AP Statistics (AP Stats): t?Distributions and Degrees of Freedom

AP Statistics – t?Distributions and Degrees of Freedom

AP Statistics: t-Distributions and Degrees of Freedom – Exam-Ready Study Guide

What This Is

The t-distribution is a probability distribution used when estimating a population mean (?) from a small sample or when the population standard deviation (?) is unknown. Unlike the normal (z) distribution, the t-distribution accounts for extra uncertainty by adjusting its shape based on degrees of freedom (df = n – 1). This is essential for confidence intervals and hypothesis tests about means (e.g., testing whether a new teaching method improves test scores, estimating the average battery life of smartphones, or comparing the effectiveness of two drugs). The AP exam frequently tests your ability to choose between z and t, compute df, and interpret results in context.

Key Terms & Formulas

• t-distribution: A symmetric, bell-shaped distribution with heavier tails than the normal distribution; used when-is unknown and sample size is small (n < 30) or moderate.
• Degrees of freedom (df): For a one-sample t-test or CI, df = n – 1. Determines the shape of the t-distribution.
• One-sample t-test for ?:
• Test statistic: ( t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} )
  • ( \bar{x} ) = sample mean, ( \mu_0 ) = hypothesized population mean, ( s ) = sample standard deviation, ( n ) = sample size.
• Hypotheses: ( H_0: \mu = \mu_0 ) vs. ( H_a: \mu \neq \mu_0 ) (or one-sided).
• Calculator: T-Test (STAT-TESTS-2:T-Test).
• One-sample t-interval for ?:
• Formula: ( \bar{x} \pm t^* \cdot \frac{s}{\sqrt{n}} )
  • ( t^* ) = critical t-value (from invT).
• Calculator: TInterval (STAT-TESTS-8:TInterval).
• Critical t-value (t*): Use invT(area to left, df) on TI-84. For a 95% CI with df=19, use invT(0.975, 19).
• Conditions for t-procedures:
• Random: Data comes from a random sample or randomized experiment.
• Independent: Sample size ( n \leq 10\% ) of population (if sampling without replacement).
• Normal/Large Sample: Population is normal or ( n \geq 30 ) (Central Limit Theorem). For ( n < 30 ), check for skewness/outliers in a graph.
• Matched pairs t-test: Treat differences as a single sample. df = n_pairs – 1.
• Two-sample t-test for ( \mu_1 - \mu_2 ):
• Test statistic: ( t = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{ \sqrt{ \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} } } )
• df: Use calculator (conservative df = smaller of ( n_1 - 1 ) or ( n_2 - 1 )).
• Calculator: 2-SampTTest (STAT-TESTS-4:2-SampTTest).
• Two-sample t-interval for ( \mu_1 - \mu_2 ):
• Formula: ( (\bar{x}_1 - \bar{x}_2) \pm t^* \cdot \sqrt{ \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} } )
• Calculator: 2-SampTInt (STAT-TESTS-0:2-SampTInt).

Step-by-Step / Process Flow

Solving a One-Sample t-Test FRQ

1. State hypotheses:
2. ( H_0: \mu = \mu_0 ) (e.g., "The true mean battery life is 10 hours").
3. ( H_a: \mu \neq \mu_0 ) (or ( < ) or ( > ) for one-sided tests).
4. Check conditions:
5. Random: "The batteries were randomly selected."
6. Independent: ( n \leq 10\% ) of all batteries (if sampling without replacement).
7. Normal/Large Sample: "The sample size is small (n=20), but a histogram of the data shows no strong skewness or outliers."
8. Compute test statistic:
9. Use ( t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} ) or T-Test on calculator.
10. Find p-value:
11. Use tcdf(lower, upper, df) for two-sided tests (multiply by 2 if needed).
12. Example: For ( t = 2.3 ) and df=19, p-value = 2 * tcdf(2.3, 1E99, 19)-0.033.
13. Make conclusion in context:
14. Compare p-value to ( \alpha ) (e.g., 0.05). If p-value < ( \alpha ), reject ( H_0 ).
15. "Since the p-value (0.033) is less than 0.05, we reject ( H_0 ). There is convincing evidence that the true mean battery life differs from 10 hours."

Solving a One-Sample t-Interval FRQ

1. State procedure: "We will construct a one-sample t-interval for ?."
2. Check conditions (same as above).
3. Compute interval:
4. Use ( \bar{x} \pm t^* \cdot \frac{s}{\sqrt{n}} ) or TInterval on calculator.
5. Example: For ( \bar{x} = 9.8 ), ( s = 1.2 ), ( n = 20 ), 95% CI = (9.21, 10.39).
6. Interpret in context:
7. "We are 95% confident that the true mean battery life is between 9.21 and 10.39 hours."

Common Mistakes

• Mistake: Using z-procedures instead of t when-is unknown.

• Correction: Always use t for means when-is unknown, regardless of sample size. The z-distribution assumes-is known (rare in real-world problems).

• Mistake: Forgetting to check the Normal/Large Sample condition for small samples.

• Correction: For ( n < 30 ), always graph the data (histogram, boxplot) to check for skewness/outliers. If the data is skewed, t-procedures may not be valid.

• Mistake: Miscalculating degrees of freedom for two-sample t-tests.

• Correction: Use the calculator’s df (not ( n_1 + n_2 - 2 )) unless the problem states equal variances. For matched pairs, df = ( n_{\text{pairs}} - 1 ).

• Mistake: Interpreting a confidence interval as "the probability that-is in the interval."

• Correction: The interval either contains-or it doesn’t. Correct interpretation: "We are 95% confident that the true mean [context] is between [lower] and [upper]."

• Mistake: Using the wrong t* for a confidence interval (e.g., using invT(0.95, df) for a 90% CI).

• Correction: For a C% CI, use invT((1 + C)/2, df). For 90% CI, use invT(0.95, df).

AP Exam Insights

• z vs. t: The AP exam loves to test whether you use z or t. Use t when-is unknown (almost always). Use z only for proportions or when-is given (rare).
• Degrees of freedom: Memorize df for common scenarios:
• One-sample: ( n - 1 )
• Matched pairs: ( n_{\text{pairs}} - 1 )
• Two-sample: Use calculator (or conservative df = smaller of ( n_1 - 1 ) or ( n_2 - 1 )).
• Calculator pitfalls:
• For TInterval, enter ( \bar{x} ), ( s ), and ( n ) (not ?).
• For 2-SampTTest, choose "Pooled: No" unless the problem states equal variances.
• FRQ setups:
• Often involve matched pairs (e.g., "before and after" measurements) or two-sample comparisons (e.g., "treatment vs. control").
• May ask you to justify conditions (e.g., "Explain why the Normal condition is met").

Quick Check Questions

1. Multiple Choice: A researcher tests ( H_0: \mu = 50 ) vs. ( H_a: \mu > 50 ) using a sample of 25 observations. The sample mean is 52, and the sample standard deviation is 8. What is the test statistic?
2. (A) ( \frac{52 - 50}{8 / \sqrt{25}} )
3. (B) ( \frac{52 - 50}{8 / \sqrt{24}} )
4. (C) ( \frac{52 - 50}{8} )

5. Answer: (A). The test statistic is ( t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} ), and df = 24 (but not needed for the test statistic).

6. FRQ Part: A 95% confidence interval for the mean weight loss (in pounds) of a diet program is (3.2, 8.6). Interpret this interval in context.

7. Answer: "We are 95% confident that the true mean weight loss for all participants in the diet program is between 3.2 and 8.6 pounds." (Must include "true mean," "confidence," and context.)

8. Multiple Choice: Which of the following is not a condition for using a one-sample t-test?

9. (A) The data comes from a random sample.
10. (B) The sample size is less than 10% of the population.
11. (C) The population standard deviation is known.
12. (D) The data shows no strong skewness or outliers (for small samples).
13. Answer: (C). The t-test is used because-is unknown.

Last-Minute Cram Sheet

1. t-distribution: Use when-is unknown; shape depends on df = n – 1.
2. One-sample t-test: ( t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} ); T-Test on calculator.
3. One-sample t-interval: ( \bar{x} \pm t^* \cdot \frac{s}{\sqrt{n}} ); TInterval.
4. df for one-sample: ( n - 1 ).
5. df for matched pairs: ( n_{\text{pairs}} - 1 ).
6. Two-sample t-test: 2-SampTTest; df from calculator.
7. Conditions: RIN (Random, Independent, Normal/Large Sample).
8. Normal condition: For ( n < 30 ), check graph for skewness/outliers.
9. t* for CI: invT((1 + C)/2, df) (e.g., 95% CI-invT(0.975, df)).
10. Never use z for means when-is unknown! Always check the 10% condition for independence.