Source: Moore, McCabe & Craig, Introduction to the Practice of Statistics, 10th Ed., Ch. 5–6
Tags: sampling distribution, central limit theorem, CLT, standard error, confidence interval, margin of error, hypothesis test, null hypothesis, p-value, significance level, z-test, ECON 2500
Difficulty: Intermediate | Prerequisites: Ch. 1-4 notes (distributions, probability, random variables)
Chapters 5 and 6 are the heart of the course. Chapter 5 introduces sampling distributions, the idea that a statistic computed from a random sample is itself a random variable with a predictable distribution. The Central Limit Theorem (CLT) is the single most important result: it tells you that the distribution of sample means is approximately normal for large n, regardless of the population shape. Chapter 6 uses that result to build the two main tools of statistical inference, confidence intervals and hypothesis tests. Everything from Chapter 7 onward is an application of these ideas to specific settings.
Chapter 5 shows that if you repeatedly sample from a population, the sample mean has a distribution centred at the population mean with spread that shrinks as n grows, and that distribution is approximately normal for large n (CLT). Chapter 6 uses this to construct confidence intervals (a range of plausible values for a parameter) and hypothesis tests (a formal way to decide whether data provide evidence against a claim).
Parameter
A number that describes the population (e.g. the population mean mu or the population proportion p). Fixed but usually unknown.
Statistic
A number computed from the sample (e.g. x-bar or p-hat). It varies from sample to sample.
Sampling distribution
The distribution of a statistic over all possible samples of the same size from the same population. In simple terms, it answers: "If I drew this sample a thousand times, what pattern would my statistic follow?"
Central Limit Theorem (CLT)
For large n, the sampling distribution of the sample mean x-bar is approximately Normal, regardless of the shape of the population distribution. The mean of this distribution is mu (the population mean) and the standard deviation is sigma / square root of n.
Standard error
The standard deviation of a statistic's sampling distribution. For x-bar when sigma is known: sigma / square root of n. It measures how much the statistic typically varies from sample to sample.
Confidence interval
A range of values computed from the data that is likely to contain the true parameter. A 95% confidence interval means: if we repeated the sampling process many times, about 95% of the resulting intervals would contain the true parameter.
Margin of error
The plus-or-minus part of a confidence interval. For a z-interval: z* x (sigma / square root of n).
Confidence level (C)
The percentage of intervals that would capture the true parameter in repeated sampling. Common levels: 90%, 95%, 99%.
Null hypothesis (H0)
The claim being tested, typically a statement of "no effect" or "no difference." In simple terms, the default assumption you are trying to find evidence against.
Alternative hypothesis (Ha)
The claim you suspect is true and hope the data will support. Can be one-sided (Ha: mu > mu0) or two-sided (Ha: mu ≠ mu0).
Test statistic
A standardised value that measures how far the sample result is from the null hypothesis value. For a z-test: z = (x-bar - mu0) / (sigma / square root of n).
P-value
The probability, assuming H0 is true, of observing a result as extreme as or more extreme than what the data showed. A small p-value means the data are unlikely under H0.
Significance level (alpha)
The threshold for rejecting H0. If the p-value is less than or equal to alpha, reject H0. Common choice: alpha = 0.05.
Type I error
Rejecting H0 when it is true (a false positive). The probability of a Type I error equals alpha.
Type II error
Failing to reject H0 when it is false (a false negative).
Power
The probability of correctly rejecting H0 when it is false: Power = 1 - P(Type II error). Higher power means a better chance of detecting a real effect.
If a population has mean mu and standard deviation sigma, and you draw an SRS of size n:
The mean of the sampling distribution of x-bar is mu (unbiased).
The standard deviation of x-bar is sigma / square root of n. Larger samples produce less variable estimates.
If the population itself is Normal, then x-bar is exactly Normal for any n.
If the population is not Normal, the Central Limit Theorem says x-bar is approximately Normal for large n (typically n ≥ 30 is a rough guideline, though it depends on how non-normal the population is).
The CLT is the reason inference works. It means you can use Normal-based methods (z-intervals, z-tests) for the sample mean even when the population is skewed or otherwise non-normal, as long as the sample is large enough. Without the CLT, you would need to know the exact population distribution to do inference.
For a sample proportion p-hat from an SRS of size n:
The mean of the sampling distribution of p-hat is p (the population proportion).
The standard deviation of p-hat is square root of [p(1 - p) / n].
For large n, p-hat is approximately Normal, provided np ≥ 10 and n(1 - p) ≥ 10.
A level C confidence interval for mu when sigma is known:
x-bar ± z* x (sigma / square root of n)
where z* is the critical value from the standard Normal distribution corresponding to confidence level C (e.g. z* = 1.96 for 95%).
Interpretation: "We are 95% confident that the true population mean lies between [lower bound] and [upper bound]." This means that 95% of intervals constructed this way, across many samples, would capture mu.
To achieve a desired margin of error m:
n = (z* x sigma / m) squared
Round up to the next whole number.
Steps:
State H0 (e.g. H0: mu = mu0) and Ha (one-sided or two-sided).
Compute the test statistic: z = (x-bar - mu0) / (sigma / square root of n).
Find the p-value from the standard Normal distribution.
Compare the p-value to alpha. If p-value ≤ alpha, reject H0.
State the conclusion in context.
Two-sided (Ha: mu ≠ mu0): p-value = 2 x P(Z ≥ |z|). You are testing for any difference.
One-sided (Ha: mu > mu0 or Ha: mu < mu0): p-value = P(Z ≥ z) or P(Z ≤ z). You are testing for a difference in a specific direction.
A two-sided test at significance level alpha rejects H0 exactly when the value mu0 falls outside the corresponding (1 - alpha) confidence interval. They always agree.
A result can be statistically significant (small p-value) without being practically important. With a very large sample, even a tiny difference from mu0 can be significant. Always consider the size of the effect, not just whether it is significant.
Standard deviation of x-bar: sigma / square root of n
Standard deviation of p-hat: square root of [p(1 - p) / n]
z confidence interval for mu (sigma known): x-bar ± z* x (sigma / square root of n)
Common z values:* 90% CI → z* = 1.645; 95% CI → z* = 1.96; 99% CI → z* = 2.576
Sample size for a desired margin of error m: n = (z* x sigma / m) squared, rounded up
z test statistic for mu: z = (x-bar - mu0) / (sigma / square root of n)
P-value (two-sided): 2 x P(Z ≥ |z|)
P-value (one-sided, Ha: mu > mu0): P(Z ≥ z)
P-value (one-sided, Ha: mu < mu0): P(Z ≤ z)
Polling organisations report margins of error alongside election forecasts: "The candidate leads with 52% ± 3%." That ± 3% is a confidence interval margin of error built from the sampling distribution of p-hat. Central banks use hypothesis testing logic when deciding whether observed inflation differs from their target. In pharmaceutical trials, the p-value from a hypothesis test determines whether a new drug is approved. Every time you see a claim backed by data in economics, the reasoning behind it traces back to these two chapters.
Students often interpret a 95% confidence interval as "there is a 95% probability that the true mean lies in this interval." That is subtly wrong. The true mean is fixed; the interval is random. The correct interpretation is about the method: 95% of intervals constructed this way would capture the true mean.
Students confuse the p-value with the probability that H0 is true. The p-value is the probability of observing data this extreme assuming H0 is true, not the probability that H0 is true.
Students think failing to reject H0 means H0 is proven true. It does not. It means the data did not provide strong enough evidence to reject it. Absence of evidence is not evidence of absence.
Students forget that the standard error shrinks with the square root of n, not n itself. Doubling the sample size does not halve the standard error; you need to quadruple n to halve it.
⚠️ State the Central Limit Theorem precisely: it applies to the distribution of x-bar (or p-hat), not to the population itself. The population does not become normal as n increases.
⚠️ Memorise the three common z* critical values (1.645, 1.96, 2.576) and know which confidence level each corresponds to.
⚠️ When constructing a confidence interval, always check conditions: SRS, independence, and either known sigma or large enough n for the CLT.
⚠️ In hypothesis testing, always state H0 and Ha, compute the test statistic, find the p-value, make a decision, and state the conclusion in context. Exam answers that skip steps lose marks.
⚠️ Know the difference between statistical and practical significance. A question may ask you to comment on whether a statistically significant result matters in practice.
True or false: The Central Limit Theorem says the population distribution becomes normal as n increases. Answer: False. The CLT says the sampling distribution of x-bar becomes approximately normal. The population stays as it is.
Fill in the blank: To halve the margin of error of a confidence interval (same confidence level), you need to multiply the sample size by ___. Answer: 4 (because margin of error is proportional to 1 / square root of n).
True or false: A p-value of 0.03 means there is a 3% chance that H0 is true. Answer: False. It means there is a 3% chance of observing data this extreme if H0 were true.
Fill in the blank: z* for a 95% confidence interval is ___. Answer: 1.96.
True or false: If a 95% CI for mu is (48, 52), then a two-sided test of H0: mu = 53 at alpha = 0.05 would reject H0. Answer: True. 53 lies outside the interval, so the test rejects.
Q: A population has mean 100 and standard deviation 15. You draw an SRS of n = 36. What are the mean and standard deviation of the sampling distribution of x-bar? What is the approximate shape?
A: The mean of x-bar is 100. The standard deviation (standard error) is 15 / square root of 36 = 15 / 6 = 2.5. By the CLT, the distribution is approximately Normal (n = 36 is large enough).
Q: A 99% confidence interval for the mean weight of a product is (498.2g, 501.8g). The quality standard requires the mean to be 500g. Does this interval provide evidence that the mean differs from 500g at the 1% significance level?
A: No. The value 500 lies inside the 99% confidence interval, so a two-sided test at alpha = 0.01 would not reject H0: mu = 500.
Q: A researcher tests H0: mu = 50 vs. Ha: mu > 50 and obtains z = 2.33. Find the p-value and state whether you reject H0 at alpha = 0.01.
A: P-value = P(Z ≥ 2.33) = 0.0099. Since 0.0099 < 0.01, reject H0. There is significant evidence that mu > 50 at the 1% level.
Q: Explain, in one sentence, why failing to reject H0 does not prove H0 is true.
A: The test may simply lack the power (e.g. too small a sample) to detect a real difference, so the absence of evidence against H0 does not confirm it.
Chapter 5's sampling distributions underpin every inference procedure in the rest of the course. Chapter 7 extends these z-procedures to the more realistic case where sigma is unknown (using the t-distribution). Chapter 8 applies the same logic to proportions. The hypothesis-testing framework from Chapter 6 reappears in every inference chapter; once you master it here, the later chapters are variations on the same theme.
Related Terms / Search Tags
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