How do you test a claim about data and estimate a population value with a confidence interval?
Carrying out and interpreting hypothesis tests (t-tests and z-tests), using the p-value and significance level to reach a conclusion, constructing and interpreting confidence intervals, and recognising errors in statistical testing.
A focused answer to the SQA Higher Applications of Mathematics inferential statistics content, covering null and alternative hypotheses, p-values and significance levels, t-tests and z-tests, confidence intervals, and errors in statistical testing.
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What this dot point is asking
The SQA wants you to test a claim about a population using a hypothesis test, interpret the p-value against a significance level, draw a conclusion in context, construct and interpret a confidence interval, and recognise the errors that statistical testing can make. Software computes the test statistic and p-value; the examined skill is setting up, interpreting and concluding correctly.
The logic of a hypothesis test
A hypothesis test decides whether sample data give enough evidence to reject a default claim.
You set a significance level (commonly , written ) before testing. This is the threshold of evidence required to reject .
The p-value and the decision
Software returns a p-value for the test. Interpreting it correctly is the heart of the topic.
A small p-value means the observed data would be very unlikely if were true, which is evidence against . Always state the conclusion in context, for example "there is evidence that the mean weight differs from g", not just "reject ".
t-tests and z-tests
Both tests compare a sample mean with a claimed value, but they differ in what is known about the spread.
In practice at Higher you will most often run a t-test, because the population standard deviation is rarely known. The software handles the distribution; you choose the right test and interpret the output.
Confidence intervals
A confidence interval estimates a population parameter (such as the mean) with a range rather than a single number, attaching a confidence level.
Errors in statistical testing
Testing can reach a wrong conclusion by chance.
Try this
Q1. A test gives a p-value of at the significance level. State the conclusion. [2 marks]
- Cue. Since , reject ; the result is statistically significant.
Q2. State whether a t-test or a z-test is used when the population standard deviation is unknown. [1 mark]
- Cue. A t-test, because the standard deviation is estimated from the sample.
Q3. A confidence interval for a mean is . What happens to its width at the level? [2 marks]
- Cue. It becomes wider, because greater confidence requires a larger range of plausible values.
Exam-style practice questions
Practice questions written in the style of SQA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
SQA Higher Apps style5 marksA manufacturer claims its bags of crisps contain g on average. A sample is tested and software returns a p-value of for the test against this claim. Using a significance level, state the hypotheses and the conclusion, and explain what the p-value means.Show worked answer →
The null hypothesis is : the mean weight is g; the alternative is : the mean weight is not g (1 mark).
The p-value is the probability of obtaining a sample result at least as extreme as the one observed if were true (1 mark).
Since the p-value is less than the significance level , the result is statistically significant, so we reject (2 marks).
There is evidence that the true mean weight differs from the claimed g (1 mark). Markers reward clear hypotheses, the comparison of p-value with the significance level, and a conclusion stated in context.
SQA Higher Apps style4 marksA confidence interval for the mean daily screen time of teenagers is hours. Interpret this interval, and explain what a interval from the same data would look like by comparison.Show worked answer →
We are confident that the true mean daily screen time for the population of teenagers lies between and hours (2 marks).
A confidence interval would be wider than the interval, because demanding greater confidence requires capturing a larger range of plausible values (1 mark).
So raising the confidence level increases the interval width for the same sample, trading precision for certainty (1 mark). Markers reward a correct interpretation referring to the population mean and the point that higher confidence widens the interval.
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