AP Statistics (AP Stats): Hypothesis Test for One Mean (One?Sample t?test)

AP Statistics – Hypothesis Test for One Mean (One?Sample t?test)

AP Statistics: Hypothesis Test for One Mean (One-Sample t-test) – Exam-Ready Study Guide

What This Is

A one-sample t-test determines whether a sample mean significantly differs from a hypothesized population mean when the population standard deviation (?) is unknown (and estimated by the sample standard deviation, s). This is essential on the AP exam because it’s one of the most common inference procedures, appearing in FRQs and MCQs every year. Real-world example: A school administrator claims the average SAT score of students at their school is 1200. You collect a random sample of 30 students and want to test if the true mean score differs from 1200.

Key Terms & Formulas

• Null Hypothesis (H?): ? = (The population mean equals the hypothesized value.)
• Alternative Hypothesis (H?): ?- (two-tailed), ? > (right-tailed), or ? < (left-tailed).
• Test Statistic (t): t = (x? – ) / (s / ?n)
• x? = sample mean
• = hypothesized population mean
• s = sample standard deviation
• n = sample size
• Degrees of Freedom (df): df = n – 1 (for one-sample t-test).
• P-value: Probability of observing a test statistic as extreme as (or more extreme than) the one calculated, assuming H? is true. Found using tcdf(lower, upper, df) on TI-84.
• Significance Level (?): Threshold for rejecting H? (commonly ? = 0.05).
• Conditions for Inference (LINER):
• Linear (data roughly symmetric, no outliers)
• Independent (10% condition: n-0.10N if sampling without replacement)
• Normal (or n-30 by CLT, or check normality with a graph)
• Experimental/Observational: Data comes from a random sample or randomized experiment.
• Random: Data is collected randomly.
• TI-84 Commands:
• T-Test: STAT-TESTS-2:T-Test (enter , x?, s, n, and H?).
• tcdf: 2nd-DISTR-6:tcdf(lower, upper, df) (for p-value).
• invT: 2nd-DISTR-4:invT(area, df) (for critical t-values).

Step-by-Step / Process Flow

Follow these steps for any one-sample t-test FRQ:

1. State Hypotheses

2. Write H? and H? in context (e.g., H?:-= 1200 vs. H?:-? 1200, where ? = true mean SAT score).

3. Check Conditions (LINER)

4. Random: “The problem states the sample was randomly selected.”
5. Independent: “The sample size (30) is less than 10% of all students at the school.”

6. Normal: “The sample size is ?30, so the sampling distribution of x? is approximately normal by the CLT.” (Or: “A boxplot shows no strong skewness or outliers.”)

7. Calculate Test Statistic

8. Use the formula t = (x? – ) / (s / ?n) or T-Test on TI-84.

9. Example: If x? = 1180, s = 100, n = 30, and = 1200, then t = (1180 – 1200) / (100 / ?30)--1.095.

10. Find P-value

11. For H?:-? , use 2 * tcdf(|t|, 1E99, df) (two-tailed).
12. For H?:-> , use tcdf(t, 1E99, df).
13. For H?:-< , use tcdf(-1E99, t, df).

14. Example: For t = -1.095 and df = 29, 2 * tcdf(1.095, 1E99, 29)-0.282.

15. Make a Conclusion

16. Compare p-value to ? (usually 0.05):
    • If p-value-?: “Reject H?. There is convincing evidence that [H? in context].”
    • If p-value > ?: “Fail to reject H?. There is not convincing evidence that [H? in context].”
17. Example: “Since 0.282 > 0.05, we fail to reject H?. There is not convincing evidence that the true mean SAT score differs from 1200.”

Common Mistakes

• Mistake: Using z instead of t when-is unknown. Correction: Always use t for means when-is unknown (even if n is large). The z-test is only for proportions or when-is known.

• Mistake: Forgetting to check the 10% condition for independence. Correction: If sampling without replacement, verify n-0.10N (e.g., “30 students is less than 10% of 1000 students at the school”).

• Mistake: Misinterpreting the p-value as the probability H? is true. Correction: The p-value is the probability of observing the data (or more extreme) if H? is true, not the probability H? is true.

• Mistake: Skipping the normality check for small samples (n < 30). Correction: For small samples, always check a graph (boxplot, histogram) for skewness/outliers. If skewed, the t-test may not be valid.

• Mistake: Writing conclusions without context. Correction: Always state conclusions in terms of the problem (e.g., “There is not convincing evidence that the mean SAT score differs from 1200”).

AP Exam Insights

• z vs. t: The AP exam loves testing this distinction. Use z for proportions or when-is known; use t for means when-is unknown.
• Two-tailed vs. One-tailed: FRQs often specify H?:-? (two-tailed). If not, default to two-tailed unless the problem implies directionality.
• Calculator Pitfalls:
• T-Test gives the p-value directly—don’t recalculate it manually unless asked.
• For two-tailed tests, remember to double the p-value from tcdf.
• Context Matters: Always define ? in hypotheses (e.g., ? = true mean [variable] for [population]).

Quick Check Questions

1. MCQ: A researcher tests H?:-= 50 vs. H?:-> 50 using a sample of n = 25. The test statistic is t = 1.8. What is the p-value?
2. (A) tcdf(1.8, 1E99, 24)
3. (B) 2 * tcdf(1.8, 1E99, 24)
4. (C) tcdf(-1E99, 1.8, 24)

5. (D) 1 - tcdf(1.8, 1E99, 24) Answer: (A). For H?:-> , the p-value is the area to the right of t = 1.8.

6. FRQ Part: A sample of 40 batteries has a mean lifetime of 12.2 hours and a standard deviation of 1.5 hours. Test H?:-= 12 vs. H?:-? 12 at ? = 0.05.

7. (a) State the conditions for inference.
8. (b) Calculate the test statistic and p-value.
9. (c) State your conclusion in context. Answer:
10. (a) Random: Assume random sample. Independent: 40-10% of all batteries. Normal: n = 40-30 (CLT applies).
11. (b) t = (12.2 – 12) / (1.5 / ?40)-0.843; df = 39; p-value = 2 * tcdf(0.843, 1E99, 39)-0.404.
12. (c) Since 0.404 > 0.05, fail to reject H?. There is not convincing evidence that the true mean battery lifetime differs from 12 hours.

Last-Minute Cram Sheet

1. Formula: t = (x? – ) / (s / ?n) (df = n – 1).
2. Conditions: LINER (Linear, Independent, Normal, Experimental, Random).
3. 10% Condition: n-0.10N if sampling without replacement. Always check!
4. Normality Check: For n < 30, graph data; for n-30, CLT applies.
5. TI-84: T-Test for test statistic/p-value; tcdf for manual p-values.
6. Two-tailed p-value: 2 * tcdf(|t|, 1E99, df).
7. Conclusion Template: “Since p-value [?/>] ?, we [reject/fail to reject] H?. There [is/is not] convincing evidence that [H? in context].”
8. H? vs. H?: H? is always =; H? is ?, <, or >.
9. Don’t use z for means unless-is known.
10. Define-in hypotheses (e.g., ? = true mean [variable] for [population]).