One-Sample Z-Test

Comparing Two Means

• Compares the means of two groups where the outcome variable is interval or ratio scale and the predictor is binary
• A t-test is standard for comparing means; variations exist based on the question at hand
• Includes one-sample, independent samples, and paired samples t-tests
• Cohen’s d also covered as a standard measure of effect size for a t-test

The One-Sample Z-Test

• A test of limited real-world use, mainly serving as a transition to the more versatile t-test

Inference Problem Addressed

• Dr. Zeppo's class average is 67.5 with a standard deviation of 9.5
• A sample of 20 psychology students has a mean grade of 72.3
• It addresses the question: do psychology students score differently (higher or lower) than the average?

Constructing the Hypothesis Test

• Null hypothesis (H0): The true population mean for psychology student grades is 67.5

• Alternative hypothesis (H1): The population mean is not 67.5

• Assumes psychology grades are normally distributed

• Assumes scores have a known standard deviation is 𝛔 = 9.5

Diagnostic Test Statistic

• The difference serves as a starting point since hypotheses refer to population mean
• Calculate 𝑋 − 𝜇0, the difference between the sample mean 𝑋 and the hypothesized population mean 𝜇0
• If zero or close, the null hypothesis is supported
• Relies on knowing the raw data are normally distributed with population standard deviation 𝜎

Population Distribution

• If the null hypothesis is valid, the data follows a normal distribution: 𝑋 ~ Normal(𝜇0, 𝜎2)
• The sampling distribution of the mean 𝑋 is also normal and has mean 𝜇
• Standard error of mean is: 𝑆𝐸(𝑋) = 𝜎/√𝑁

Standard Score Conversion

• Converts sample mean 𝑋 into a standard score 𝑧𝑋
• Formula: 𝑧𝑋 = (𝑋 − 𝜇0) / 𝑆𝐸(𝑋) or 𝑧𝑋 = (𝑋 − 𝜇0) / (𝜎/√𝑁)
• This z-score is the test statistic, and like all z-scores, follows a standard normal distribution: 𝑍𝑋 ~ Normal(0, 1)

Critical Regions

• The 5% critical region for the z-test remains constant regardless of the population parameters for raw scores
• Researchers can calculate their z-statistic and reference a textbook table

Worked Example Using R

• Calculating sample mean: sample.mean <- mean(grades)
• Define variables for population standard deviation (σ = 9.5) and null hypothesis population mean (μ0 = 67.5): > mu.null <- 67.5 > sd.true <- 9.5
• Variable for sample size > N <- length(grades)
• Calculate standard error of the mean: > sem.true <- sd.true / sqrt(N)
• The z-score calculation: >z.score <- (sample.mean - mu.null) / sem.true
• p-value calculation
• Calculating area under the curve (from 2.26 upwards): > upper.area <- pnorm(q = z.score, lower.tail = FALSE)
• Calculating area from -2.26 downwards > lower.area <- pnorm(q = -z.score, lower.tail = TRUE)
• The p-value is then > p.value <- lower.area + upper.area
• Can report as z = 2.26, N = 20, p < .05).

Assumptions of the z-test

• Normality: Assumes the true population distribution is normal, but can be checked
• Independence: Assumes observations aren't related or correlated, but relies on good experimental design
• Known Standard Deviation: Assumes population standard deviation is known, typically not true in real data analysis

The One-Sample T-Test

• An alternative when population standard deviation (σ) is unknown
• Uses the estimated standard deviation
• Requires adjustment due to reliance on the estimate

Introducing The T-Test

• The ambiguity in the z-test caused by relying on true 𝛔 to be resolved by William Sealy Gosset in 1908
• Gossett, working for Guinness brewery, published under pseudonym "A Student"
• Recognized how the sampling distribution changes, t distribution and the t-statistic is calculated

T-statistic

• Calculated the same way with true mean is μ
• Sample mean 𝑋 and population standard deviation is Ô 𝑡 = 𝑋 − 𝜇 / 𝛔𝑁
• Sampling distribution turns into a t-distribution with N - 1 df