Math stats external report

AS91584 (Evaluate statistically based reports) worth 4 credits.

What is a population?

The entire target group you want to draw conclusions about.

What is a sample?

A subset of the population that is actually observed or surveyed.

What is a parameter?

A value that describes the population.

What is a statistic?

A value calculated from a sample.

What is sampling variability?

The fact that different random samples can produce slightly different results.

What is a categorical variable?

A variable that places observations into categories.

What is a quantitative variable?

A variable measured numerically, such as age or income.

What is a nominal categorical variable?

Categories with no natural order, e.g. preferred political party.

What is an ordinal categorical variable?

Categories with a meaningful order, e.g. dissatisfied → neutral → satisfied.

Is 'percentage supporting' a variable?

No. It is a summary measure calculated from individual responses.

What is simple random sampling?

Individuals are randomly selected from the population or sampling frame.

What is stratified sampling?

The population is divided into groups (strata), then a random sample is taken from each group.

Why use stratified sampling?

It can ensure important groups are represented in the sample.

What is systematic sampling?

Selecting every kth person after a random starting point.

What is voluntary-response sampling?

People choose themselves to participate, which can create bias.

What is a major problem with an online poll where readers choose to participate?

Voluntary-response bias and possible coverage bias.

What is a target population?

The specific population the report wants to make conclusions about.

Why must a report state its target population?

So we know who the results can reasonably be generalised to.

What is a sampling frame?

The list or method used to identify members of the population who can be selected.

Why does a low response rate matter?

Non-responders may differ systematically from responders, creating non-response bias.

How can question wording create bias?

Leading or loaded wording can influence how people respond.

Why does the date of data collection matter?

Opinions can change over time, especially during events such as elections.

How do you calculate a sample proportion?

number with the characteristic ÷ total sample × 100%.

What is a percentage point increase?

The direct difference between two percentages.

Support increases from 52% to 61%. What is the percentage-point increase?

9 percentage points.

How do you calculate relative percentage increase?

(new − original) ÷ original × 100%.

Support rises from 52% to 61%. What is the relative increase?

About 17.3%.

What is the approximate 95% margin of error for a proportion?

MOE ≈ 1/√n.

What happens to margin of error when sample size increases?

It decreases.

What happens to the margin of error if the sample size is multiplied by 4?

The margin of error is halved.

What is the MOE for n = 1,000?

About 3.2%.

What is the MOE for n = 400?

5%.

What is the approximate 95% interval for 58% with n = 1,000?

54.8% to 61.2%.

What is a confidence interval?

A range of plausible values for the population parameter, based on the sample and statistical method.

When can you support a claim that a majority of the population agrees?

When the entire interval is above 50%.

If an interval is (48%, 55%), can you conclude that a majority agrees?

No. The interval includes values below 50%.

If an interval is (55%, 65%), can you conclude that a majority agrees?

Yes. The entire interval is above 50%.

What does a difference interval represent?

The plausible values for the difference between two population proportions.

For A − B, what does a positive interval mean?

A is estimated to be higher than B.

For A − B, what does a negative interval mean?

A is estimated to be lower than B.

What does it mean if an A − B interval contains 0?

There is no clear evidence of a population difference.

Interpret (0.08, 0.16) for A − B.

A is estimated to be 8–16 percentage points higher than B.

Interpret (−0.12, −0.04) for A − B.

A is estimated to be 4–12 percentage points lower than B.

Interpret (−0.03, 0.05) for A − B.

The interval includes 0, so there is no clear evidence of a population difference.

If Poll A is 51% ± 3% and Poll B is 55% ± 3%, has public opinion definitely changed?

No. Sampling variability could explain the difference.

Why shouldn't you rely only on overlapping intervals?

Overlap is only a rough guide; an interval for the difference is more appropriate.

Why can't polls with different target populations be directly compared?

They estimate different population parameters.

Why is an opt-in website poll potentially biased?

It can have voluntary-response and coverage bias.

What is a non-sampling error?

An error caused by the way data is collected, measured, processed, or represented rather than random sampling variation.

What is coverage error?

Some members of the target population are missing or under-represented in the sampling frame.

What is non-response bias?

Responders differ systematically from people who do not respond.

What is measurement error?

Incorrect or inaccurate responses caused by misunderstanding, poor recall, or socially desirable answers.

What is processing error?

Errors when data is entered, coded, weighted, calculated, or processed.

Does increasing sample size remove a biased question?

No. A larger sample reduces sampling error but does not remove systematic bias.

Does an observational study prove causation?

Generally, no. It can show an association but not establish cause and effect.

What is a confounding variable?

A variable that is related to both the explanatory variable and the outcome and may explain the observed association.

What type of study best supports a causal conclusion?

A suitable experiment.

Why is random allocation important in an experiment?

It helps create comparable groups and reduces the effect of confounding variables.

Instead of saying 'Social media causes anxiety,' what could you say about observational data?

'Social media use is associated with anxiety.'

Why can a truncated vertical axis be misleading?

It can exaggerate the apparent size of differences.

What should you check when a graph uses percentages?

Check what the percentages are percentages of, and whether the denominator is appropriate.

Why should a report identify its data source?

So the reliability and origin of the data can be evaluated.

What is selective presentation?

Presenting data in a way that emphasises a particular message while potentially hiding relevant information.

Why is 'Exactly 58% of all voters support the policy' inappropriate?

58% is a sample statistic, not necessarily the exact population proportion.

How should you describe a sample result cautiously?

Use wording such as 'the sample estimated…' or 'the results suggest…'.

Why is 'Group A is definitely better because 54% is above 51%' inappropriate?

The difference may be due to sampling variability.

Why is 'The poll proves the advertisement changed opinions' inappropriate?

A poll/observational study does not necessarily establish causation.

If an interval is (48%, 55%), can you say 'most people agree'?

No. The interval includes values below 50%.

Can a sample from one school automatically represent everyone?

No. Generalisation depends on the target population and sampling method.

What are the main things to check when evaluating a statistical report?

Population, sample, sampling method, response rate, question wording, timing, uncertainty, data source, and possible bias.

What is sampling error?

Random variation that occurs because a sample rather than the entire population is studied.

What is the key difference between sampling and non-sampling error?

Sampling error is random sample-to-sample variation; non-sampling errors are problems such as coverage, non-response, measurement, or processing errors.