8.2 A 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:
8.3 A Confidence Interval for A Population Proportion
p′= 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:
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 = = 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
= the formula used to determine the sample size if the random variable is binary