STAT1008 Chap.10 Sampling Distributions and the Central Limit Theorem
Sampling Distributions and the Central Limit Theorem
Sampling Distributions and the Central Limit Theorem treats a statistic as a random variable whose value changes from sample to sample. The sample mean is unbiased for the population mean, with standard error sigma over the square root of n. For a normal population the sample mean is exactly normal; by the Central Limit Theorem it is approximately normal for large i.i.d.
samples, with n of at least 30 as the course’s guide. The same logic gives the sampling distribution of the sample proportion. Assessment questions here usually supply a population mean and standard deviation, or a population proportion, and ask for the probability that a sample statistic falls in some range. The skill being tested is choosing the right standard deviation and justifying normality for the statistic.
What this chapter covers
- 01
Repeated sampling and the sampling distribution
- 02
Unbiasedness of the sample mean
- 03
Standard error and sample size
- 04
Sampling from a normal population
- 05
The Central Limit Theorem
- 06
Sampling distribution of a proportion
Worked example · free
Probability that a sample mean of grocery spending exceeds $195
- 1n = 36 ≥ 30 and households are sampled independently, so the CLT applies.
- 1Standard error = 45/√36 = $7.50, so the sample mean is approximately Normal(180, 7.5).
- 2z = (195 − 180)/7.5 = 2.00; the table gives 0.9772 below 2.00, so the upper tail is 0.0228.
Key terms
- Sampling distribution
- A sampling distribution describes how the values of a statistic are spread across repeated samples of the same size.
- Standard error
- The standard error is the standard deviation of a statistic’s sampling distribution.
- Central Limit Theorem
- The Central Limit Theorem states that the mean of a large i.i.d. sample is approximately normal whatever the population shape.
- Unbiasedness
- Unbiasedness is the property of an estimator whose expected value equals the parameter being estimated.
- Sample proportion
- The sample proportion is the number of sampled units with a characteristic divided by the sample size.
Sampling Distributions and the Central Limit Theorem FAQ
Does the Central Limit Theorem make the data normal?
No. It describes the distribution of the sample mean, not of individual observations. A skewed population stays skewed, but the average of a large independent sample from it is approximately normal.
How large must a sample be for the Central Limit Theorem?
The course uses n of at least 30 as a working guide. The further the population is from normal, the larger the sample needs to be, and for a population already known to be normal any sample size gives an exactly normal mean.
What conditions apply to the sampling distribution of a proportion?
The sample proportion is approximately normal when np and n(1 minus p) both exceed 5. Its mean is p and its standard error is the square root of p(1 minus p) divided by n.
What is the difference between a standard deviation and a standard error?
The standard deviation describes how individual observations vary around the population mean. The standard error describes how a statistic, such as the sample mean, varies from one sample to the next; for the mean it equals sigma divided by the square root of n.
Why does a larger sample give a more precise mean?
The standard error of the mean shrinks with the square root of the sample size, so sample means from larger samples cluster more tightly around the population mean. Quadrupling the sample size halves the standard error.
Exam move
For each practice question, first decide whether it concerns one observation or a sample mean, and write the relevant standard deviation before any calculation. Compute standard errors for several sample sizes to feel the square-root effect. Then pair every mean question with a proportion version, checking the np and n(1 minus p) conditions with numbers.
Practise with pairs of questions that share a population: one about a single observation and one about a sample mean. Seeing the two probabilities side by side, for example 0.37 for one household against 0.02 for the mean of 36, fixes the difference between sigma and the standard error firmly.
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