AP Statistics (AP Stats): Conditions for Inference (Random, Normal, Independent)

AP Statistics – Conditions for Inference (Random, Normal, Independent)

AP Statistics: Conditions for Inference – Exam-Ready Study Guide

What This Is

Conditions for inference are the "green light" checks that allow us to use confidence intervals and significance tests. Without verifying these, our results may be invalid. For example, if a pharmaceutical company tests a new drug on 50 volunteers but the sample isn’t random, the results can’t be generalized to the entire population. Similarly, if we survey 1000 voters but the data isn’t independent (e.g., surveying entire households), our margin of error will be misleading. The AP exam always expects you to check these conditions before performing inference.

Key Terms & Formulas

• Random Condition: Data must come from a well-designed random sample or randomized experiment. Without randomness, results may be biased.

• Example: "Subjects were randomly assigned to treatment/control groups."

• Independent Condition (10% Rule):

• For sampling without replacement, check if n-0.10N (sample size-10% of population).
• Ensures observations are independent (no "double-counting").

• Often forgotten in FRQs!

• Normal Condition (for means):

• Large sample (n-30): Central Limit Theorem (CLT) applies; sampling distribution of x? is approx. normal.
• Small sample (n < 30): Check if population is normal (given in problem) or if sample data is roughly symmetric with no outliers (use histogram/boxplot).

• Formula: If population is normal, x? ~ N(?, ?/?n).

• Normal Condition (for proportions):

• np-10 and n(1-p)-10 (for confidence intervals, use p?; for tests, use p?).
• Ensures sampling distribution of p? is approx. normal.

• Example: For a 95% CI with p? = 0.4 and n = 50, check 50(0.4)-10 and 50(0.6)-10.

• One-Sample t-Test for ?:

• H?:-= (null hypothesis)
• H?:-? (or <, > for one-sided)
• Test statistic: t = (x? – ) / (s/?n), df = n – 1

• Calculator: T-Test (STAT-TESTS-2)

• One-Sample z-Test for p:

• H?: p = p?
• H?: p-p? (or <, >)
• Test statistic: z = (p? – p?) / ?(p?(1-p?)/n)

• Calculator: 1-PropZTest (STAT-TESTS-5)

• Confidence Interval for? (t-interval):

• x? ± t*(s/?n), df = n – 1

• Calculator: TInterval (STAT-TESTS-8)

• Confidence Interval for p (z-interval):

• p? ± z* ?(p?(1-p?)/n)

• Calculator: 1-PropZInt (STAT-TESTS-A)

• BINS Mnemonic (for proportions):

• Binary (success/failure)
• Independent (10% rule)
• Normal (np-10, n(1-p)-10)

• Sample size (n-30 for means, or check normality)

• LINER Mnemonic (for regression inference):

• Linear relationship (check scatterplot/residuals)
• Independent observations (10% rule)
• Normal residuals (check histogram/Q-Q plot)
• Equal variance (residuals have consistent spread)
• Random sampling/experiment

Step-by-Step / Process Flow

How to tackle an FRQ asking for inference (e.g., "Is there convincing evidence that the true mean differs from 50?")

1. State Hypotheses (for tests) or Parameter (for intervals):
2. Test: H?:-= 50, H?:-? 50

3. Interval: "We want to estimate the true mean ?."

4. Check Conditions (Write them out!):

5. Random: "The problem states the sample was randomly selected."
6. Independent: "n = 40-10% of all [population], so observations are independent."

7. Normal:

   • For means: "n = 40-30, so CLT applies (or: sample data is roughly symmetric with no outliers)."
   • For proportions: "np? = 20-10 and n(1-p?) = 20-10."

8. Name the Procedure:

9. "We will use a one-sample t-test for? (or z-test for p)."

10. Compute Test Statistic / Interval:

11. Calculator: Use T-Test or 1-PropZTest (for tests) or TInterval/1-PropZInt (for intervals).

12. By hand: Plug into formulas (rare on FRQs, but know them!).

13. Find p-value / Critical Value:

14. Calculator: Reports p-value automatically.

15. By hand: Use tcdf or normalcdf (2nd-VARS).

16. Conclusion in Context:

17. Test: "Since p-value = 0.02 <-= 0.05, we reject H?. There is convincing evidence that the true mean differs from 50."
18. Interval: "We are 95% confident the true mean is between 48 and 52."

Common Mistakes

• Mistake: Forgetting to check the 10% condition for independence.

• Correction: Always write, "n-10% of N" (even if the problem doesn’t mention population size). The AP exam loves to test this.

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

• Correction: For means, always use t unless-is given (rare on AP). For proportions, always use z.

• Mistake: Checking np-10 with p? instead of p? in a hypothesis test.

• Correction: For tests, use p? (null value). For intervals, use p?.

• Mistake: Saying "the data is normal" instead of "the sampling distribution is approximately normal."

• Correction: Conditions are about the sampling distribution, not the sample data (though we use sample data to check).

• Mistake: Skipping the random condition because the problem says "random sample."

• Correction: Still write it out! The AP rubric requires you to state all conditions explicitly.

AP Exam Insights

• FRQs often combine conditions with inference: You’ll be asked to "check conditions" before computing a test/interval. Always write them out in order (Random-Independent-Normal).
• Tricky distinction: z vs. t
• z: Proportions, or means when-is known (rare).
• t: Means when-is unknown (almost always).
• Normal condition for small samples: If n < 30, you must justify normality (e.g., "The problem states the population is normal" or "The sample data is roughly symmetric with no outliers").
• Calculator pitfall: TInterval and T-Test require df = n – 1. If you forget, the calculator won’t remind you!
• Confidence level vs. confidence interval: The level (e.g., 95%) is the success rate of the method; the interval is the range of plausible values.

Quick Check Questions

1. Multiple Choice: A researcher wants to test H?: p = 0.6 vs. H?: p < 0.6 using a sample of 50 people, where 24 had a positive response. Which condition is not met?
2. (A) Random
3. (B) Independent
4. (C) Normal
5. (D) All conditions are met.

Answer: (C) Normal. Check: np? = 50(0.6) = 30-10, but n(1-p?) = 50(0.4) = 20-10. However, p? = 24/50 = 0.48, so np? = 24 < 10 (for intervals), but for tests, we use p?. Correction: Actually, np? = 30-10 and n(1-p?) = 20-10, so the condition is met. The correct answer is (D). (This is a trick question to test your understanding of p? vs. p?!)

1. FRQ Part: A factory claims its lightbulbs last 1000 hours on average. A consumer group tests 25 bulbs and finds x? = 990 hours, s = 20 hours. They perform a test at-= 0.05.
2. Part (a): State the hypotheses.
3. Part (b): Check conditions for inference.

Answer: - (a) H?:-= 1000, H?:-? 1000. - (b) - Random: "The problem states the bulbs were randomly selected." - Independent: "25-10% of all bulbs produced, so observations are independent." - Normal: "n = 25 < 30, but the problem states the population is normal (or: the sample data is roughly symmetric with no outliers)."

1. Multiple Choice: Which of the following is not a condition for a one-sample t-interval for
2. (A) The sample is random.
3. (B) The population standard deviation is known.
4. (C) The sample size is large or the population is normal.
5. (D) The sample size is less than 10% of the population.

Answer: (B). The t-interval is used when-is unknown.

Last-Minute Cram Sheet

1. Conditions for inference: Random, Independent (10% rule), Normal (np-10, n-30, or check normality).
2. BINS: Binary, Independent, Normal, Sample size (for proportions).
3. LINER: Linear, Independent, Normal residuals, Equal variance, Random (for regression).
4. z vs. t: z for proportions/known ?; t for means/unknown ?.
5. 10% rule: n-0.10N ( always check, even if not mentioned!).
6. Normal condition for means: n-30 (CLT) or population normal.
7. Normal condition for proportions: np-10 and n(1-p)-10 (use p? for tests, p? for intervals).
8. Calculator commands:
9. T-Test (means), 1-PropZTest (proportions)
10. TInterval, 1-PropZInt
11. invT(area, df) for critical t*
12. Degrees of freedom: df = n – 1 for one-sample t.
13. Always state conditions explicitly in FRQs! The AP rubric requires it.