MEDD8815 Chap.4 Sampling Distributions, Confidence Intervals and Tests
Sampling Distributions, Confidence Intervals and Tests
Why this chapter is the hinge of the course
Everything before Session 4 describes data; everything after it draws conclusions about populations. The bridge is the sampling distribution: the pattern of values a statistic takes across repeated samples.
This chapter builds that idea for a sample mean and a sample proportion, then uses it to construct confidence intervals and significance tests in the simplest setting the course offers, a mean when the population standard deviation is treated as known.
Sampling distributions and the central limit theorem
The sample mean is centred on the population mean, and its standard deviation is the population standard deviation divided by the square root of the sample size.
If the population is Normal, so is the sample mean; if not, the central limit theorem makes it approximately Normal in large samples. The same reasoning applies to a sample proportion, which is simply the mean of zeros and ones.
A computed figure shows the sampling distribution narrowing as the sample grows.
Confidence intervals
A confidence interval is an estimate plus or minus a margin of error, and its confidence level is the long-run success rate of the method rather than a probability about one interval. A simulation of many intervals makes this visible.
The chapter works a 95 per cent and a 90 per cent interval from one sample, shows how the margin depends on the confidence level, the spread and the sample size, and solves for the sample size needed to reach a chosen margin.
Significance tests, errors and power
A test states hypotheses about a parameter, computes a standardised test statistic, finds a P-value and concludes.
The chapter follows those four steps on the same running sample, shows that a one-sided and a two-sided analysis of the same data can reach different verdicts, and connects two-sided tests to confidence intervals. It closes with the two ways a test can be wrong and with power, computed for a specific alternative, as the planning question every study should ask.
What this chapter covers
- 01
Population distributions and sampling distributions
- 02
The centre and spread of a sample mean
- 03
The central limit theorem
- 04
The sampling distribution of a sample proportion
- 05
Confidence intervals and what confidence means
- 06
Margin of error and choosing a sample size
- 07
Hypotheses, test statistics and P-values
- 08
Type I and Type II errors and power
Worked example · free
Choose the sample size before collecting the data
- 1Write the sample-size formula and the critical value for 95 per cent.
- 2Substitute and round in the correct direction.
- 1Explain the effect of halving the margin.
Key terms
- Central Limit Theorem
- The result that the mean of a large random sample is approximately Normally distributed, whatever the shape of the population it came from.
- Confidence Interval
- An interval computed from sample data by a method that captures the parameter in a stated proportion of repeated samples, the confidence level.
- Margin of Error
- The amount added to and subtracted from an estimate to form a confidence interval; it covers random sampling error only.
- Null Hypothesis
- The statement being tested in a significance test, usually that there is no effect or no difference, written in terms of a parameter.
- P-value
- The probability, computed assuming the null hypothesis is true, of a test statistic at least as extreme as the one observed.
- Significance Level
- The value alpha fixed before a test, below which a P-value is called statistically significant; it also fixes the chance of a Type I error.
- Statistical Power
- The probability that a test rejects the null hypothesis when a particular alternative value of the parameter is true.
Sampling Distributions, Confidence Intervals and Tests FAQ
What does 95 per cent confident actually mean?
It describes the method, not the single interval you computed. If you drew many samples and built an interval from each in the same way, about 95 per cent of those intervals would capture the true parameter. Your one interval either contains it or does not.
Why do we never say we accept the null hypothesis?
A large P-value shows only that the data are consistent with the null hypothesis, not that it is true; a small study can easily miss a real effect. The correct wording is that we fail to reject it, and a confidence interval shows which values remain plausible.
When should the alternative hypothesis be one-sided?
Only when the direction of interest was stated before the data were seen, for example a suspicion that a new timetable reduced homework time. Choosing the direction after looking at the data halves the P-value unfairly, and the chapter shows a case where that choice changes the verdict.
Assessment move
Carry one small example through all four uses on a single sheet: its sampling distribution, a 95 per cent interval, a one-sided test and the power against one alternative. Seeing the same standard error appear in all four is the fastest way to understand the chapter, and every later session reuses that sheet with a different standard error.
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