) needed to achieve a desired margin of error at a given level of confidence

General form of a confidence interval

(lower value, upper value) = (point estimate error bound, point estimate + error bound)

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

error bound = upper value point estimate or error bound = upper valuelower value2.upper valuelower value2.

Single population mean, known standard deviation, normal distribution

Use the normal distribution for means; population standard deviation is known: EBM = zα2σn α2σn..

The confidence interval has the format (x¯x¯EBM, x¯x¯ + EBM).

8.2 A Single Population Mean Using the Student's t-Distribution

s = the standard deviation of sample values

t= x¯μsnt= x¯μsn 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

EBM=tα2snEBM=tα2sn = the error bound for the population mean when the population standard deviation is unknown

tα2tα2 is the t-score in the Student’s t-distribution with area to the right equal to α2.α2.

The general form for a confidence interval for a single mean, population standard deviation unknown, Student's t is given by

(lower bound, upper bound) = (point estimate – EBM, point estimate + EBM)

= (x¯tsn,x¯tsn).= (x¯tsn,x¯tsn).

8.3 A Population Proportion

p′ = x / n, where x represents the number of successes 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'

p~N(p,pqn)p~N(p,pqn) The variable p′ has a binomial distribution that can be approximated with the normal distribution shown here,

EBP = the error bound for a proportion = zα2pqn.EBP = the error bound for a proportion = zα2pqn.

Confidence interval for a proportion:

(lower bound, upper bound)=(p′ EBP,p+EBP)=(p′ zpqn, p+zpqn)(lower bound, upper bound)=(pEBP,p+EBP)=(pzpqn, p+zpqn)

n= zα22pqEBP2n= zα22pqEBP2 provides the number of participants needed to estimate the population proportion with confidence 1 – α and margin of error EBP.

Use the normal distribution for a single population proportion p =xn.p =xn.

EBP=(zα2)pqn p+q=1EBP=(zα2)pqn p+q=1

The confidence interval has the format (p′EBP, p′ + EBP).

x¯x¯ is a point estimate for μ.

p′ is a point estimate for ρ.

s is a point estimate for σ.