PT SET 1 — Vocab, Identification & Rules

Probability Density Function

Function showing probabilities for a continuous variable

Density curve rules

Total area = 1 and a PDF cannot be negative

Normal distribution

Symmetric bell shaped distribution centered at μ

Standard normal distribution

μ = 0 and σ = 1

Z score

Number of standard deviations a value is from μ

Individual Z score

z = x − μ divided by σ

Unusual Z score

Usually an absolute Z score greater than 2

Sampling distribution

Distribution of a statistic from repeated samples

Sampling distribution of x̄

Distribution of sample means with mean μx̄ = μ

Standard error of x̄

σx̄ = σ divided by √n

Sample mean Z score

z = x̄ − μ divided by σ divided by √n

Central Limit Theorem

As n increases x̄ becomes more nearly normal

CLT rule

n ≥ 30 means x̄ approaches normal regardless of population shape

Point estimate

Single sample value used to estimate a population parameter

Confidence interval

Range of plausible values for a population parameter

Margin of error

Maximum distance from the point estimate

Confidence interval structure

Point estimate ± E

Interval rules

Lower = estimate − E and upper = estimate + E

Find E or point estimate

E = upper − lower divided by 2 and estimate = upper + lower divided by 2

Confidence interval width

Width = 2E

Confidence level rule

Higher confidence gives larger E and a wider interval

Sample size rule

Larger n gives smaller E a narrower interval and greater precision

Standard deviation rule

Larger σ or s gives larger E and a wider interval

Known σ

Use Z for a mean confidence interval

Unknown σ

Use T for a mean confidence interval

Mean margin of error

Known σ uses E = zα/2 times σ divided by √n and unknown σ uses E = tα/2 times s divided by √n

T distribution rule

Unknown σ means use s and T with df = n − 1

Proportion confidence interval

Use p̂ and a Z critical value

Proportion margin of error

E = zα/2 times √[p̂ times 1 − p̂ divided by n]

Plus 4 method

Use p̃ = x + 2 divided by n + 4 and replace p̂ with p̃

Sample size formulas

Mean uses [zα/2 times σ divided by E]² and proportion uses p̂ times 1 − p̂ times [zα/2 divided by E]²

No prior p̂

Use p̂ = 0.50

Sample size rounding

Always round the required sample size UP

Key symbols

μ = population mean x̄ = sample mean σ = population SD s = sample SD p = population proportion p̂ = sample proportion n = sample size

Parameter vs statistic

Parameter describes a population while statistic describes a sample

Z vs T recognition

Known σ = Z while unknown σ or given s = T

Problem type recognition

Range = CI maximum error = E how many needed = sample size and center of interval = point estimate

Confidence level and alpha

α = 1 − confidence level and two tailed intervals use α divided by 2

Critical value tables

Z uses the Z table and T uses the T table