Summary of Symbols and Definitions; Confidence Intervals for the Population Mean; Confidence Interval and Hypothesis Testing for Sample Mean and Proportion

1. 1 Inferential Statistics http://www.slideshare.net/amenning/documents
Inferential Statistics: Confidence Intervals and Hypothesis Testing
1. Summary of Symbols and Definitions
Symbol Definition Symbol Definition
μ (mu) Population mean x̄ (x-bar) Sample mean
σ2
(sigma square) Population variance s2
Sample variance
σ (sigma) Population standard deviation s (also SD) Sample standard deviation
n Sample size df Degrees of freedom
σx̄
(sigma x-bar)
Standard deviation of the
population of sample means
SE (for pop.
mean)
Standard error: SE=SD/√n
SE is our best to estimate of σx̄
p Population proportion p̂ (p-hat) Sample proportion
(1-α)100% Confidence level α (alpha) Significance level
z Test statistic based on normal
distribution
t Test statistic based on
t-distribution
zα Normal point with right-hand
tail area α : P(z ≤ zα) = 1-α
tα
tα, df
t-point with right-hand tail area α
(… and degrees of freedom df)
H0 Null hypothesis Ha Alternative hypothesis
μ0 Hypothesized value of the
population mean
p0 Hypothesized value of the
population proportion
2. Confidence Intervals for the Population Mean
All problems are based on a sample of size n randomly drawn from a population. The population mean μ
is unknown. The sample mean x̄ is used to estimate the population mean and the sample standard
deviation s is used to estimate the population standard deviation σ (in rare cases, σ may be known from
prior experiments). It is conventional to refer to the (exact or estimated) standard deviation as SD.
In all problems, the first step will be to calculate the standard error SE=SD/√n, which estimates the
standard deviation of the sampling distribution and is a measure of the accuracy of the sample mean. A
confidence Interval is always centered on the sample mean; its margin of error is a multiple of the
standard error.
Large Sample (n ≥ 30) Confidence Interval for population Mean:
 CI at 95%: [x̄ ± 𝟏. 𝟗𝟔 𝑺𝑬] = [x̄ ± 𝟏. 𝟗𝟔 𝑺𝑫/√ 𝒏]
 CI at 99%: [x̄ ± 𝟐. 𝟓𝟕𝟓 𝑺𝑬] = [x̄ ± 𝟐. 𝟓𝟕𝟓 𝑺𝑫/√ 𝒏]
 CI at (1-α)100%: [x̄ ± zα/2 𝑺𝑬] = [x̄ ± zα/2 𝑺𝑫/√ 𝒏]
Small Sample CI: use the t-point tα/2 instead of the normal point zα/2 except if σ is known. The population
must be mound-shaped (approximately normal).

2. 2 Inferential Statistics http://www.slideshare.net/amenning/documents
3. Confidence Interval and Hypothesis Testing for Sample Mean and Proportion
Sample Mean Sample Proportion
Basics
Estimate x̄ p̂
Conditions (valid if) n ≥ 30 or population distribution
normal (mound-shaped)
np̂ ≥ 5 and n(1-p̂) ≥ 5
Standard error 𝑺𝑬 = 𝑺𝑫/√ 𝒏
Substitute SD = σ if known,
otherwise SD = s
𝑺𝑬 = √𝒑(𝟏 − 𝒑)/𝒏
For CI substitute p = p̂
For HT substitute p = p0
Degrees of freedom Df = n-1 (for small sample)
Confidence interval (CI)
Confidence Level (1-α)100% (1-α)100%
Margin of Error E = tα/2, df SE or
E = zα/2 SE if σ known or n large
E = zα/2 SE
CI at (1-α)100% CI = [x̄ ± E] CI = [p̂ ± E]
Hypothesis Testing (HT)
Significance level α α
Null Hypothesis H0 : μ= μ0 or μ- μ0 = 0 H0 : p= p0 or p - p0 = 0
Alternative Hypothesis Ha : μ≠ μ0 (two-sided)
Ha : μ< μ0 (one-sided)
Ha : μ> μ0 (one-sided)
Ha : p≠ p0 (two-sided)
Ha : p< p0 (one-sided)
Ha : p> p0 (one-sided)
Critical value tα/2, df (two-sided)
tα, df (one-sided)
Use z-point for large sample or if σ known
zα/2 (two-sided)
zα (one-sided)
Effect size x̄- μ0 p̂ - p0
Statistically significant
effect / deviation
tα/2, df SE or zα/2 SE (two-sided)
tα, df SE or zα SE (one-sided)
zα/2 SE (two-sided)
zα SE (one-sided)
Test statistic t = (x̄ - μ0 )/SE z = (p̂ - p0)/SE
Hypothesis test H0 is rejected if the effect size, i.e. the deviation of the parameter
estimate from the hypothesized parameter, is statistically
significant at significance level α. Otherwise H0 is not rejected.
Equivalently: H0 is rejected if the absolute value of the test statistic
exceeds the critical value. For one-sided test, first check the sign
of the deviation; use common sense!