AP Statistics (AP Stats): Confidence Interval for One Mean (x? ± t* × s/?n)

AP Statistics – Confidence Interval for One Mean (x? ± t* × s/?n)

AP Statistics: Confidence Interval for One Mean (x? ± t* × s/?n) – Exam-Ready Study Guide

What This Is

A confidence interval for one mean estimates the true population mean (?) using sample data when the population standard deviation (?) is unknown. This is essential on the AP exam because real-world data (e.g., average battery life of smartphones, SAT scores of a school district, or recovery time after a medical treatment) rarely comes with a known ?. Instead, we use the t-distribution to account for additional uncertainty from estimating-with the sample standard deviation (s). The interval is calculated as x? ± t* × (s/?n), where t* is the critical t-value for the desired confidence level.

Real-world example: A school administrator wants to estimate the average time (in minutes) students spend on homework per night. She surveys 30 students, finds a sample mean (x?) of 45 minutes and a sample standard deviation (s) of 12 minutes. A 95% confidence interval helps her estimate the true average homework time for all students in the district.

Key Terms & Formulas

• Confidence Interval (CI) for? (? unknown): x? ± t* × (s/?n)
• x? = sample mean
• t* = critical t-value (from t-distribution)
• s = sample standard deviation
• n = sample size

• Degrees of freedom (df): n – 1

• t-distribution: A symmetric, bell-shaped distribution used when-is unknown. It has heavier tails than the normal distribution, especially for small n. As n increases, the t-distribution approaches the normal distribution.

• t* (critical t-value): The multiplier for the margin of error. Found using invT(area to left, df) on the TI-84.

• For a 95% CI, area to left = 0.975 (two-tailed).

• Example: For 90% CI with n = 25, df = 24, use invT(0.95, 24).

• Margin of Error (ME): t* × (s/?n) The maximum expected difference between x? and-for the given confidence level.

• Confidence Level (e.g., 95%): The long-run success rate of the method. If we took many samples and built a 95% CI from each, ~95% would capture ?.

• Conditions for Inference (LINER):

• Large sample or Normal population: n-30 (Central Limit Theorem) OR the population is normal (check with a histogram/boxplot).
• Independent observations: Random sampling AND n-10% of the population (10% condition).

• Random sampling: Data must come from a random sample or randomized experiment.

• TI-84: TInterval STAT-TESTS-8:TInterval

• Input: Stats or Data (if using a list).

• Output: Confidence interval and x?, s, n.

• Interpretation of CI: "We are [C]% confident that the true mean [context] is between [lower bound] and [upper bound]."

• z vs. t:

• Use z if-is known (rare on AP exam).
• Use t if-is unknown (almost always on AP exam).

Step-by-Step / Process Flow

How to solve a typical AP FRQ for a confidence interval for ?:

1. State the parameter and confidence level:

2. "We want to estimate the true mean [context] with [C]% confidence."

3. Check conditions (LINER):

4. Random: "The problem states the sample was randomly selected."
5. Independent: "The sample size (n = [X]) is less than 10% of the population."

6. Normal/Large: "Since n-30, the sampling distribution of x? is approximately normal OR the population is normal (show a histogram/boxplot)."

7. Name the procedure:

8. "We will use a one-sample t-interval for ?."

9. Calculate the interval:

10. By hand: x? ± t × (s/?n*)
    • Find t* using invT(area, df) (e.g., for 95% CI, area = 0.975).

11. TI-84: TInterval (input x?, s, n, C-level).

12. Interpret the interval in context:

13. "We are [C]% confident that the true mean [context] is between [lower] and [upper]."

14. Answer the question (if applicable):

15. Example: "Based on the interval, we cannot conclude that the mean [context] is greater than [value] because [value] is inside the interval."

Common Mistakes

• Mistake: Using the z-distribution instead of t when-is unknown. Correction: Always use t for confidence intervals when-is unknown (which is almost always on the AP exam). The z-distribution is only for known-or proportions.

• Mistake: Forgetting to check the 10% condition for independence. Correction: Even if the problem doesn’t mention it, always verify that n-10% of the population when sampling without replacement. Example: If the population is 5,000 students and n = 500, the 10% condition is violated.

• Mistake: Misinterpreting the confidence level as the probability that-is in the interval. Correction: The confidence level is about the method, not the interval. Say: "We are 95% confident that the interval captures ?," not "There is a 95% chance-is in the interval."

• Mistake: Using n instead of n – 1 for degrees of freedom. Correction: df = n – 1 for a one-sample t-interval. Example: For n = 25, df = 24.

• Mistake: Rounding t too early in calculations. Correction: Keep t to at least 3 decimal places until the final answer to avoid rounding errors.

AP Exam Insights

• Tricky Distinction: Confidence level vs. confidence interval
• Level (e.g., 95%): Describes the method’s success rate over many samples.

• Interval (e.g., 42 to 48): The specific range calculated from one sample.

• Common FRQ Setup:

• The problem gives x?, s, and n (or a data list) and asks for a confidence interval.

• Often includes a follow-up: "Does this interval provide evidence that the mean is greater than [value]?" (Answer: Only if the entire interval is above the value.)

• Calculator Pitfalls:

• TInterval vs. ZInterval: Always use TInterval unless-is given (rare).
• Inputting data: If using a list, make sure the list is correctly entered in L1 (or another list).

• Degrees of freedom: The calculator automatically uses n – 1, but you must state df in your work.

• What’s Frequently Tested:

• Checking conditions (especially the 10% condition).
• Interpreting the interval in context (AP loves to ask for this!).
• Comparing the interval to a hypothesized value (e.g., "Is there evidence the mean is greater than 50?").

Quick Check Questions

1. Multiple Choice: A random sample of 20 students has a mean GPA of 3.2 with a standard deviation of 0.4. Which of the following is the correct 95% confidence interval for the true mean GPA? (A) 3.2 ± 1.96 × (0.4/?20) (B) 3.2 ± 2.093 × (0.4/?20) (C) 3.2 ± 1.96 × (0.4/?19) (D) 3.2 ± 2.093 × (0.4/?19)

Answer: (B) Explanation: Use t-distribution (? unknown), df = 19, so t = 2.093 (from invT(0.975, 19)). The standard error is s/?n* = 0.4/?20.

1. FRQ Part: A researcher wants to estimate the average height (in inches) of 10th-grade students in a large school district. She randomly selects 50 students and measures their heights. The sample mean is 65.2 inches, and the sample standard deviation is 3.1 inches. (a) Construct and interpret a 99% confidence interval for the true mean height of 10th-grade students in the district. (b) The researcher claims that the true mean height is greater than 64 inches. Does the interval support this claim? Explain.

Answer: (a) Conditions: Random sample, n = 50-10% of all 10th graders, n-30 (normality satisfied). Interval: 65.2 ± 2.678 × (3.1/?50) = (64.1, 66.3) Interpretation: We are 99% confident that the true mean height of 10th-grade students is between 64.1 and 66.3 inches. (b) Yes, the interval supports the claim because the entire interval (64.1, 66.3) is above 64 inches.

Last-Minute Cram Sheet

1. Formula: x? ± t × (s/?n*) (use t, not z, when-is unknown).
2. df = n – 1 (for one-sample t-interval).
3. Conditions (LINER): Large/Normal, Independent (10% condition), Random.
4. TI-84: TInterval (Stats or Data)-input x?, s, n, C-level.
5. t*: invT(area to left, df) (e.g., 95% CI-area = 0.975).
6. Interpretation: "We are [C]% confident that the true mean [context] is between [lower] and [upper]."
7. Always check the 10% condition (even if the problem doesn’t mention it).
8. Use t, not z, unless-is given (rare on AP exam).
9. Margin of Error = *t × (s/?n) (larger n-smaller ME).
10. If the interval contains a value, we cannot reject it (e.g., if 0 is in the interval, no evidence the mean is different from 0).