#AP Statistics: T-Tests - Your Ultimate Study Guide 🚀

Hey there, future AP Stats master! This guide is designed to be your go-to resource for t-tests, especially as you gear up for the exam. Let's break down these concepts and make sure you're feeling confident and ready. Remember, you've got this!

#Introduction to T-Tests

T-tests are all about comparing means. Specifically, they help us determine if the difference between a sample mean and a hypothesized population mean is statistically significant. This is a BIG topic, so let's get started!

#

A t-score is your test statistic when you're working with a sample mean. It tells you how many standard errors away your sample mean is from the hypothesized population mean. The bigger the t-score, the more significant the difference. Think of it as a measure of "how unusual" your sample mean is if the null hypothesis is true.

#Example: Ricardo's Oranges 🍊

Ricardo has 30 oranges. The bag says they average 4.5 oz, but his sample averages 4.65 oz with a standard deviation of 0.8 oz. Let's calculate the t-score:

$$t = \frac{4.65 - 4.5}{\frac{0.8}{\sqrt{30}}} = 1.027$$

#Calculating P-Values

The p-value tells you the probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true. It's the key to making a decision about your hypothesis!

#One-Tailed vs. Two-Tailed Tests 🦊

• One-Tailed Test: Use this when your alternative hypothesis is directional (e.g., $\mu > X$ or $\mu < X$). You're only looking for evidence in one "tail" of the distribution.
• Two-Tailed Test: Use this when your alternative hypothesis is non-directional (e.g., $\mu \neq X$). You're looking for evidence in either tail of the distribution.

#Using the T-Table

1. Degrees of Freedom (df): $df = n - 1$. For Ricardo's oranges, $df = 30 - 1 = 29$
2. Find Your T-Score: Locate your t-score on the t-table using the correct df.
3. Estimate P-Value: The t-table gives you a range for the p-value. If your t-score falls between two values on the table, your p-value falls between the corresponding probabilities.

#Example: Finding the P-Value for Ricardo's Oranges

Our t-score is 1.027, and df = 29. Looking at the t-table, we find that the p-value is approximately 0.15 for a one-tailed test. For a two-tailed test, we double this to get 0.3.

#Using Your Calculator 📱

Your TI-84 calculator is a lifesaver for t-tests! Here’s how to use it:

1. STAT Menu: Go to STAT, then TESTS.
2. Select T-Test: Choose option 2: T-Test.
3. Input Data: You can either enter summary stats (like we did with Ricardo's oranges) or input raw data into lists.
4. Calculate: The calculator will give you the t-score and p-value. Don't forget to include the degrees of freedom in your written work!

#Drawing Conclusions Using P-Values 🍀

The p-value is your guide to making a decision about the null hypothesis. Compare your p-value to your significance level ($\alpha$, usually 0.05).

• p < <math-inline>\alpha: Reject the null hypothesis ($H_0$). You have sufficient evidence to support the alternative hypothesis ($H_a$).
• p > <math-inline>\alpha: Fail to reject the null hypothesis ($H_0$). You do not have sufficient evidence to support the alternative hypothesis ($H_a$).

#Templates for Conclusions

• p < <math-inline>\alpha: "Since p < <math-inline>\alpha, we reject $H_0$. We have convincing evidence at the <math-inline>\alpha level that (context of $H_a$)."
• p > <math-inline>\alpha: "Since p > <math-inline>\alpha, we fail to reject $H_0$. We do not have convincing evidence at the <math-inline>\alpha level that (context of $H_a$)."

#Final Exam Focus

Okay, let's talk about what's most important for the exam:

• T-Test for Means: This is a core concept. Make sure you understand how to calculate t-scores, p-values, and draw conclusions.
• One-Tailed vs. Two-Tailed Tests: Know when to use each and how it affects your p-value.
• Calculator Skills: Be proficient with your calculator for t-tests. It's a time-saver!
• Context: Always interpret your results in the context of the problem. Don't just give numbers; explain what they mean.
• Assumptions: Remember to check the conditions for t-tests (randomness, independence, normality).

#Practice Questions

Practice Question

Multiple Choice Questions

1. A researcher is testing the hypothesis that the average height of adult women is greater than 5'4". They collect a random sample of 100 women and find a sample mean height of 5'5" with a standard deviation of 3 inches. What is the t-score for this test? (A) 1.00 (B) 2.00 (C) 3.00 (D) 4.00 (E) 5.00

2. A one-sample t-test is performed with a sample size of 25. The calculated t-statistic is 2.3. What is the p-value if it is a two-tailed test? (A) Between 0.01 and 0.02 (B) Between 0.02 and 0.05 (C) Between 0.05 and 0.10 (D) Between 0.10 and 0.20 (E) Greater than 0.20

3. A company claims that their batteries last an average of 50 hours. A consumer group tests a sample of 36 batteries and finds a sample mean of 48 hours with a sample standard deviation of 5 hours. What is the p-value for testing the claim that the average battery life is less than 50 hours? (A) 0.01 (B) 0.02 (C) 0.04 (D) 0.05 (E) 0.06

Free Response Question

A researcher wants to determine if the mean weight of a certain species of fish in a particular lake is different from 10 pounds. They collect a random sample of 40 fish and find a sample mean weight of 10.5 pounds with a standard deviation of 2 pounds.

(a) State the null and alternative hypotheses. (b) Calculate the test statistic. (c) Determine the p-value. (d) State the conclusion in the context of the problem, using a significance level of 0.05. Scoring Rubric

(a) Hypotheses (1 point)

• 1 point for correct null and alternative hypotheses.
  • $H_0: \mu = 10$
  • $H_a: \mu \neq 10$

(b) Test Statistic (1 point)

• 1 point for correctly calculating the t-score.
  • $t = \frac{10.5 - 10}{\frac{2}{\sqrt{40}}} = 1.58$

(c) P-Value (1 point)

• 1 point for correctly determining the p-value.
  • $df = 39$
  • p-value is approximately 0.122 (two-tailed)

(d) Conclusion (1 point)

• 1 point for correct conclusion in context.
  • Since p > 0.05, we fail to reject the null hypothesis. We do not have convincing evidence that the mean weight of the fish is different from 10 pounds.

Remember, you've got this! Keep practicing, stay confident, and you'll do great on the AP Statistics exam. Good luck! 🎉