S5 Statistical hypothesis testing — 9MA0 spec coverage

S5.1

Apply the language of hypothesis testing via a binomial model: null/alternative hypothesis, significance level, test statistic, 1- and 2-tail tests, critical value/region, acceptance region, p-value; extend to correlation coefficients.

The null hypothesis $H_0$ gives the reference parameter value; the alternative $H_1$ states the direction or difference supported by the claim being tested.
Use a one-tailed test for a pre-specified directional alternative and a two-tailed test for any change; choose this before observing the data.
The critical region contains outcomes sufficiently unlikely under $H_0$ ; the acceptance region is its complement, and the p-value is the probability under $H_0$ of an outcome at least as extreme as observed.
For a correlation test, use $H_0:\rho=0$ and compare the sample product-moment correlation coefficient with the supplied critical value; significance does not establish causation.

Tier 1 · Easy

2 marks

ORIGINAL

A company claims that the probability $p$ of a customer choosing its premium plan has increased from $0.40$ . State suitable hypotheses and identify the number of tails.

Tier 2 · Standard

4 marks

ORIGINAL

Under $H_0$ , $X\sim\operatorname{B}(20,0.2)$ . For an upper-tailed test at the $5\%$ level, $P(X\geq7)=0.0867$ and $P(X\geq8)=0.0321$ . State the critical region, the critical value and the acceptance region.

Tier 3 · Hard

6 marks

ORIGINAL

For a sample of $18$ paired observations, a two-tailed $5\%$ correlation test has critical values $-0.468$ and $0.468$ . The sample product-moment correlation coefficient is $r=-0.520$ , with p-value $0.026$ . State the hypotheses, carry out the test and interpret the p-value without claiming causation.

S5.2

Conduct a hypothesis test for the proportion in the binomial distribution and interpret results in context; a sample makes an inference about the population; the significance level is the probability of incorrectly rejecting H0.

Model the number of sample successes by $X\sim\operatorname{B}(n,p_0)$ under $H_0:p=p_0$ , provided the binomial assumptions are defensible.
Calculate the probability, under $H_0$ , of the observed result or one more extreme in the direction specified by $H_1$ ; double an appropriate tail for a symmetric two-tailed binomial test.
Reject $H_0$ when the p-value is at most the significance level; otherwise say there is insufficient evidence to reject $H_0$ , not that $H_0$ has been proved.
The significance level is the probability of incorrectly rejecting $H_0$ when it is true; discreteness often makes the actual probability of the critical region smaller than the nominal level.

Tier 1 · Easy

4 marks

ORIGINAL

A coin is tested with $H_0:p=0.5$ against $H_1:p>0.5$ . It lands heads $10$ times in $12$ tosses. Given $P(X\geq10)=0.0193$ for $X\sim\operatorname{B}(12,0.5)$ , conduct the test at the $5\%$ level.

Tier 2 · Standard

5 marks

ORIGINAL

A process is tested using $H_0:p=0.5$ against $H_1:p\neq0.5$ . In $20$ independent trials there are $5$ successes. Given $P(X\leq5)=0.02069$ under $X\sim\operatorname{B}(20,0.5)$ , conduct a two-tailed test at the $5\%$ level.

Tier 3 · Hard

6 marks

ORIGINAL

A one-tailed test uses $H_0:p=0.2$ against $H_1:p>0.2$ with a sample of $15$ . Under $H_0$ , $P(X\geq6)=0.0611$ and $P(X\geq7)=0.0181$ . Find the $5\%$ critical region. If $6$ successes are observed, state the conclusion and explain the actual probability of a Type I error.

S5.3

Conduct a statistical hypothesis test for the mean of a Normal distribution with known, given or assumed variance and interpret the results in context.

For a Normal population with known or assumed standard deviation $\sigma$ , the sample mean satisfies $\overline{X}\sim\operatorname{N}(\mu,\sigma^2/n)$ .
Under $H_0:\mu=\mu_0$ , standardise the observed mean using $Z=\frac{\overline{x}-\mu_0}{\sigma/\sqrt n}$ .
Use the tail or tails specified by $H_1$ , compare the p-value with the significance level, and give the conclusion in the language of the population mean.
The method relies on a random, independent sample from a Normal population and uses a known, given or assumed variance rather than estimating it within this specified test.

Tier 1 · Easy

5 marks

ORIGINAL

A Normal population has known standard deviation $12$ . Test $H_0:\mu=100$ against $H_1:\mu>100$ using a random sample of $36$ with mean $104$ , at the $5\%$ level.

Tier 2 · Standard

6 marks

ORIGINAL

A Normal population has known standard deviation $5$ . For a sample of $25$ , test $H_0:\mu=50$ against $H_1:\mu\neq50$ at the $1\%$ level. The critical standard Normal values are $\pm2.576$ . Find the critical values of the sample mean and decide what to conclude if $\overline{x}=47.3$ .

Tier 3 · Hard

7 marks

ORIGINAL

A Normal population has known standard deviation $6$ . To test $H_0:\mu=80$ against $H_1:\mu>80$ at the $5\%$ level, find the smallest sample size $n$ for which an observed mean of $82$ would lead to rejection. Use the critical value $1.645$ , then find the p-value for this minimum $n$ .