COMM5000 Chap.3 Sampling Distributions and Estimation
Sampling Distributions and Estimation
Separate population parameters, sample statistics and sampling distributions, then use standard errors and the central limit theorem to describe estimator precision. The population contains the units relevant to the business question and has an unknown parameter such as a mean or proportion. One sample provides observations and produces a statistic.
If the sampling process were repeated, the statistic would change; the distribution of those hypothetical statistics is the sampling distribution. Its centre reveals whether an estimator is unbiased under the design, and its spread is the standard error. For independent observations, the standard error of a sample mean decreases with the square root of sample size.
Quadrupling the sample therefore halves, rather than quarters, this source of uncertainty. The central limit theorem supports approximate normal inference for sample means under suitable conditions as samples grow. It does not make the original data normal and does not erase dependence, severe selection or measurement error. Sample proportions have their own standard error based on p and one minus p.
In practice p is unknown and later interval methods estimate the standard error with the sample proportion. Precision and representativeness answer different questions. A large convenience sample can estimate the wrong population quantity with a very small standard error. Report the sampling method, target population and exclusions beside the estimate so the reader does not confuse numerical precision with validity.
What this chapter covers
- 01
Population and sample
- 02
Point estimators
- 03
Sampling distributions
- 04
Standard errors
- 05
Central limit theorem
Worked example · free
Probability for a sample mean
- +1The four marks used here are not an official university marking scheme. The mean of the sampling distribution is 12.
- +1Standard error = 4/sqrt(64) = 0.5.
- +1z = (13 - 12)/0.5 = 2.
- +1The right-tail probability is about 0.0228.
Key terms
- Population parameter
- A numerical feature of the full population targeted by an analysis.
- Point estimator
- A statistic used to estimate an unknown population parameter.
- Standard error
- The standard deviation of an estimator across repeated samples.
Sampling Distributions and Estimation FAQ
What does the central limit theorem say?
Under suitable conditions, the sampling distribution of the mean becomes approximately normal as sample size grows. It does not claim that the individual observations become normal.
Does a large sample remove bias?
Not necessarily. A large sample reduces random standard error under the design, but systematic exclusion, nonresponse or measurement problems can remain and may be estimated very precisely.
How should I explain standard error to a manager?
Describe it as the expected repeated-sample variation of an estimator under the stated sampling process. It is not the typical distance of individual observations from the mean. Show how sample size enters through a square root and explain that quadrupling an independent sample halves the standard error.
Then separate precision from representation: a small standard error can coexist with coverage, nonresponse or measurement bias. Pair the numerical value with the population, sampling method and dependence assumptions so the manager understands what the precision statement includes and which sources of uncertainty remain outside it.
Assessment move
Draw three boxes labelled population, sample and sampling distribution. Put every symbol and number in the correct box before using a formula. For a sample mean, state the centre of its sampling distribution and compute the standard error from the individual standard deviation and sample size. For a proportion, identify p, one minus p and n separately.
Practise a direction check: an average of many independent observations should vary less than one observation, so its standard error must be smaller than the population standard deviation. Interpret probabilities about sample statistics with explicit repeated-sample wording.
Then challenge the sampling design by asking who could not enter the sample, whether observations are dependent, and whether the process changed from the period used to set model inputs. Compare two sample sizes to see the square-root precision rule and explain why more data do not automatically repair bias. In written work, distinguish an estimate, its standard error and the variability of individual values.
End each solution with a sentence on representation and dependence, because those conditions determine whether the theoretical sampling distribution is relevant to the business population. Simulate the logic by imagining ten repeated samples and sketching how their means would cluster more tightly than the raw observations. Explain why quadrupling n halves the standard error and use that rule as a numerical check.
For clustered observations, name the shared source of dependence and describe why the effective information may be smaller than the row count. Keep bias and precision in separate columns of a revision table. Under bias, list frame coverage, nonresponse and measurement; under precision, list variability and sample size.
When reporting an estimator, pair the numerical standard error with a description of the design that makes it meaningful. If the design is uncertain, state that limitation rather than allowing a small standard error to imply confidence in population coverage.
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