AP Statistics (AP Stats): Sampling Distribution of a Sample Mean (x?) – Central Limit Theorem

AP Statistics – Sampling Distribution of a Sample Mean (x?) – Central Limit Theorem

AP Statistics Study Guide: Sampling Distribution of a Sample Mean (x?) – Central Limit Theorem (CLT)

What This Is

The sampling distribution of a sample mean (x?) describes the distribution of all possible sample means from repeated samples of size n from a population. The Central Limit Theorem (CLT) states that, regardless of the population’s shape, the sampling distribution of x? will be approximately normal if the sample size is large enough (n-30 or the population is already normal). This is essential for constructing confidence intervals and hypothesis tests about a population mean (?). Real-world example: A factory tests whether the average weight of cereal boxes meets the advertised 16 oz. by taking random samples of 50 boxes—even if individual box weights vary, the sample mean will follow a predictable, normal distribution.

Key Terms & Formulas

• Sampling Distribution of x?: The distribution of all possible sample means from samples of size n from a population.
• Central Limit Theorem (CLT): If n is large (?30) or the population is normal, the sampling distribution of x? is approximately N(?, ?/?n).
• Mean of the Sampling Distribution (?): Always equals the population mean (?). Formula: ? = ?
• Standard Deviation of the Sampling Distribution (?): Measures spread of sample means. Formula: ? = ?/?n (called the standard error of the mean).
• Use ? = s/?n only if-is unknown and n is large (or population is normal).
• z-score for x?: Measures how many standard errors x? is from ?. Formula: z = (x? – ?) / (?/?n)
• Normal Condition for x?:
• If population is normal, sampling distribution of x? is normal for any n.
• If population is not normal, n-30 ensures normality (CLT).
• 10% Condition: If sampling without replacement, n-10% of the population to ensure independence.
• Random Condition: Sample must be randomly selected (SRS).
• Calculator Command for Probabilities (NormalCDF):
• normalcdf(lower, upper, ?, ?/?n)-Finds P(lower < x? < upper).
• Example: normalcdf(15, 17, 16, 0.5)-P(15 < x? < 17) for ?=16, ?=0.5.
• Calculator Command for Critical Values (invNorm):
• invNorm(area to left, ?, ?/?n)-Finds x? value for a given percentile.
• Example: invNorm(0.975, 16, 0.5)-97.5th percentile of x?.

Step-by-Step / Process Flow

How to solve an FRQ about the sampling distribution of x?:

1. State the Parameter & Statistic:
2. Parameter: Population mean (?).

3. Statistic: Sample mean (x?) from a sample of size n.

4. Check Conditions (RIN):

5. Random: Sample is randomly selected (SRS).
6. Independent (10% Condition): If sampling without replacement, n-10% of population.

7. Normal: Either:

   • Population is normal (stated or assumed), or
   • n-30 (CLT applies).

8. Describe the Sampling Distribution:

9. Shape: Normal (if conditions met).
10. Center: ? = ?.

11. Spread: ? = ?/?n (or s/?n if-is unknown).

12. Calculate Probabilities or Critical Values:

13. Use normalcdf or invNorm with ? =-and ? = ?/?n.

14. Example: Find P(x? > 105)-normalcdf(105, 1E99, 100, 15/?50).

15. Interpret in Context:

16. Example: "There is a 2.5% chance that a random sample of 50 boxes will have an average weight more than 105 oz if the true mean is 100 oz."

Common Mistakes

• Mistake: Forgetting to divide-by ?n when calculating ?.

• Correction: Always use ? = ?/?n (not ?). The standard error decreases as n increases.

• Mistake: Assuming the sampling distribution is normal for small n when the population is skewed.

• Correction: Only use normality if n-30 (CLT) or the population is normal. For small n from skewed populations, the sampling distribution is not normal.

• Mistake: Using-instead of s when-is unknown and n is small.

• Correction: If-is unknown and n < 30, use a t-distribution (not covered here, but important for inference).

• Mistake: Ignoring the 10% condition when sampling without replacement.

• Correction: Always check n-10% of the population to ensure independence.

• Mistake: Confusing the sampling distribution of x? with the population distribution.

• Correction: The sampling distribution is narrower (?/?n) and more normal (CLT) than the population.

AP Exam Insights

• Frequently Tested:
• Calculating probabilities for x? (e.g., "What’s the probability the sample mean is less than 50?").
• Comparing ? for different sample sizes (e.g., "How does the standard error change if n increases from 25 to 100?").

• Interpreting the CLT (e.g., "Why can we assume normality for x? even if the population is skewed?").

• Tricky Distinctions:

• z vs. t: Use z when-is known (rare on AP exam) or n is large. Use t when-is unknown and n is small (covered in inference).

• Population vs. Sampling Distribution: The population distribution can be any shape, but the sampling distribution of x? is normal (if conditions met).

• Common FRQ Setups:

• Given a population mean (?) and standard deviation (?), find the probability that x? falls in a certain range.
• Given a sample mean (x?) and n, find the probability of observing a value as extreme or more extreme (for hypothesis tests).

• Comparing two sampling distributions (e.g., "How does the standard error change if n doubles?").

• Calculator Pitfalls:

• Forgetting to use ?/?n (not ?) in normalcdf/invNorm.
• Using normalcdf with incorrect bounds (e.g., 1E99 for "greater than" or -1E99 for "less than").
• Mixing up-and x? in interpretations.

Quick Check Questions

1. MCQ: A population has-= 80 and-= 10. What is the standard error of the mean for a sample of size 25?
2. (A) 0.4
3. (B) 2
4. (C) 10
5. (D) 40

6. Answer: (B) 2. Explanation: ? = ?/?n = 10/?25 = 2.

7. FRQ Part: A factory produces bags of chips with a mean weight of 16 oz and a standard deviation of 0.5 oz. A quality control inspector takes a random sample of 40 bags.

8. (a) Describe the sampling distribution of the sample mean weight.
9. (b) Find the probability that the sample mean weight is less than 15.9 oz.

10. Answer:

    • (a) Shape: Approximately normal (CLT, n = 40-30). Center: ? = 16 oz. Spread: ? = 0.5/?40-0.079 oz.
    • (b) normalcdf(-1E99, 15.9, 16, 0.079)-0.1038 (about 10.4%).

11. MCQ: Which of the following is not a condition for the Central Limit Theorem to apply?

12. (A) The sample is randomly selected.
13. (B) The population is normally distributed.
14. (C) The sample size is at least 30.
15. (D) The 10% condition is satisfied.
16. Answer: (B) The population is normally distributed. Explanation: The CLT applies regardless of the population’s shape if n-30.

Last-Minute Cram Sheet

1. CLT: Sampling distribution of x? is normal if n-30 or population is normal.
2. ? = ? (center of sampling distribution = population mean).
3. ? = ?/?n (standard error decreases as n increases).
4. Conditions (RIN): Random, Independent (10% condition), Normal (n-30 or population normal).
5. Calculator: Use normalcdf(lower, upper, ?, ?/?n) for probabilities.
6. Calculator: Use invNorm(area, ?, ?/?n) for critical values.
7. Always check the 10% condition when sampling without replacement.
8. Never use-for ?—always divide by ?n.
9. Interpretation: "There is a [probability] chance that a random sample of [n] will have a mean between/above/below [value]."
10. Doubling n halves ? (since ? = ?/?n).