22576 Chap.7 Sampling, Standard Error and Confidence Intervals
Sampling, Standard Error and Confidence Intervals
Week 5 asks whether conclusions from a sample hold for the population. This chapter separates sampling error, which a larger sample reduces, from sampling bias, which only better selection fixes, and defines the standard error as the variation of the sample mean from sample to sample.
It builds interval estimates at the 90, 95 and 99% levels and closes with the random, stratified and equal-size sampling practised in KNIME.
These calculations are short but easy to get wrong under time pressure, so the chapter shows one survey worked at all three confidence levels.
The key habit is to compute the standard error once, then change only the multiplier, and to write every conclusion as a range with its confidence level and units rather than as a single figure.
What this chapter covers
- 01
Statistical inference from samples
- 02
Sampling error versus sampling bias
- 03
Standard error of the mean
- 04
Interval estimates and margins of error
- 05
Confidence levels and their multipliers
- 06
Random, stratified and equal-size sampling
Worked example · free
Build three confidence intervals from one café survey
- 1Standard error: 6.40 ÷ √64 = 6.40 ÷ 8 = $0.80.
- 190%: 18.40 ± 1.645 × 0.80 = 18.40 ± 1.316, giving $17.08 to $19.72.
- 195%: 18.40 ± 1.960 × 0.80 = 18.40 ± 1.568, giving $16.83 to $19.97.
- 199%: 18.40 ± 2.576 × 0.80 = 18.40 ± 2.061, giving $16.34 to $20.46.
Key terms
- Sampling bias
- Distortion that arises when the selection process favours certain members of the population, making the sample unrepresentative.
- Interval estimate
- A range for a population value, the sample mean plus or minus a margin of error.
- Stratified sampling
- Sampling the same percentage from each group of a chosen variable so group shares are kept.
- Sampling error
- The natural variation that arises from using a sample instead of the whole population.
- Confidence level
- The chosen level of certainty, such as 90, 95 or 99%, that sets the width of an interval.
Sampling, Standard Error and Confidence Intervals FAQ
Does a larger sample fix a biased sample?
No. A larger sample reduces sampling error, the natural variation from using a sample, but if some customers are more likely to be selected, such as only unhappy customers answering a survey, the bias remains however many replies arrive. Random selection is the remedy for bias.
Why is a 99% interval wider than a 95% one?
Higher confidence needs a larger multiplier on the standard error, 2.576 instead of 1.960, so the range must widen to be more certain of containing the true mean. The only way to narrow an interval without lowering confidence is a larger sample, which shrinks the standard error.
How do I write the conclusion of an interval estimate?
State both bounds, the confidence level and the units, for example that the true average spend is likely to lie between two dollar amounts at 95% confidence. Avoid claiming the sample mean is the true value, and avoid saying the interval is more accurate at higher confidence.
How is the standard error calculated?
Divide the sample standard deviation by the square root of the sample size. A survey of 100 customers with a standard deviation of $20 has a standard error of 20 divided by 10, which is $2; quadrupling the sample to 400 halves the standard error to $1.
What does equal size sampling do?
It draws the same number of observations from each category of a chosen variable, which corrects an over-represented group before groups are compared. In the Week 6 tutorial it is the method that best addressed a sample dominated by married home owners, checked afterwards with a composition analysis.
What is the difference between random and stratified sampling?
A random sample selects observations independently, so the mix of groups follows the original data by chance. A stratified sample takes the same percentage from each group of a chosen variable, so every group keeps its share exactly. In KNIME both are options of the Table Partitioner node.
Assessment move
Memorise the three multipliers and practise intervals with square-number sample sizes until each takes under a minute. Then rerun the Week 6 sampling exercise in KNIME and use the Pivot node to compare the composition each sampling method produces. Write the 90, 95 and 99% multipliers at the top of your rule sheet and check every interval by confirming it is centred on the sample mean.
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