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Inferential Statistics Calculators

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Symbols and Meanings
CL = confidence level or level of confidence or degree of confidence or confidence coefficient
CI = confidence interval or interval estimate
$\alpha$ = level of significance or significance level
z_α/2 = critical z value separating an area or probability of ^α⁄2 in the right tail
−z_α/2 = critical z value separating an area or probability of ^α⁄2 in the left tail
z_α = critical z value separating an area or probability of α in the right tail
−z_α = critical z value separating an area or probability of α in the left tail
p̂ = sample proportion or estimated proportion of successes
q̂ = estimated proportion of failures
p = population proportion
SE = standard error
SE_est = estimated standard error
x = number of individuals in the sample with the specified characteristic
n = sample size
N = population size
E = margin of error or maximum error of estimation or error bound
x̄ = sample mean
μ = population mean
s = sample standard deviation
s² = sample variance
σ = population standard deviation
σ² = population variance
t_α/2 = critical t value separating an area or probability of ^α⁄2 in the right tail (use for one-tailed tests)
t_α = critical t value separating an area or probability of α in the right tail (use for two-tailed tests)
df = degrees of freedom
Χ² = Chi-Square distribution
Χ²_R = right-tailed (upper-tail) critical values of the Chi-Square distribution
Χ²_L = left-tailed (lower-tail) critical values of the Chi-Square distribution
CI for Population Proportion in Plus-Minus Notation = p̂ ± E
CI for Population Proportion in Interval Notation = (p̂ - E, p̂ + E)
CI for Population Proportion in Trilinear Inequality = p̂ - E < p < p̂ + E
CI for Population Mean in Plus-Minus Notation = x̄ ± E
CI for Population Mean in Interval Notation = (x̄ - E, x̄ + E)
CI for Population Mean in Trilinear Inequality = x̄ - E < μ < x̄ + E
min = minimum data value
max = maximum data value
R = range (we shall use it typically for the Range Rule of Thumb)

Critical Values

Given: CL
To Find: α, z_α/2
Requirements: population is normally distributed OR $n \gt 30$; AND $\sigma$ is given

The confidence level is in

Given: α
To Find: CL, z_α/2
Requirements: population is normally distributed OR $n \gt 30$; AND $\sigma$ is given

The level of significance is in

Given: CL, df
To Find: α, critical t
Requirements: population is normally distributed OR $n \gt 30$ AND s is given

The confidence level is in

The degrees of freedom is

The test is

Given: α, df
To Find: CL, critical t
Requirements: population is normally distributed OR $n \gt 30$ AND s is given

The level of significance is in

The degrees of freedom is

The test is

Given: CL, n
To Find: α, critical t
Requirements: population is normally distributed OR $n \gt 30$ AND s is given

The confidence level is in

The sample size is

The test is

Given: α, n
To Find: CL, critical t
Requirements: population is normally distributed OR $n \gt 30$ AND s is given

The level of significance is in

The sample size is

The test is

Given: CL, df
To Find: α, critical Χ²
Requirements: population is normally distributed

The confidence level is in

The degrees of freedom is

The test is

Given: α, df
To Find: CL, critical Χ²
Requirements: population is normally distributed

The level of significance is in

The degrees of freedom is

The test is

Estimating Population Proportion

Given: CI
To Find: p̂, E
The lower confidence limit is

The upper confidence limit is

Given: CI, n
To Find: p̂, E, x
The lower confidence limit is

The upper confidence limit is

The sample size is

Given: p̂, n, CL
To Find: q̂, E, CI
Requirements: np̂ ≥ 5, nq̂ ≥5, np̂q̂ ≥ 10

The confidence level is in

The sample proportion is in

The sample size is

Given: CL, x, n
To Find: p̂, q̂, E, CI
Requirements: np̂ ≥ 5, nq̂ ≥ 5, np̂q̂ ≥ 10

The confidence level is in

The number of individuals is

The sample size is

Given: CL, p̂, E
To Find: q̂, n
The confidence level is in

The sample proportion is in

The margin of error is in

Given: CL, E
To Find: n
The confidence level is in

The margin of error is in

Given: z_α/2, p̂, E
To Find: q̂, n
The critical value is

The sample proportion is in

The margin of error is in

Given: z_α/2, E
To Find: n
The critical value is

The margin of error is in

Given: p̂, E, CL (Use 95% if not given)
To Find: CI
The sample proportion is in

The margin of error is

The confidence level is in

Estimating Population Mean

Given: CI
To Find: x̄, E
The lower confidence limit is

The upper confidence limit is

Given: x̄, E, CL (Use 95% if not given)
To Find: CI
The sample mean is

The margin of error is

The confidence level is in

Given: CL, σ, n, x̄
To Find: E, CI
When requirements are met: population is normally distributed OR $n \gt 30$

The confidence level is in

The population standard deviation is

The sample size is

The sample mean is

Given: CL, s, n, x̄
To Find: E, CI
When requirements are met: population is normally distributed OR $n \gt 30$

The confidence level is in

The test is

The sample standard deviation is

The sample mean is

The sample size is

Given: σ, CL, E
To Find: n
The population standard deviation is

The confidence level is in

The margin of error is in

Given: s, CL, df, E
To Find: n
The sample standard deviation is

The confidence level is in

The degrees of freedom is

The margin of error is in

Given: min, max, α, E, sample size
To Find: R, s, df, n
The minimum data value is

The maximum data value is

The significance level is in

The margin of error is in

The initial/first sample size is

Given: min, max, CL, E
To Find: R, σ, n
The minimum data value is

The maximum data value is

The confidence level is in

The margin of error is in

Given: Raw Dataset (from Sample)
To Find: x̄, s
n =

x̄ =

s =

Estimating Population Variance and Population Standard Deviation

Given: CL, n, s
To Find: df, Χ²_L, Χ²_R, CI
The confidence level is in

The sample size is

The sample standard deviation is

Expected Counts

Given: i, p_i, n
To Find: E_i
i =

p_i =

The number of trials is

E_i =