Introductory Statistics: Inference Hypothesis Tests One-Sample t-Test Test Statistic t x-μ₀sn p-value Decision

What Is This?

A one-sample t-test is a statistical procedure used to determine whether a sample mean is statistically different from a known population mean. It appears in exams to test your understanding of hypothesis testing, statistical significance, and the application of the t-distribution. Typical questions involve calculating the test statistic, determining the p-value, and making a decision based on the results.

Why It Matters

This topic is commonly tested in statistics exams, particularly in introductory and intermediate-level courses. It frequently appears in multiple-choice and short-answer questions, carrying moderate to high marks. The skill being tested is your ability to perform hypothesis testing and interpret statistical results, which is crucial for data analysis and decision-making in various fields.

Core Concepts

1. Hypothesis Testing: Understand the null hypothesis (H₀) and alternative hypothesis (H₁). The null hypothesis typically states that the sample mean (x̄) is equal to the population mean (μ₀).
2. Test Statistic: The formula for the test statistic is ( t = \frac{(\bar{x} - \mu_0)}{(s / \sqrt{n})} ), where x̄ is the sample mean, μ₀ is the population mean, s is the sample standard deviation, and n is the sample size.
3. p-value: The p-value is the probability of observing a test statistic as extreme as the one computed, assuming the null hypothesis is true. It helps decide whether to reject the null hypothesis.
4. Decision Rule: Compare the p-value to the significance level (α). If p-value < α, reject the null hypothesis; otherwise, do not reject it.
5. Degrees of Freedom: For a one-sample t-test, the degrees of freedom (df) is n - 1, where n is the sample size.

Prerequisites

1. Basic Statistics: Understand mean, standard deviation, and the concept of a normal distribution.
2. Hypothesis Testing: Know the basics of hypothesis testing, including null and alternative hypotheses.
3. t-Distribution: Be familiar with the t-distribution and how it differs from the normal distribution, especially in smaller sample sizes.

The Rule-Book (How It Works)

Primary Rule

The one-sample t-test compares a sample mean to a known population mean to determine if there is a significant difference. The test statistic is calculated using the formula:

[ t = \frac{(\bar{x} - \mu_0)}{(s / \sqrt{n})} ]

Sub-rules and Exceptions

1. Assumptions: The sample should be randomly selected, and the population should be normally distributed or the sample size should be large (n > 30).
2. Degrees of Freedom: Use n - 1 for the degrees of freedom.
3. Two-Tailed vs. One-Tailed Test: Decide whether the test is two-tailed (H₁: μ ≠ μ₀) or one-tailed (H₁: μ > μ₀ or μ < μ₀).

Visual Pattern

Think of the t-test as a way to measure how far the sample mean is from the population mean in terms of standard errors. The larger the t-value, the stronger the evidence against the null hypothesis.

Exam / Job / Audit Weighting

• Frequency: Common
• Difficulty Rating: Intermediate
• Question Type or Real-World Task Type: Multiple-choice, short-answer, data analysis tasks

Difficulty Level

Intermediate

Must-Know Rules, Formulas, Standards, or Principles

1. Test Statistic Formula: ( t = \frac{(\bar{x} - \mu_0)}{(s / \sqrt{n})} )
2. p-value Interpretation: If p-value < α, reject H₀.
3. Degrees of Freedom: df = n - 1

Worked Examples (Step-by-Step)

Easy

Question: A sample of 25 observations has a mean of 50 and a standard deviation of 10. Test the hypothesis that the population mean is 45 at a 5% significance level.

1. State Hypotheses: H₀: μ = 45, H₁: μ ≠ 45
2. Calculate Test Statistic:
   [ t = \frac{(50 - 45)}{(10 / \sqrt{25})} = \frac{5}{2} = 2.5 ]
3. Determine p-value: Using a t-table or calculator with df = 24, find the p-value for t = 2.5.
4. Decision: If p-value < 0.05, reject H₀.

Answer: Reject H₀ if p-value < 0.05.

Medium

Question: A researcher claims that the average height of adult males in a city is 175 cm. A random sample of 30 males has a mean height of 178 cm with a standard deviation of 8 cm. Test this claim at a 1% significance level.

1. State Hypotheses: H₀: μ = 175, H₁: μ ≠ 175
2. Calculate Test Statistic:
   [ t = \frac{(178 - 175)}{(8 / \sqrt{30})} = \frac{3}{1.46} \approx 2.05 ]
3. Determine p-value: Using a t-table or calculator with df = 29, find the p-value for t = 2.05.
4. Decision: If p-value < 0.01, reject H₀.

Answer: Do not reject H₀ if p-value > 0.01.

Hard

Question: A company claims that the average life of their light bulbs is 1000 hours. A consumer group tests 40 bulbs and finds an average life of 980 hours with a standard deviation of 120 hours. Test the company's claim at a 5% significance level.

1. State Hypotheses: H₀: μ = 1000, H₁: μ ≠ 1000
2. Calculate Test Statistic:
   [ t = \frac{(980 - 1000)}{(120 / \sqrt{40})} = \frac{-20}{19.0} \approx -1.05 ]
3. Determine p-value: Using a t-table or calculator with df = 39, find the p-value for t = -1.05.
4. Decision: If p-value < 0.05, reject H₀.

Answer: Do not reject H₀ if p-value > 0.05.

Common Exam Traps & Mistakes

1. Mistake: Forgetting to divide by the standard error.
2. Wrong Answer: Using ( t = \frac{(\bar{x} - \mu_0)}{s} )

3. Correct Approach: Use ( t = \frac{(\bar{x} - \mu_0)}{(s / \sqrt{n})} )

4. Mistake: Misinterpreting the p-value.

5. Wrong Answer: Rejecting H₀ when p-value > α.

6. Correct Approach: Reject H₀ only if p-value < α.

7. Mistake: Incorrect degrees of freedom.

8. Wrong Answer: Using df = n.

9. Correct Approach: Use df = n - 1.

10. Mistake: Not recognizing the type of test (one-tailed vs. two-tailed).

11. Wrong Answer: Using a two-tailed test when a one-tailed test is appropriate.
12. Correct Approach: Identify the type of test based on the alternative hypothesis.

Shortcut Strategies & Exam Hacks

1. Memory Aid: Remember the formula as "sample mean minus population mean, divided by standard error."
2. Elimination Strategy: If the p-value is very high, you can quickly eliminate options that suggest rejecting H₀.
3. Pattern Recognition: Look for keywords like "significance level," "sample mean," and "population mean" to identify the type of question.

Question-Type Taxonomy

1. Multiple-Choice: Choose the correct test statistic or p-value interpretation.
2. Example: What is the test statistic for a sample mean of 50, population mean of 45, sample standard deviation of 10, and sample size of 25?

3. Favored By: Introductory statistics exams.

4. Short-Answer: Calculate the test statistic and p-value.

5. Example: Given a sample mean of 178 cm, population mean of 175 cm, sample standard deviation of 8 cm, and sample size of 30, calculate the test statistic and decide whether to reject the null hypothesis at a 1% significance level.

6. Favored By: Intermediate statistics exams.

7. Data Analysis: Interpret the results of a one-sample t-test in a real-world context.

8. Example: A company claims their product lasts 1000 hours. A sample of 40 products has an average life of 980 hours with a standard deviation of 120 hours. Should the company's claim be rejected at a 5% significance level?
9. Favored By: Advanced statistics and business analytics exams.

Practice Set (MCQs)

Question 1

Question: A sample of 36 observations has a mean of 80 and a standard deviation of 15. The population mean is 75. What is the test statistic for a one-sample t-test? - A: 2.0 - B: 3.0 - C: 4.0 - D: 5.0

Correct Answer: A Explanation: ( t = \frac{(80 - 75)}{(15 / \sqrt{36})} = \frac{5}{2.5} = 2.0 ) Why the Distractors Are Tempting: B, C, and D are plausible but incorrect calculations of the test statistic.

Question 2

Question: If the p-value for a one-sample t-test is 0.03 and the significance level is 0.05, what should you do? - A: Reject the null hypothesis - B: Do not reject the null hypothesis - C: Increase the sample size - D: Recalculate the test statistic

Correct Answer: A Explanation: If p-value < α, reject H₀.
Why the Distractors Are Tempting: B suggests the opposite decision, C and D are irrelevant actions.

Question 3

Question: What is the degrees of freedom for a sample size of 20? - A: 19 - B: 20 - C: 21 - D: 18

Correct Answer: A Explanation: df = n - 1 = 20 - 1 = 19 Why the Distractors Are Tempting: B, C, and D are close but incorrect values.

Question 4

Question: A sample of 50 observations has a mean of 150 and a standard deviation of 20. The population mean is 145. What is the test statistic for a one-sample t-test? - A: 2.5 - B: 3.5 - C: 4.5 - D: 5.5

Correct Answer: A Explanation: ( t = \frac{(150 - 145)}{(20 / \sqrt{50})} = \frac{5}{2.83} \approx 1.77 ) Why the Distractors Are Tempting: B, C, and D are plausible but incorrect calculations of the test statistic.

Question 5

Question: If the p-value for a one-sample t-test is 0.10 and the significance level is 0.05, what should you do? - A: Reject the null hypothesis - B: Do not reject the null hypothesis - C: Increase the sample size - D: Recalculate the test statistic

Correct Answer: B Explanation: If p-value > α, do not reject H₀.
Why the Distractors Are Tempting: A suggests the opposite decision, C and D are irrelevant actions.

30-Second Cheat Sheet

• Test Statistic Formula: ( t = \frac{(\bar{x} - \mu_0)}{(s / \sqrt{n})} )
• p-value Interpretation: If p-value < α, reject H₀.
• Degrees of Freedom: df = n - 1
• Hypothesis Testing: Understand null and alternative hypotheses.
• Decision Rule: Compare p-value to α.
• Assumptions: Random sample, normal distribution or large sample size.
• Two-Tailed vs. One-Tailed Test: Identify based on H₁.

Learning Path

1. Beginner Foundation: Review basic statistics, including mean, standard deviation, and normal distribution.
2. Core Rules: Learn the one-sample t-test formula, p-value interpretation, and decision rules.
3. Practice: Solve multiple practice problems, focusing on calculating the test statistic and interpreting p-values.
4. Timed Drills: Practice under exam conditions to improve speed and accuracy.
5. Mock Tests: Take full-length mock exams to simulate the real exam environment.

Related Topics

1. Two-Sample t-Test: Compares means of two independent samples.
2. Relation: Both involve hypothesis testing and the t-distribution.
3. Paired t-Test: Compares means of the same group under different conditions.
4. Relation: Similar to one-sample t-test but involves paired data.
5. ANOVA: Compares means of three or more groups.
6. Relation: Extends the concept of hypothesis testing to multiple groups.