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

s = the standard deviation of sample values.

t= x−−μsn√

$�= \frac{\overline{�}-�}{\frac{�}{\sqrt{�}}}$ 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~t_df 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¯−tv,α(sn√)≤μ≤x¯+tv,α(sn√)

$\overline{�}–{�}_{\mathrm{v,\alpha }}\left(\frac{�}{\sqrt{�}}\right)\le �\le \overline{�}+{�}_{\mathrm{v,\alpha }}\left(\frac{�}{\sqrt{�}}\right)$ 

8.3 A Confidence Interval for A Population Proportion

p′= xn

$\frac{�}{�}$ 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’−Zαp’q’n−−−√≤p≤p’+Zαp’q’n−−−√

$\mathrm{p’}–{�}_{�}\sqrt{\frac{\mathrm{p’q’}}{�}}\le �\le \mathrm{p’}+{�}_{�}\sqrt{\frac{\mathrm{p’q’}}{�}}$ 

n= Zα22p′q′e2

$�= \frac{{�}_{\frac{�}{2}}{}^2{�}^{\prime }{�}^{\prime }}{{�}^2}$ 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.

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

n = Z2σ2(x¯−μ)2

$\frac{{�}^2{�}^2}{(\overline{�}–�{)}^2}$ = 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=Z2αpqe2

$�=\frac{{�}_{�}^2\mathrm{pq}}{{�}^2}$ = the formula used to determine the sample size if the random variable is binary