Hypothesis testing (binomial, correlation, Normal mean) – A-level Maths exam questions with mark scheme

1StandardCalculator5 marks

A supermarket claims that 25% of its customers use the self-checkout tills. The manager of one branch believes that the proportion at her branch is lower than this. She takes a random sample of 30 customers and finds that 3 of them used the self-checkout tills.

Test, at the 5% level of significance, whether there is evidence to support the manager's belief. State your hypotheses clearly. [5]

$\mathrm{H}_0: p=0.25,\quad \mathrm{H}_1: p<0.25$B1
$X\sim\mathrm{B}(30,\,0.25)$ usedM1
$\mathrm{P}(X\le 3)=$ awrt 0.0374 (or critical region $X\le 3$ with $\mathrm{P}(X\le3)=0.0374$, $\mathrm{P}(X\le4)=0.0979$)A1
compares with 0.05 (or 3 in CR) and draws a correct non-contextual inference: reject $\mathrm{H}_0$ / significantM1
conclusion in context: there is evidence that the proportion of customers using self-checkout at this branch is less than 25% (the manager's belief is supported)A1
Answer: $\mathrm{P}(X\le3)=0.0374<0.05$: reject $\mathrm{H}_0$ – evidence the proportion is lower than 25%
Examiner tip: Hypotheses must use the parameter $p$, and the final A1 needs a non-assertive conclusion in context ('there is evidence that…').

2HardCalculator6 marks

In a large population, 20% of people are left-handed. A researcher wants to test whether the proportion of left-handed people among musicians is different from 20%. She takes a random sample of 40 musicians.

(a) Using a 5% level of significance, find the critical region for a two-tailed test of whether the proportion is different from 20%. The probability in each tail should be as close as possible to, but not more than, 2.5%. [3]

(b) Find the actual significance level of this test. [1]

In her sample, 13 musicians are left-handed.

(a) $X\sim\mathrm{B}(40,0.2)$ with evidence of using cumulative probabilities in a tail, e.g. $\mathrm{P}(X\le2)=0.0079$ or $\mathrm{P}(X\ge14)=0.0194$M1
(a) $X\le 2$A1
(a) $X\ge 14$A1
(b) $0.0079+0.0194=$ awrt 0.0273 (2.73%)B1
(c) 13 is not in the critical region / do not reject $\mathrm{H}_0$M1
(c) insufficient evidence that the proportion of left-handed musicians is different from 20%A1
Answer: (a) $X\le2$ or $X\ge14$; (b) 0.0273; (c) not significant – no evidence the proportion differs from 20%
Examiner tip: For the upper tail use $\mathrm{P}(X\ge14)=1-\mathrm{P}(X\le13)$ – an off-by-one error here loses the A1.

3StretchCalculator8 marks

A manufacturer of light bulbs states that 10% of its bulbs are faulty.

(a) A quality inspector selects a random sample of $n$ bulbs. Find the smallest value of $n$ for which the probability that none of the bulbs is faulty is less than 0.01. [3]

After a change in the production process, the manufacturer claims the proportion of faulty bulbs has decreased. A random sample of 50 bulbs is taken and 1 is found to be faulty.

(b) Test, at the 5% level of significance, whether there is evidence to support the manufacturer's claim. State your hypotheses clearly. [5]

(a) $0.9^n<0.01$M1
(a) $n>\frac{\ln 0.01}{\ln 0.9}$ $(=43.7\ldots)$, or trial of $0.9^{43}=0.0108$ and $0.9^{44}=0.0097$M1
(a) $n=44$A1
(b) $\mathrm{H}_0:p=0.1,\quad \mathrm{H}_1:p<0.1$B1
(b) $X\sim\mathrm{B}(50,0.1)$ and attempts $\mathrm{P}(X\le1)$M1
(b) awrt 0.0338A1
(b) $0.0338<0.05$ so reject $\mathrm{H}_0$ (significant)M1
(b) there is evidence that the proportion of faulty bulbs has decreased (supports the claim)A1
Answer: (a) $n=44$; (b) $\mathrm{P}(X\le1)=0.0338<0.05$ – evidence the proportion has decreased
Examiner tip: Dividing by $\ln 0.9$ (a negative number) flips the inequality – check your $n$ by substituting back.

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4StandardCalculator5 marks

Using the large data set, a student takes a random sample of 20 days from one UK weather station. She calculates the product moment correlation coefficient between daily mean pressure and daily total rainfall and obtains $r=-0.412$.

(a) Interpret this value of $r$ in context. [1]

(b) Test, at the 5% level of significance, whether there is evidence of negative correlation between daily mean pressure and daily total rainfall. State your hypotheses clearly. [4]

(a) negative correlation in context, e.g. days with higher (mean) pressure tend to have less rainfall (allow 'weak' or 'moderate' negative; reject 'strong'; reject answers stating that pressure causes less rainfall)B1
(b) $\mathrm{H}_0:\rho=0,\quad \mathrm{H}_1:\rho<0$ (both, in terms of $\rho$)B1
(b) critical value $-0.3783$ (allow $0.3783$; accept CR $r<-0.3783$)B1
(b) correct comparison, e.g. $-0.412<-0.3783$ (or $0.412>0.3783$), so reject $\mathrm{H}_0$ / significant / in critical regionM1
(b) there is evidence of negative correlation between daily mean pressure and daily total rainfall (must be in context; do not accept 'pressure causes less rainfall')A1
Answer: (b) $r=-0.412 < -0.3783$: reject $\mathrm{H}_0$ – evidence of negative correlation
Examiner tip: Hypotheses must use $\rho$ (the population PMCC), not $r$ – writing $\mathrm{H}_0:r=0$ loses the B1.

5StretchCalculator7 marks

The lengths, $X$ mm, of rods made by a machine are normally distributed with standard deviation 3 mm. Before a service, the mean length was 25 mm. After the service, an engineer believes the mean length has increased. She takes a random sample of 16 rods.

(a) Write down suitable hypotheses for a test of the engineer's belief. [1]

(b) Find the critical region for $\bar{X}$, the sample mean, at the 5% level of significance. [3]

The sample mean length of the 16 rods is 26.4 mm.

(d) Write down the probability of incorrectly concluding that the mean has increased when it has not. [1]

(a) $\mathrm{H}_0:\mu=25,\quad \mathrm{H}_1:\mu>25$B1
(b) $\bar X\sim\mathrm{N}\!\left(25,\frac{3^2}{16}\right)$, i.e. standard deviation 0.75M1
(b) $\frac{\bar x-25}{0.75}>1.6449$M1
(b) $\bar X>26.23$ (awrt 26.2)A1
(c) 26.4 is in the critical region (or $z=1.87>1.6449$, or $p=0.031<0.05$), so reject $\mathrm{H}_0$M1
(c) there is evidence that the mean length of rods has increased after the serviceA1
(d) 0.05B1
Answer: (b) $\bar X>26.23$; (c) reject $\mathrm{H}_0$ – evidence the mean has increased; (d) 0.05
Examiner tip: The sample mean has standard deviation $\sigma/\sqrt{n}=0.75$, not 3 – using 3 is the most common error.

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