Stats Exam 2 Symbols/Formulas to know

μ

The population mean

σ

The population standard deviation

p

population proportion

N

population size

σ^2

population variance

xˉ

The sample mean, used as a point estimator for the population mean μ

s

The sample standard deviation, used to estimate σ when it is unknown

pˉ

The sample proportion, used as a point estimator for the population proportion p

n

sample size, sample statistics

df

Degrees of freedom, calculated as n−1 when using the t-distribution

μxˉ​

The mean of the sampling distribution of xˉ, which is equal to μ

σxˉ​

The standard deviation of the sampling distribution of xˉ, also known as the standard error of the mean

σ^xˉ​

The estimated standard error of the mean, used when the population standard deviation σ is unknown and s is substituted

σpˉ

The standard error of the proportion

σ^pˉ​

The estimated standard error of the proportion

α

The significance level, which represents the probability of making a Type I error (rejecting a true null hypothesis)

1−α

The confidence level (e.g., 0.95 for a 95% interval)

z

The z-score or z-test statistic, used when the population standard deviation is known or for proportions

β

The probability of making a Type II error (failing to reject a false null hypothesis)

t

The t-test statistic, used when the population standard deviation is unknown

zα/2​ or tα/2

The critical value (multiplier) for a specific confidence level

ME

The Margin of Error, which is the amount added to and subtracted from a point estimate to form an interval

UCL or LCL

The Upper Confidence Limit and Lower Confidence Limit of an interval

H0

The null hypothesis, representing the status quo or a prior belief

H1

The alternative hypothesis, representing the statement opposite to the null hypothesis

μH0​​

The hypothesized value of the population mean

pH0

The hypothesized value of the population proportion

Formula: Mean of the Sampling Distribution of xˉ

μxˉ​=μ

Formula: Standard Error of the Mean

σxˉ​=σ/sqrt(n)

Formula: z-Score for the Sample Mean

zxˉ​=x-bar - μxˉ​ / σx-bar

formula: sample proportion

p-bar = x / n

formula: Mean of the Sampling Distribution of p-bar

μpˉ​​=p

formula: Standard Error of the population:

σpˉ​​ = sqrt (p(1-p)/n)

formula: z-Score for the Sample Proportion

z p-bar = p bar - p / σpˉ​​

formula: CI for mean (σ Known)

xˉ±zα/2​σxˉ​

formula: Estimated Standard Error of the Mean (used when σ is unknown)

σ^xˉ​= s / sqrt(n)

formula: CI for mean (σ is unknown)

xˉ±tα/2​σ^xˉ

formula: Estimated Standard Error of the Proportion

σ^pˉ​​ = sqrt(pˉ​(1−pˉ​)​​/n)

formula: CI for Population Proportion

pˉ​±zα/2​σ^pˉ​​

formula: Test Statistic for Mean (σ Known)

zxˉ​ = x-bar - μH0 / σ/sqrt(n)

formula: Test Statistic for Mean (σ Unknown)

txˉ​= ​xˉ−μH0​​​ / s/sqrt(n)

formula: Test Statistic for Proportion

zp = pˉ​−pH0 / sqrt(pH0(1-pH0)/n)

x

z-Score for a single value: Used specifically when finding the probability of a randomly selected individual rather than a sample mean

Binomial Approximation Conditions:

np≥5 and n(1−p)≥5, to justify using the normal distribution for proportions