The Single-Sample t Test: Concepts, Calculations, and Interpretation

Chapter 9: The Single-Sample t Test

Introduction to the Single-Sample t Test

The single-sample t test is a statistical method used to determine whether the mean of a single sample differs significantly from a known or hypothesized population mean when the population standard deviation is unknown. This test is fundamental in inferential statistics and is widely used in behavioral sciences and general chemistry for data analysis.

• Purpose: To compare a sample mean to a population mean when the population standard deviation is not available.

• Application: Used when only sample data are available and the population parameters must be estimated.

Wider and Flatter t Distributions

t distributions are similar to the normal distribution but are wider and have heavier tails, especially with smaller sample sizes. As the sample size increases, the t distribution approaches the normal (z) distribution.

• Key Point: The shape of the t distribution depends on the degrees of freedom (df), which is related to sample size.

• Example: A t distribution with 2 individuals is much wider and flatter than one with 30 individuals.

The t Statistic

The t statistic is used to test hypotheses about means when the population standard deviation is unknown. As sample size increases, the sample standard deviation (s) approaches the population standard deviation (σ), making t and z statistics more similar.

• Definition: The t statistic measures the difference between the sample mean and the population mean, scaled by the standard error.

• Formula: $t = \frac{M - \mu_M}{S_M}$

The t Distributions

t distributions are used when the population parameters are not known. They allow estimation of the population standard deviation from the sample.

• Population Standard Deviation: $SD = \sqrt{\sum (X - \mu)^2 / N}$

• Sample Standard Deviation: $s = \sqrt{\sum (X - M)^2 / (N-1)}$

Calculating the Estimated Population Standard Deviation

To estimate the population standard deviation from a sample, follow these steps:

1. Step 1: Calculate the sample mean ($M$).

2. Step 2: Use the sample mean in the corrected standard deviation formula.

• Formula: $s = \sqrt{\sum (X - M)^2 / (N-1)}$

• Example Calculation: $M = \frac{8 + 12 + 16 + 12 + 14}{5} = 12.4$ $s = \sqrt{35.2 / (5-1)} = \sqrt{8.8} = 2.97$

Calculating Standard Error for the t Statistic

The standard error of the mean quantifies the variability of the sample mean.

• Formula: $S_M = \frac{s}{\sqrt{N}}$

• t Statistic Formula: $t = \frac{M - \mu_M}{S_M}$

• Example: $S_M = \frac{2.97}{\sqrt{5}} = 1.33$ $t = \frac{12.4 - 11}{1.33} = 1.05$

Hypothesis Tests: The Single-Sample t Test

The single-sample t test is used when the population mean is known but the standard deviation is not. The degrees of freedom (df) are calculated as $N-1$.

• Degrees of Freedom: $df = N - 1$

Excerpt from the t Table

The t table provides critical values for different degrees of freedom and significance levels. These values are used to determine whether the calculated t statistic is significant.

df	One-Tailed Tests: 0.10	One-Tailed Tests: 0.05	One-Tailed Tests: 0.01	Two-Tailed Tests: 0.10	Two-Tailed Tests: 0.05	Two-Tailed Tests: 0.01
1	3.078	6.314	31.821	6.314	12.706	63.657
2	1.886	2.920	6.965	2.920	4.303	9.925
3	1.638	2.353	4.541	2.353	3.182	5.841
4	1.533	2.132	3.747	2.132	2.776	4.604
5	1.476	2.015	3.365	2.015	2.571	4.032

The Six Steps of Hypothesis Testing

Hypothesis testing using the single-sample t test involves six systematic steps:

1. Identify populations, the distribution, and assumptions.

2. State the null and research hypotheses.

3. Determine characteristics of the comparison distribution.

4. Determine critical values, or cutoffs.

5. Calculate the test statistic.

6. Make a decision.

Step-by-Step Example: Single-Sample t Test

Step 1: Identify Populations, Distribution, and Assumptions

• Population 1: All clients at a counseling center who sign a contract to attend at least 10 sessions.

• Population 2: All clients who do not sign the contract.

• Comparison Distribution: Distribution of means.

• Test Used: Single-sample t test.

Step 2: State the Hypotheses

• Null Hypothesis ($H_0$): $\mu_1 = \mu_2$

• Alternative Hypothesis ($H_1$): $\mu_1 \neq \mu_2$

Step 3: Determine Characteristics of the Comparison Distribution

• Sample Mean: $\mu_M = 4.6$

• Sample Standard Deviation: $s = 1.114$

• Standard Error: $S_M = \frac{2.490}{\sqrt{5}} = 1.114$

Step 4: Determine Critical Values

• Degrees of Freedom: $df = N - 1 = 5 - 1 = 4$

• Critical Values: For a two-tailed test at 95% confidence, $\pm 2.776$

Step 5: Calculate the Test Statistic

• Formula: $t = \frac{M - \mu_M}{S_M}$

• Example Calculation: $t = \frac{7.8 - 4.6}{1.114} = 2.873$

Step 6: Make a Decision

• If $|t|$ exceeds the critical value, reject the null hypothesis.

• In this example, $2.873 > 2.776$, so the null hypothesis is rejected.

Calculating Confidence Intervals for a Single-Sample t Test

Confidence intervals provide a range of values within which the population mean is likely to fall.

1. Step 1: Draw a picture of the t distribution including the confidence interval.

2. Step 2: Indicate the bounds of the confidence interval.

3. Step 3: Look up the t statistics for the middle 95%.

4. Step 4: Convert the t value into raw means: $M_{lower} = M_{sample} - t_{critical} \times S_M$ $M_{upper} = M_{sample} + t_{critical} \times S_M$

5. Step 5: Check that the confidence interval makes sense.

• Example Calculation: $M_{lower} = 7.8 - 2.776 \times 1.114 = 4.71$ $M_{upper} = 7.8 + 2.776 \times 1.114 = 10.89$

• Interpretation: The sample mean falls exactly in the middle of the interval, and the interval ranges from 3.09 below to 3.09 above the sample mean.

Interpretation of Confidence Interval

If we were to repeatedly sample five students from the same population, the 95% confidence interval would include the population mean 95% of the time.

Calculating Effect Size

Effect size quantifies the magnitude of the difference between the sample mean and the population mean.

• Formula: $d = \frac{M - \mu}{s}$

• Example Calculation: $d = \frac{7.8 - 4.6}{2.490} = 1.29$

Additional info: These notes are based on behavioral science statistics but the single-sample t test is a core concept in general chemistry and other sciences for analyzing experimental data.