Chapter 11 built the theoretical machinery of sampling distributions — how a statistic like the sample mean or sample proportion varies from sample to sample, and how the Central Limit Theorem lets us treat that variation as approximately normal for large samples. Chapter 12 puts that machinery to work for the first time: this is where statistics actually starts answering real questions about an unknown population using nothing but sample data. Statistical inference has two main branches — estimation (this chapter) and hypothesis testing (the next chapter) — and both rest entirely on the sampling-distribution theory just developed.
This chapter covers both flavours of estimation: point estimation, which produces a single best-guess number for an unknown parameter, and interval estimation, which produces a range of plausible values together with a stated level of confidence. Interval estimation — confidence intervals — is the centrepiece of the chapter, covering the population mean, the population proportion, and the difference between two means or two proportions, across a range of practical situations (known/unknown variance, large/small samples, normal/non-normal populations).
Learning Objectives
- Distinguish statistical inference, estimation, and hypothesis testing, and explain the role each plays
- Distinguish a point estimator from a point estimate, and construct point estimators for the mean, variance, and proportion
- Explain unbiasedness and determine whether a given estimator is biased or unbiased
- Identify the best (minimum variance unbiased) estimator for common population parameters
- Construct pooled estimators for the mean, variance, and proportion from two samples
- Explain what a confidence interval is and interpret the meaning of the confidence coefficient
- Construct confidence intervals for a population mean under known and unknown variance, and for large and small samples (Z and t)
- Construct confidence intervals for a population proportion
- Construct confidence intervals for the difference between two population means and two population proportions
Key Concepts
12.1-12.2 Statistical Inference and Estimation
Statistical inference is the field concerned with drawing conclusions about a population's distribution by using observed sample data governed by that distribution. It has two main branches: estimation of parameters (determining likely values for an unknown population parameter) and testing of hypotheses (deciding between competing claims about a parameter, covered in the next chapter). Statistical estimation is the procedure of making a judgment about the unknown value of a population parameter using sample observations — since examining the entire population is usually impracticable, we instead estimate its parameters from a sample, with the accuracy of the estimate controllable in part by increasing the sample size.
Estimation is further divided into two types. Point estimation produces a single number, intended as the best guess for the unknown parameter. Interval estimation instead produces a range of values, together with a stated degree of confidence that the range actually contains the true parameter — and it is interval estimation that gives us information about the PRECISION of an estimate, something a single point estimate can never convey on its own.
12.3-12.4 Point Estimator, Point Estimate, and Unbiasedness
A point estimator is a sample statistic used to estimate the unknown true value of a population parameter — it is always a function of the sample observations and, being computed from a random sample, is itself a random variable with its own probability distribution; it is denoted by a capital letter such as T. A point estimate is the specific numerical value of the estimator computed from one particular observed sample — a fixed number, denoted by a small letter such as t. This distinction (estimator = random variable / formula; estimate = the actual number you get) is fundamental and recurs throughout the chapter.
Unbiasedness is the most important desirable property of an estimator: the distribution of a good estimator should be centred at the true value of the parameter it estimates. An estimator T is unbiased for a parameter theta if E(T) = theta — meaning that, averaged across every possible sample, T gives the correct value with no systematic tendency to over- or under-estimate. If E(T) is not equal to theta, T is biased, and the amount of the bias is Bias = E(T) – theta. The sample mean X-bar is always an unbiased estimator of the population mean mu. Interestingly, the sample variance S-squared = [sum(X_i – X-bar)^2]/n (divide by n) is a BIASED estimator of sigma-squared — its expected value is [(n-1)/n].sigma-squared, slightly less than the truth — which is exactly why the alternative version S-hat-squared = [sum(X_i – X-bar)^2]/(n-1) (divide by n-1) is preferred: it is unbiased, E(S-hat-squared) = sigma-squared.
12.5-12.6 Best Estimator and Pooled Estimators
Among all unbiased estimators of a parameter, the best (or minimum variance) estimator is the one whose sampling distribution has the smallest variance — i.e., no other unbiased estimator is more precise. For a population with unknown mean mu and unknown variance sigma-squared, the best estimators are the sample mean X-bar (for mu) and the unbiased sample variance S-hat-squared = [sum(X_i – X-bar)^2]/(n-1) (for sigma-squared). For a population with unknown proportion pi, the best estimator is the sample proportion P = X/n.
When two independent samples are available from the SAME population, their information can be combined into a single, more precise pooled estimator. The pooled estimator of the mean is X-bar-p = (n1.X-bar-1 + n2.X-bar-2)/(n1+n2) — essentially a weighted average of the two sample means, weighted by sample size. The pooled estimator of the variance is S-p-squared = [(n1-1).S-hat-1-squared + (n2-1).S-hat-2-squared]/(n1+n2-2) — a weighted average of the two sample variances. The pooled estimator of a proportion is pi-hat = (X1+X2)/(n1+n2) = (n1.P1 + n2.P2)/(n1+n2).
12.7 The Concept of Interval Estimation
A point estimate alone tells you nothing about how far it might be from the truth — however good the estimator, a single number can never be expected to exactly equal the parameter, and looking at just that one value gives no sense of its precision. Interval estimation solves this by extending the idea of an error bound to produce a whole INTERVAL of plausible values, likely (with a stated, known probability) to contain the true parameter. This is the concept of the confidence interval: an interval (L, U), computed from sample data, such that BEFORE sampling, P(L < theta < U) = 1 – alpha for a specified high probability 1 – alpha.
The interval (L, U) is then called a 100(1-alpha)% confidence interval for theta, and 1-alpha is called the confidence coefficient (or confidence level). L and U — the lower and upper confidence limits — are themselves random variables, since they are computed from sample data; theta itself is the fixed, constant, unknown quantity. The width U-L measures the precision of the estimate: a narrower interval is more precise. Precision can be increased either by decreasing the standard error of the estimate (i.e., increasing the sample size) or by decreasing the confidence coefficient — but these two goals pull in opposite directions: for any fixed sample size, demanding higher confidence necessarily widens the interval. It is crucial to interpret a confidence interval correctly: 1-alpha is the long-run proportion of intervals (across many repeated samples) that would contain theta — it is NOT the probability that theta lies in this one particular already-computed interval, since theta is a fixed constant, not a random variable.
12.8 Confidence Interval for the Population Mean, mu
The exact formula for a confidence interval for mu depends on three things: whether the population is normal, whether sigma-squared is known, and whether the sample size is large or small. When the population is normal with KNOWN variance sigma-squared, the sampling distribution of X-bar is exactly normal, so the two-sided 100(1-alpha)% confidence interval is x-bar +/- z_(1-alpha/2).(sigma/sqrt(n)), where z_(1-alpha/2) is the standard normal value cutting off alpha/2 in the upper tail.
When the population is NOT normal (or its shape is unknown), but the sample is large (conventionally n>30), the Central Limit Theorem justifies treating X-bar as approximately normal regardless of the population's shape, giving the same-looking approximate interval x-bar +/- z_(1-alpha/2).(sigma/sqrt(n)) — and when sigma is also unknown (the usual real-world case), it is simply replaced by the sample estimate s-hat, since for large n this substitution barely affects the result: x-bar +/- z_(1-alpha/2).(s-hat/sqrt(n)). When sampling is done WITHOUT replacement from a finite population and the sampling fraction n/N exceeds 5%, the finite population correction is added: x-bar +/- z_(1-alpha/2).(sigma/sqrt(n)).sqrt[(N-n)/(N-1)] — this fpc can be safely ignored when n is less than 5% of N.
When the population IS normal but sigma-squared is UNKNOWN and the sample is SMALL (n<=30), the statistic T = (X-bar – mu)/(S-hat/sqrt(n)) follows Student's t-distribution with v = n-1 degrees of freedom rather than the standard normal distribution — reflecting the extra uncertainty introduced by estimating sigma from a small sample. The resulting confidence interval is x-bar +/- t_(v,1-alpha/2).(s-hat/sqrt(n)), using the t-table instead of the Z-table. For large degrees of freedom, the t-distribution converges to the standard normal, so the two approaches agree as n grows.
12.9 Confidence Interval for the Population Proportion, pi
When estimating a population proportion pi (e.g., the fraction of defective items, or the fraction of voters favouring a candidate), the sampling distribution of the sample proportion P=X/n is approximately normal for large n (with pi not too close to 0 or 1), by the same Central Limit Theorem logic. The resulting two-sided 100(1-alpha)% confidence interval is p +/- z_(1-alpha/2).sqrt[p(1-p)/n], where p is the observed sample proportion. As with the mean, if sampling is done without replacement from a finite population and n exceeds 5% of N, the finite population correction sqrt[(N-n)/(N-1)] is multiplied in; otherwise it may be dropped.
12.10-12.11 Comparative Studies and CI for the Difference Between Two Means
Many practical questions involve comparing two populations rather than describing just one — is a new fertilizer more effective than the old one, does a new machine pack more precisely than an old one, do two departments differ in output. Two samples used for such comparisons are independent if the selection from one population has no bearing on the selection from the other; they are dependent (or paired/matched) if each observation in one sample is deliberately paired with a corresponding observation in the other.
For two INDEPENDENT samples from normal populations with KNOWN variances (any sample size), X-bar-1 – X-bar-2 is exactly normal with mean mu1-mu2 and variance sigma1-squared/n1 + sigma2-squared/n2, giving the confidence interval (x-bar-1 – x-bar-2) +/- z_(1-alpha/2).sqrt[sigma1^2/n1 + sigma2^2/n2]. When both samples are large (n1,n2 > 30), the same formula works approximately for ANY population shape (by the CLT), with the sample variances s-hat-1-squared and s-hat-2-squared substituted for the unknown sigma1-squared and sigma2-squared.
When both samples are SMALL, populations are normal, and the two (unknown) population variances are assumed EQUAL, a pooled estimate of the common variance is first computed, S-p-squared = [(n1-1).S-hat-1^2 + (n2-1).S-hat-2^2]/(n1+n2-2), and then the confidence interval uses the t-distribution with v=n1+n2-2 degrees of freedom: (x-bar-1 – x-bar-2) +/- t_(v,1-alpha/2).s-p.sqrt[1/n1 + 1/n2].
12.12 Confidence Interval for the Difference Between Two Proportions
To compare the incidence rate of some characteristic across two populations (e.g., defect rates from two production lines, approval rates in two regions), independent samples of sizes n1 and n2 give sample proportions P1=X1/n1 and P2=X2/n2. For large samples, the CLT ensures P1-P2 is approximately normal with mean pi1-pi2 and (estimated) standard error sqrt[P1(1-P1)/n1 + P2(1-P2)/n2], giving the two-sided confidence interval (p1-p2) +/- z_(1-alpha/2).sqrt[p1(1-p1)/n1 + p2(1-p2)/n2].
Important Definitions
What is the difference between a point estimator and a point estimate?
A point estimator is a sample statistic (a formula/random variable, e.g., X-bar) used to estimate a parameter; a point estimate is the specific numerical value that estimator takes for one particular observed sample.
What does it mean for an estimator to be unbiased?
An estimator T is unbiased for a parameter theta if E(T) = theta — i.e., averaged across all possible samples, it gives the correct value with no systematic over- or under-estimation.
What is bias?
The amount by which a biased estimator's expected value differs from the true parameter: Bias = E(T) – theta.
What is the best (minimum variance) estimator?
Among all unbiased estimators of a parameter, the one whose sampling distribution has the smallest variance — i.e., the most precise unbiased estimator available.
What is a pooled estimator?
An estimator that combines information from two independent samples of the same population into a single, more precise estimate, e.g., the pooled sample mean or pooled sample variance.
What is a confidence interval?
An interval (L,U), computed from sample data, such that prior to sampling, P(L<theta<U) = 1-alpha for a specified high probability 1-alpha, called the confidence coefficient.
What is the confidence coefficient?
The probability 1-alpha that the confidence interval procedure produces an interval containing the true parameter; commonly 90%, 95%, 98%, or 99%.
How should a confidence interval be correctly interpreted?
As a long-run statement: in repeated sampling, 100(1-alpha)% of all such intervals constructed would contain the true parameter — it is NOT the probability that the parameter lies in this one specific already-computed interval.
What is the t-distribution used for in this chapter?
For constructing confidence intervals for a mean when the population is normal, the variance is unknown, and the sample size is small (n<=30); it accounts for the extra uncertainty of estimating sigma from a small sample.
What is the finite population correction and when is it used?
The factor sqrt[(N-n)/(N-1)], applied to the standard error when sampling without replacement from a finite population and the sample is more than 5% of the population (n>0.05N); it may be ignored otherwise.
Key Facts and Relations
| Topic | Key Fact / Relation |
|---|---|
| Point estimators (mean, variance, proportion) | X-bar = (sum Xi)/n; S-hat^2 = [sum(Xi-Xbar)^2]/(n-1); P = X/n |
| Unbiasedness / Bias | Unbiased: E(T) = theta. Bias = E(T) – theta |
| Pooled mean / variance / proportion | Xbar_p = (n1.Xbar1+n2.Xbar2)/(n1+n2); Sp^2 = [(n1-1)S1^2+(n2-1)S2^2]/(n1+n2-2); pi-hat = (X1+X2)/(n1+n2) |
| CI for mu, sigma known | x-bar +/- z_(1-a/2) . sigma/sqrt(n) |
| CI for mu, large n, sigma unknown | x-bar +/- z_(1-a/2) . s-hat/sqrt(n) |
| CI for mu, without replacement (fpc) | x-bar +/- z_(1-a/2) . (sigma or s-hat)/sqrt(n) . sqrt[(N-n)/(N-1)] |
| CI for mu, small n, normal, sigma unknown | x-bar +/- t_(v,1-a/2) . s-hat/sqrt(n), v = n-1 |
| CI for proportion, pi | p +/- z_(1-a/2) . sqrt[p(1-p)/n] |
| CI for mu1-mu2, known variances / large samples | (xbar1-xbar2) +/- z_(1-a/2) . sqrt[sigma1^2/n1 + sigma2^2/n2] |
| CI for mu1-mu2, small samples, equal unknown variance | (xbar1-xbar2) +/- t_(v,1-a/2) . sp.sqrt[1/n1+1/n2], v=n1+n2-2 |
| CI for pi1-pi2 | (p1-p2) +/- z_(1-a/2) . sqrt[p1(1-p1)/n1 + p2(1-p2)/n2] |
| Confidence coefficient breakdown | 1-alpha = confidence level; alpha/2 in each tail; common z-values: 90%->1.645, 95%->1.960, 98%->2.326, 99%->2.576 |
Diagrams
Repeated Confidence Intervals for mu: A simulation of 25 different 95% confidence intervals, each computed from a different random sample of the same population, plotted as horizontal line segments against a fixed vertical line at the true population mean — illustrating that about 95% of such intervals capture the true mean while a few, purely by chance, miss it
Effect of Confidence Level on Interval Width: Three nested confidence intervals (90%, 95%, 99%) computed from the same sample data and centred at the same sample mean, showing how demanding a higher confidence level necessarily widens the interval
Effect of Sample Size on Interval Width: A line chart showing how the width of a 95% confidence interval for the mean shrinks as the sample size n increases, illustrating that larger samples give more precise (narrower) interval estimates
Short Questions & Answers
Distinguish between a point estimate and an interval estimate.
A point estimate is a single number intended as the best guess for a parameter and gives no information about its precision; an interval estimate is a range of values, together with a stated confidence level, that conveys both an estimate and its precision.
Why is the sample variance S^2 = sum(Xi-Xbar)^2/n considered a biased estimator, and what is used instead?
Because E(S^2) = [(n-1)/n].sigma^2, which is systematically less than the true sigma^2; the unbiased alternative S-hat^2 = sum(Xi-Xbar)^2/(n-1), which satisfies E(S-hat^2)=sigma^2, is used instead.
What does a 95% confidence interval actually mean?
That if the same sampling and interval-construction procedure were repeated many times, approximately 95% of the resulting intervals would contain the true population parameter — not that there is a 95% chance the parameter is in this one specific interval.
How can the precision (width) of a confidence interval be improved?
By increasing the sample size (which reduces the standard error) or by decreasing the confidence coefficient (accepting a lower level of confidence) — these two levers work in opposite directions.
When should the t-distribution be used instead of the standard normal (Z) distribution for a confidence interval on the mean?
When the population is normal, the population variance is unknown, and the sample size is small (typically n<=30); the t-distribution accounts for the added uncertainty of estimating sigma from limited data.
When is the finite population correction factor included in a confidence interval formula, and when can it be dropped?
It is included when sampling without replacement and the sample is more than 5% of the population (n>0.05N); it can safely be dropped when the sample is a small fraction of the population (n<0.05N).
Long Questions & Answers
Explain the concepts of point estimation and unbiasedness in detail, including the distinction between an estimator and an estimate, and explain with an example why the sample variance divided by n is a biased estimator while the version divided by n-1 is unbiased.
What is the difference between a point estimator and a point estimate?
A point estimator is a sample statistic, T = g(X1, X2, …, Xn), used to estimate an unknown population parameter; because it is a function of a random sample, the estimator T is itself a random variable with its own sampling distribution, denoted with a capital letter. A point estimate is the specific numeric value that estimator produces once one particular sample is actually observed, calculated from the observed data and denoted with a lowercase letter. For example, the formula X-bar is the estimator of the population mean mu, while a specific computed number like 28 is the estimate.
What does it mean for an estimator to be unbiased?
A good estimator's distribution should be centred on the true value of the parameter it estimates. Formally, an estimator T is unbiased if E(T) = theta, meaning that averaged across every conceivable sample that could be drawn, the resulting estimate lands exactly on the true parameter value theta, with no systematic tendency to over- or under-estimate. If instead E(T) does not equal theta, the estimator is biased, and the size of this discrepancy is its bias, calculated as Bias = E(T) – theta.
Why is the sample variance formula dividing by n a biased estimator of the population variance?
The formula S^2 = sum(Xi – X-bar)^2 / n mirrors the population variance formula but can be proven biased: E(S^2) = ((n-1)/n) * sigma^2, which is always strictly less than sigma^2 for any finite n. This happens because X-bar, by its own construction, fits that particular sample's data more closely than the true (unknown) population mean mu ever could, which systematically shrinks the squared deviations — and therefore the resulting variance — to a value that runs too low on average, as shown in the textbook's worked example using the population 2, 4, 6, 8, 10.
Why is dividing by (n-1) instead of n the preferred unbiased formula for sample variance?
The alternative formula, S-hat^2 = sum(Xi – X-bar)^2 / (n-1), was introduced specifically to correct S^2's downward bias. It can be proven, and is demonstrated in the same worked textbook example, that E(S-hat^2) = sigma^2 exactly, with no remaining systematic bias. This proven unbiasedness is exactly why S-hat^2, rather than the more natural-looking divide-by-n version, is recommended as the best estimator of population variance throughout the rest of estimation theory.
Explain what a confidence interval is, how the confidence coefficient should be correctly interpreted, and describe how the choice of formula for a confidence interval on the population mean changes depending on whether the population variance is known, whether the sample is large or small, and whether the population itself is normal.
What is a confidence interval, and what does the confidence coefficient (1 – alpha) mean?
A confidence interval for an unknown parameter theta is an interval [L, U], calculated from sample data, constructed so that before the sample is drawn, the probability that the interval will contain the true value of theta equals a chosen level, 1 – alpha. This quantity is called the confidence coefficient or confidence level, and is commonly expressed as a percentage such as 90%, 95%, 98%, or 99%.
What is the correct long-run interpretation of a confidence interval, and what misinterpretation should be avoided?
Theta is a fixed, unchanging quantity; it is the endpoints L and U that are random, since both are calculated from whatever sample data happened to be observed. The correct interpretation is a long-run one: if the same sampling and interval-construction procedure were repeated many times, approximately 95% of the resulting intervals would contain the true theta. It is technically incorrect to say there is a 95% probability that theta lies within one single, already-computed interval — once computed, that interval has either captured theta or it hasn't.
How is interval precision controlled, and what confidence interval formula applies when sigma is known and the population is normal?
An interval's precision is governed by its width, U – L, which can only be reduced — improving precision — by increasing the sample size (which reduces the standard error) or by accepting a lower confidence coefficient (which lowers reliability). When the population is normal and its variance sigma^2 is already known, the sampling distribution of X-bar is exactly normal for any sample size, giving the exact interval: x-bar +/- z_(1-alpha/2) * sigma/sqrt(n).
What confidence interval formula applies for large samples when sigma is unknown?
When sigma is unknown but the sample is large (conventionally n > 30), the Central Limit Theorem justifies treating X-bar as approximately normal regardless of the population's shape. The unknown sigma is simply replaced by its sample estimate, s-hat, giving the approximate large-sample interval: x-bar +/- z_(1-alpha/2) * s-hat/sqrt(n).
What confidence interval formula applies for small samples when sigma is unknown, and why is the t-distribution used?
When the sample is small (n <= 30) and sigma is unknown, the Central Limit Theorem can no longer be relied on, so the population must instead be assumed normal. Under this assumption, T = (X-bar – mu)/(s-hat/sqrt(n)) follows Student's t-distribution with v = n-1 degrees of freedom, which is more spread out than the normal distribution to reflect the extra uncertainty of estimating sigma from limited data. This gives the small-sample interval: x-bar +/- t_(v, 1-alpha/2) * s-hat/sqrt(n), using the t-table rather than the z-table.
Multiple Choice Questions (MCQs)
A point estimator is best described as: (A) A fixed constant (B) A sample statistic that is a random variable (C) The true population parameter (D) A confidence interval
Correct answer: (B) A sample statistic that is a random variable. A point estimator is a formula/statistic computed from sample data; since the sample varies, the estimator is itself a random variable.
An estimator T is unbiased for parameter theta if: (A) E(T) = 0 (B) E(T) = theta (C) Var(T) = 0 (D) T = theta always
Correct answer: (B) E(T) = theta. Unbiasedness means the expected value of the estimator equals the true parameter, E(T)=theta.
The sample variance S^2 = sum(Xi-Xbar)^2/n is: (A) An unbiased estimator of sigma^2 (B) A biased estimator of sigma^2 (C) Equal to sigma^2 always (D) Undefined
Correct answer: (B) A biased estimator of sigma^2. E(S^2) = [(n-1)/n]sigma^2, which is less than sigma^2, so S^2 (divide by n) is biased; the divide-by-(n-1) version is unbiased.
The 'best' estimator among all unbiased estimators is the one with: (A) The largest variance (B) The smallest variance (C) A bias of zero only (D) The largest sample size
Correct answer: (B) The smallest variance. The best (minimum variance) unbiased estimator is the most precise one — the unbiased estimator with the smallest possible variance.
A 95% confidence interval means: (A) There is a 95% chance the parameter lies in this specific interval (B) 95% of the population lies in the interval (C) In repeated sampling, 95% of such intervals would contain the true parameter (D) The sample mean equals the parameter 95% of the time
Correct answer: (C) In repeated sampling, 95% of such intervals would contain the true parameter. The correct long-run interpretation: 95% of intervals constructed this way, across repeated sampling, would contain the true (fixed) parameter.
Increasing the sample size, holding confidence level fixed, causes a confidence interval's width to: (A) Increase (B) Decrease (C) Stay the same (D) Become undefined
Correct answer: (B) Decrease. A larger sample size reduces the standard error, which narrows (decreases) the confidence interval.
The t-distribution is used instead of the Z-distribution for a confidence interval on the mean when: (A) The sample size is large and sigma is known (B) The population is normal, sigma is unknown, and n is small (C) The population is not normal and n is large (D) Never — Z is always used
Correct answer: (B) The population is normal, sigma is unknown, and n is small. The t-distribution applies specifically when the population is normal, sigma^2 is unknown, and the sample size is small (n<=30).
For a confidence interval on a population proportion, the standard error is estimated using: (A) sigma/sqrt(n) (B) sqrt[p(1-p)/n] (C) s-hat/sqrt(n) (D) sqrt[sigma1^2/n1+sigma2^2/n2]
Correct answer: (B) sqrt[p(1-p)/n]. The estimated standard error of the sample proportion P is sqrt[p(1-p)/n].
For the difference between two means with small samples and equal but unknown variances, the confidence interval uses: (A) The standard normal distribution only (B) The pooled variance and the t-distribution with v=n1+n2-2 (C) The F-distribution (D) The chi-square distribution
Correct answer: (B) The pooled variance and the t-distribution with v=n1+n2-2. This case uses the pooled variance estimate Sp^2 and the t-distribution with n1+n2-2 degrees of freedom.
The finite population correction factor is applied when: (A) Sampling with replacement from an infinite population (B) Sampling without replacement and the sample exceeds 5% of the population (C) The sample size is always small (D) The population variance is unknown
Correct answer: (B) Sampling without replacement and the sample exceeds 5% of the population. The fpc is applied for sampling without replacement from a finite population when n exceeds 5% of N; it can be dropped otherwise.
Quick Revision Summary
- Estimator (random variable, capital letter) vs Estimate (fixed number, lowercase letter)
- Unbiased: E(T)=theta | Bias = E(T)-theta | Best estimator = unbiased with minimum variance
- Best estimators: mu -> X-bar; sigma^2 -> S-hat^2 (divide by n-1); pi -> P=X/n
- Pooled mean: weighted average by n1,n2 | Pooled variance: Sp^2=[(n1-1)S1^2+(n2-1)S2^2]/(n1+n2-2)
- CI = point estimate +/- (table value) x (standard error); width measures precision
- 1-alpha = confidence coefficient; long-run interpretation, NOT probability for one specific interval
- CI for mu: sigma known -> Z; large n, sigma unknown -> Z with s-hat; small n, normal, sigma unknown -> t (v=n-1)
- CI for pi: p +/- z.sqrt[p(1-p)/n]
- CI for mu1-mu2: known var/large n -> Z; small n equal var -> pooled t (v=n1+n2-2)
- CI for pi1-pi2: (p1-p2) +/- z.sqrt[p1(1-p1)/n1+p2(1-p2)/n2]
- fpc = sqrt[(N-n)/(N-1)], used when sampling without replacement and n>0.05N
Exam Tips
- Always identify FIRST which of the three mean-CI cases applies: sigma known (Z, exact); large n sigma unknown (Z, approx); small n normal sigma unknown (t)
- Never confuse the confidence coefficient (1-alpha) with a probability statement about one already-computed interval — it's a long-run repeated-sampling statement
- For difference-of-means problems, check whether variances are assumed equal (pooled t) or samples are simply large (Z with separate variances)
- Remember z-values by heart: 90%->1.645, 95%->1.960, 98%->2.326, 99%->2.576 — these appear constantly
- For small-sample t-interval problems, always compute degrees of freedom v=n-1 (or n1+n2-2 for two samples) BEFORE looking up the table value
- Check the 5% rule (n vs 0.05N) before deciding whether to include the finite population correction factor