Chapter 9: The One-Sample T-Test Study Guide

Introduction to the One-Sample T-Test

• Chapter 9 transitions from the One-Sample Z-Test (introduced in Chapter 8) to the One-Sample T-Test.
• This is not an entirely new concept but rather a set of tweaks to the hypothesis testing framework already established.
• The core structure remains the same: the test follows the established four-step process of hypothesis testing.
• Instead of calculating a $Z_{obtained}$ and comparing it to a $Z_{critical}$, the process now involves a $t_{obtained}$ and a $t_{critical}$.
• T-tests will be the focus for three consecutive chapters:
  • Chapter 9: One-Sample T-test.
  • Chapter 10: Independent Samples T-test.
  • Chapter 11: Related Samples T-test.

Fundamental Differences: Z-Test vs. T-Test

• General Variability Source:
  • Z-tests require the population parameters: the mean ($\mu$) and the population standard deviation ($\sigma$).
  • T-tests are used when the population standard deviation ($\sigma$) is unknown.
• Sample Variability: T-tests rely on sample variability rather than population variability. Researchers often cannot obtain the population standard deviation, making the T-test more practical for real-world research.
• Substitution: In T-test formulas, the sample standard deviation ($s$) or sample variance ($s^2$) is plugged in where the population standard deviation would normally go.

Essential Review: Sample Variability (Chapter 4)

• Knowledge of Chapter 4 concepts (Sum of Squares, Variance, and Standard Deviation) is mandatory for success with T-tests. Specifically, students must be able to convert between these values efficiently.
• Key Terms:
  • $SS$: Sum of Squares.
  • $s^2$: Sample Variance.
  • $s$: Sample Standard Deviation.
  • $n$: Sample size.
  • $df$ ($n-1$): Degrees of freedom.

Conversion Relationships and Formulas

• From $SS$ to $s^2$: Divide $SS$ by $(n - 1)$.
  • Formula: $s^2 = \frac{SS}{n - 1}$
• From $s^2$ to $s$: Take the square root of the variance.
  • Formula: $s = \sqrt{s^2}$
• From $s$ to $s^2$: Square the standard deviation.
  • Formula: $s^2 = (s)^2$
• From $s^2$ to $SS$: Multiply the variance by $(n - 1)$.
  • Formula: $SS = s^2 \times (n - 1)$

Variance and Standard Deviation Practice Scenarios

Practice Problem 1

• Given: $SS = 160$, $n = 32$.
• Step 1: Calculate $s^2$: $160 / (32 - 1) = 160 / 31 = 5.16$.
• Step 2: Calculate $s$: $\sqrt{5.16} = 2.27$.

Practice Problem 2

• Given: $s^2 = 20$, $n = 11$.
• Step 1: Calculate $SS$: $20 \times (11 - 1) = 20 \times 10 = 200$.
• Step 2: Calculate $s$: $\sqrt{20} = 4.47$.

Practice Problem 3

• Given: $s = 1.5$, $n = 10$.
• Step 1: Calculate $s^2$: $(1.5)^2 = 2.25$.
• Step 2: Calculate $SS$: $2.25 \times (10 - 1) = 2.25 \times 9 = 20.25$.

Purpose and Logic of the One-Sample T-Test

• Goal: To compare a single sample to a general population to determine if the sample is significantly different.
• Context of Comparisons: The comparison is only meaningful if the sample is unique in some way. Usually, this uniqueness comes from a treatment (e.g., a new medication, therapy, or experimental condition).
• Control Group vs. Experimental Group:
  • In a one-sample test, the "control group" is effectively the general population ($\mu$).
  • The "experimental group" is the sample that receives the treatment.
• Conditions for Use:
  • Use a One-Sample Z-test if you know both $\mu$ and $\sigma$.
  • Use a One-Sample T-test if you know only $\mu$ and must estimate the variability using the sample.

Formulas for the One-Sample T-Test

Estimated Standard Error ($s_M$)

• In T-tests, standard error is "estimated" because we use the sample standard deviation ($s$) as an estimate for the population standard deviation ($\sigma$).
• Formula: $s_M = \frac{s}{\sqrt{n}}$

T-Obtained ($t_{obtained}$)

• The t-statistic measures the distance between the sample mean and the population mean in terms of estimated standard error units.
• Formula: $t = \frac{M - \mu}{s_M}$
• Expanded Calculation Formula: $t = \frac{M - \mu}{s / \sqrt{n}}$

Characteristics of T-Distributions

• The T-distribution represents the distribution of all possible T-statistics for a specific sample size ($n$) drawn from a population.
• Key Fact 1: Shape Comparison: A T-distribution is always flatter and wider than the normal (Z) distribution.
• Key Fact 2: Sample Size Effect: The smaller the sample size ($n$), the flatter and wider the distribution becomes. As sample size increases, the T-distribution begins to look more like a normal distribution.
• Reason for Width/Flatness: The T-formula has more variability than the Z-formula. In the Z-formula, only the sample mean ($M$) varies from sample to sample. In the T-formula, both the sample mean ($M$) and the sample standard deviation ($s$) vary, introducing extra uncertainty.

Using the T-Table and Degrees of Freedom ($df$)

• To find a critical value on the T-table, three pieces of information are required:
  1. Alpha Level ($\alpha$): Commonly $0.05$ or $0.01$.
  2. Number of Tails: One-tailed (directional) or two-tailed (non-directional).
  3. Degrees of Freedom ($df$): This identifies the specific T-distribution in the "family" of distributions.

Degrees of Freedom Formula (Chapter 9)

• For a one-sample T-test: $df = n - 1$.

Directionality and Signs

• Two-tailed test: Use both $\pm$ signs for critical values.
• One-tailed test (Predicted Increase): Critical value is positive ($+$).
• One-tailed test (Predicted Decrease): Critical value is negative ($-$).

The "Next Lower" Rule for the T-Table

• Rule: If your exact degrees of freedom ($df$) are not listed on the table, you must use the next lower listed value.
• Example: If your calculated $df = 39$ but the table only lists $df = 30$ and $df = 40$, you must select $df = 30$.
• Reasoning: This is a conservative approach. Using a higher $df$ than you actually have makes it easier to reject the null hypothesis, thereby increasing the risk of a Type I error. Using the lower value prevents this bias.