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📘 Key Concept Questions: AP Statistics

  1. What is the Central Limit Theorem (CLT)? (*****)
  1. What is a confidence interval? (*****)
  1. What is a hypothesis test?
  1. What is a P-value? (*****)
  1. What is a Type I error in hypothesis testing?
  1. What is a Type II error in hypothesis testing? (*****)
  1. What does it mean if the P-value is small (e.g., < 0.05)?
  1. What does a 95% confidence level mean?
  1. What is the standard error of a sample mean?
  1. What is the null hypothesis?
  1. What is the alternative hypothesis?
  1. What does "statistically significant" mean?
  1. What conditions must be met to use a one-sample z-test for proportions?
  1. What is the purpose of a chi-square test? (*****)
  1. When do you use a t-distribution instead of a z-distribution?
  1. What is the margin of error?
  1. What is a sampling distribution?
  1. What are the conditions for the Central Limit Theorem to apply?
  1. What does it mean when two events are independent? (*****)
  1. What is the formula for a confidence interval for a proportion?

 
 
Detailed answers:
 

📘 AP Statistics Key Concept Questions + Detailed Answers

No.
Question
Detailed Answer
1
What is the Central Limit Theorem (CLT)?
The CLT states that the sampling distribution of the sample mean will be approximately normal if the sample size is large enough (usually n≥30n ≥ 30), regardless of the population's distribution.
2
What is a confidence interval?
A range of values, based on sample data, that is likely to contain the true population parameter. It is calculated as: statistic ± margin of error.
3
What is a hypothesis test?
A method of using sample data to evaluate a claim (hypothesis) about a population. It involves a null and an alternative hypothesis.
4
What is a P-value?
The probability of observing a sample statistic as extreme as the one obtained, assuming the null hypothesis is true. A small P-value suggests strong evidence against the null.
5
What is a Type I error?
Rejecting the null hypothesis when it is actually true (false positive). Its probability is denoted by α\alpha.
6
What is a Type II error?
Failing to reject the null hypothesis when it is false (false negative). Its probability is denoted by β\beta.
7
What does it mean if the P-value is small?
It indicates strong evidence against the null hypothesis. If the P-value < α\alpha, you reject the null.
8
What does a 95% confidence level mean?
If you repeated the sampling process many times, about 95% of the resulting confidence intervals would contain the true population parameter.
9
What is the standard error of a sample mean?
It estimates the variability of the sample mean and is calculated as σn\frac{\sigma}{\sqrt{n}} (or sn\frac{s}{\sqrt{n}} if population SD is unknown).
10
What is the null hypothesis?
A claim about the population that we test against. Usually states “no effect” or “no difference” (e.g., H0:p=0.5H_0: p = 0.5).
11
What is the alternative hypothesis?
A claim we want to find evidence for. It represents an effect, difference, or change (e.g., HA:p>0.5H_A: p > 0.5).
12
What does "statistically significant" mean?
A result is statistically significant if the P-value is smaller than the chosen significance level α\alpha; this suggests strong evidence against H0H_0.
13
What conditions must be met to use a one-sample z-test for proportions?
1) Random sample, 2) 10% condition (if sampling without replacement), 3) Large counts: np≥10np ≥ 10 and n(1−p)≥10n(1-p) ≥ 10.
14
What is the purpose of a chi-square test?
To test for differences in categorical data, such as independence of variables or whether distributions match expected counts.
15
When do you use a t-distribution instead of a z-distribution?
When the population standard deviation is unknown and the sample size is small (n<30n < 30), especially for inference about means.
16
What is the margin of error?
The maximum expected difference between the sample statistic and the true population parameter; calculated using a critical value times the standard error.
17
What is a sampling distribution?
The distribution of a statistic (e.g., sample mean or sample proportion) over all possible samples of a given size from a population.
18
What are the conditions for the Central Limit Theorem to apply?
The sample must be random and either: (1) the population is normal, or (2) the sample size is large (n≥30n ≥ 30).
19
What does it mean when two events are independent?
The occurrence of one event does not affect the probability of the other. Mathematically: P(A∩B)=P(A)P(B)P(A \cap B) = P(A)P(B).
20
What is the formula for a confidence interval for a proportion?
p^±z∗p^(1−p^)n\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}, where p^\hat{p} is the sample proportion and z∗z^* is the critical value.

 

Questions for Future Math Teachers

  1. How do you distinguish between a ratio and a rate in both definition and application? Provide examples.
  1. What is the difference between a sequence and a series? Illustrate with mathematical notation and examples.
  1. State the Pythagorean Theorem. Under what conditions is it valid?
  1. Define and compare the following statistical terms: mean, median, mode, range, and interquartile range (IQR). When is each measure most appropriate to use?
  1. Translate the following mathematical terms from Chinese to English:
      • 分子
      • 分母
      • 棱形
      • 梯形
      • 平行四边形
      • 同位角
      • 内错角
      • 同旁内角
  1. What are the four types of geometric transformations commonly tested in the IGCSE curriculum?
  1. What is the Alternate Segment Theorem? State it formally and explain its geometric significance using a diagram.
  1. The last question:
notion image
 

 
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