#AP Statistics: One-Sample t-Tests - Your Ultimate Guide 🚀

Hey there, future AP Stats master! Feeling the pressure? Don't worry, we've got you covered. This guide is designed to be your go-to resource the night before the exam. Let's break down one-sample t-tests into bite-sized pieces and get you feeling confident! 💪

#Introduction to Significance Testing

#Identifying the Need for a Significance Test

• Look for these key phrases: "do the data give convincing evidence..." or "is there convincing evidence of..." 🤔
• These phrases signal that you need to perform a statistical significance test.

#Determining Data Type

• Quantitative Data: If your data is numerical (e.g., heights, weights, test scores), you'll likely be working with tests for population means.
• Categorical Data: If your data is in categories (e.g., colors, opinions, types), you'll use different tests (not covered in this guide).

#One-Sample t-Test Overview

• Used to compare a sample mean to a known or hypothesized population mean when the population standard deviation (σ) is unknown.
• We'll use t-scores because we're estimating the population standard deviation from our sample.

#Significance Level (α)

#Understanding Alpha (α)

• The significance level (α) is the probability of rejecting the null hypothesis when it's actually true. It's the risk we're willing to take of making a mistake (a Type I error).
• Think of it as the threshold for how "unusual" our sample data must be to doubt the null hypothesis. 🔝
• Commonly set at 0.05 (5%), meaning there's a 5% chance of incorrectly rejecting the null hypothesis.

#Choosing the Right α

• Too low α: Higher chance of false negatives (Type II error) - missing a real effect.
• Too high α: Higher chance of false positives (Type I error) - claiming an effect that's not there.
• 0.05 is a common balance, but the choice depends on the context of the problem.

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• The significance level (α) and confidence level are complements of each other. 🌉
• Example: α = 0.05 corresponds to a 95% confidence interval. If your sample mean falls outside the 95% CI, you'd reject the null hypothesis at α = 0.05. - If α = 0.01, it matches with a 99% confidence interval.

#Visualizing the Rejection Region

• The rejection region is the area under the t-distribution curve where, if our sample statistic falls, we reject the null hypothesis. 🙅

Caption: A visual representation of a t-distribution with a 95% confidence interval (non-shaded) and rejection regions (shaded) for a two-tailed test.

#Setting Up Hypotheses

#Null Hypothesis (H₀)

• A statement of no effect or no difference. It's what we're trying to disprove. 🥚

• Always stated as: H₀: μ = μ₀

  • μ is the population mean
  • μ₀ is the hypothesized population mean (from the claim).

• Example: If a study claims the average bag of chips has 20 chips, then H₀: μ = 20. ### Alternate Hypothesis (Hₐ)

• The opposite of the null hypothesis. It's what we're trying to find evidence for. 🐤

• Can be one of three forms:

  • Two-tailed: Hₐ: μ ≠ μ₀ (the mean is different from the hypothesized mean)
  • Left-tailed: Hₐ: μ < μ₀ (the mean is less than the hypothesized mean)
  • Right-tailed: Hₐ: μ > μ₀ (the mean is greater than the hypothesized mean)

• The choice depends on the question. If you're testing if it's different, use ≠. If you expect it to be lower, use <. If you expect it to be higher, use >.

#Example: Hypotheses in Action

• Scenario: Study claims seniors miss 5.2 school days on average. Our sample of 150 seniors has a mean of 4.1 days missed with a standard deviation of 0.4. We want to know if the true mean is less than 5.2. - Null Hypothesis (H₀): μ = 5.2
• Alternative Hypothesis (Hₐ): μ < 5.2

#Checking Conditions for Inference

Before running the test, we need to make sure our data is suitable. Just like with confidence intervals, we have three conditions. 🧪

#1. Random

• Your sample MUST be chosen randomly to make inferences about the population. 🍀
• If your sample isn't random, your results are biased and unreliable.

#2. Independence

• Samples are often taken without replacement, which technically violates independence.
• However, if the sample size is small relative to the population, it's okay to proceed. 🏁
• Rule of Thumb: The population size must be at least 10 times the sample size (10n).
• State: "It is reasonable to believe that there are at least ____ (10n) _________ (in context of our population)"

#3. Normal

• We need to ensure our sampling distribution is approximately normal so we can use the t-distribution. 🔔
• Three ways to check:
  1. Central Limit Theorem (CLT): If the sample size (n) is at least 30, the sampling distribution is approximately normal. (Most common!)
  2. Population is Normal: If the problem states the population is normally distributed, you're good to go.
  3. Sample Data: If the sample data is roughly symmetric with no outliers (check with a modified boxplot), you can assume normality.
• Only one of these needs to be met! Check in this order for efficiency.

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#High-Priority Topics

• Identifying the Correct Test: Always start by determining if you need a t-test and if it's one-sample or two-sample.
• Writing Hypotheses: Practice writing null and alternative hypotheses correctly. Pay attention to the wording of the question to choose the right alternative.
• Checking Conditions: Don't forget to check random, independent, and normal conditions. It's an easy way to get points on FRQs!
• Interpreting Results: Understand what the p-value means and how it relates to the significance level.

#Common Question Types

• Multiple Choice: Often tests your understanding of concepts and conditions.
• Free Response (FRQ): Requires you to perform the entire test, step-by-step, and interpret the results in context.

#Last-Minute Tips

• Time Management: Don't get bogged down on one question. Move on and come back if you have time.
• Common Pitfalls: Watch out for incorrect hypotheses, forgetting to check conditions, and misinterpreting p-values.
• Strategies for FRQs: Show all your work, even if you're not sure. Partial credit is your friend!

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Practice Question

Practice Questions

#Multiple Choice Questions

1. A researcher is testing the hypothesis that the mean height of adult women is 5'4" (64 inches). A random sample of 40 women is taken and the sample mean is found to be 63 inches with a standard deviation of 2.5 inches. What is the appropriate null and alternative hypothesis? (A) H₀: μ = 63, Hₐ: μ ≠ 63 (B) H₀: μ = 64, Hₐ: μ < 64 (C) H₀: μ = 64, Hₐ: μ ≠ 64 (D) H₀: μ = 63, Hₐ: μ < 63 (E) H₀: μ = 64, Hₐ: μ > 64

2. Which of the following is NOT a condition that needs to be checked before performing a one-sample t-test? (A) The data is from a random sample. (B) The population is at least 10 times the sample size. (C) The population standard deviation is known. (D) The sampling distribution is approximately normal. (E) The sample is independent.

3. A one-sample t-test is conducted with a significance level of 0.05. The p-value is calculated to be 0.02. Which of the following is the correct conclusion? (A) Fail to reject the null hypothesis; there is not enough evidence to support the alternative hypothesis. (B) Fail to reject the null hypothesis; there is enough evidence to support the alternative hypothesis. (C) Reject the null hypothesis; there is not enough evidence to support the alternative hypothesis. (D) Reject the null hypothesis; there is enough evidence to support the alternative hypothesis. (E) The test is inconclusive.

#Free Response Question

A coffee shop claims that the average amount of coffee in their "large" cup is 16 ounces. A consumer group suspects that the actual average is less than 16 ounces. They randomly sample 30 large cups of coffee and find a sample mean of 15.5 ounces with a sample standard deviation of 0.8 ounces. Conduct a hypothesis test at a significance level of 0.05 to determine if there is convincing evidence that the true average amount of coffee in a "large" cup is less than 16 ounces.

Scoring Rubric:

Part 1: Hypotheses (1 point)

• Correct null hypothesis: H₀: μ = 16
• Correct alternative hypothesis: Hₐ: μ < 16

Part 2: Conditions (3 points)

• Random: States that the sample was randomly selected (1 point)
• Independent: States that it is reasonable to assume that there are at least 300 large cups of coffee (1 point)
• Normal: States that since n = 30, the sampling distribution is approximately normal due to the Central Limit Theorem (1 point)

Part 3: Mechanics (2 points)

• Correctly calculates the t-statistic: t = (15.5 - 16) / (0.8 / sqrt(30)) = -3.42 (1 point)
• Correctly finds the p-value using a t-distribution with 29 degrees of freedom: p-value ≈ 0.001 (1 point)

Part 4: Conclusion (1 point)

• Correctly states to reject the null hypothesis because the p-value (0.001) is less than the significance level (0.05) and concludes that there is convincing evidence that the true average amount of coffee in a "large" cup is less than 16 ounces (1 point)