What is S5.1: Hypothesis testing language?

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.

How many marks are S5.1 questions worth?

S5.1 practice drills on this page are worth 2–6 marks each; marks in a real exam vary by question.

Which past exam papers has S5.1 appeared in?

We have not yet indexed a verified past-paper appearance for S5.1 — check the Edexcel A-level Maths 9MA0 past-papers hub directly.

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.

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Hypothesis testing language

Worked answers and methods for S5.1 on Edexcel A-level Maths 9MA0.

Explanation

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.
The correlation coefficient satisfies $|r|\leq1$ : $r=1$ or $-1$ means perfect positive or negative linear correlation.
For a supplied calculator test, compare the p-value with the significance level, or $|r|$ with the critical value, in the stated tail.

Worked example

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.

1.Choose the smallest upper-tail boundary whose probability under $H_0$ does not exceed $0.05$ .
2.The boundary $7$ is too liberal because $0.0867>0.05$ , while $P(X\geq8)=0.0321<0.05$ .
3.Hence the critical region starts at $8$ and its complement is $X\leq7$ .

Answer: Critical region: $X\geq8$ .; Critical value: $8$ .; Acceptance region: $X\leq7$ .

Common mistakes

Don't place non-extreme outcomes in the critical region while excluding more extreme outcomes from the same tail.
Don't choose a boundary whose tail probability exceeds the significance level, so the critical region is too large.

Exam tip

For a critical region, select the most extreme outcomes whose total null probability does not exceed the stated level.

Worked practice

Tier 1 · Easy

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.

(2)

(Total for Question 1 is 2 marks)

Mark scheme

Mark scheme for question 1
Question	Scheme	Marks
1	$H_0:p=0.40$ . $H_1:p>0.40$ . This is a one-tailed test.	2
Notes
The null uses the established value $0.40$ . The word 'increased' gives the directional alternative $p>0.40$ , so only the upper tail is relevant.

(2 marks)

Tier 2 · Standard

A hypothesis test produces a p-value of

0.037

. State the conclusion at the

5\%

significance level and at the

1\%

significance level. Explain what the significance level represents.

(4)

(Total for Question 2 is 4 marks)

Mark scheme

Mark scheme for question 2
Question	Scheme	Marks
2	Reject $H_0$ at the $5\%$ level. Do not reject $H_0$ at the $1\%$ level. The significance level is the probability, under $H_0$ , of rejecting $H_0$ when it is true.	4
Notes
Compare the p-value with each significance level. Since $0.037<0.05$ , reject $H_0$ at $5\%$ . Since $0.037>0.01$ , the evidence is not sufficient to reject $H_0$ at $1\%$ . The significance level controls the probability of a Type I error: rejecting a true null hypothesis.

(4 marks)

Tier 3 · Hard

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.

(6)

(Total for Question 3 is 6 marks)

Mark scheme

Mark scheme for question 3
Question	Scheme	Marks
3	$H_0:\rho=0$ and $H_1:\rho\neq0$ . Reject $H_0$ because $-0.520<-0.468$ (equivalently $0.026<0.05$ ). There is sufficient evidence of negative correlation in the population. If $\rho=0$ , the probability of a sample correlation at least this extreme in either direction is $0.026$ ; this does not prove causation.	6
Notes
A two-tailed association test uses $H_0:\rho=0$ against $H_1:\rho\neq0$ . The observed coefficient lies in the lower critical region, and its p-value is below $0.05$ , so reject $H_0$ . State the conclusion as evidence of population correlation, not as proof that either variable causes the other.

(6 marks)

Tier 1 · Easy

For a test based on a binomial count

X

, where

X

is the number of successes, the critical region is

X\leq2

X\geq14

. State whether the test is one-tailed or two-tailed, and state the acceptance region.

(2)

(Total for Question 4 is 2 marks)

Mark scheme

Mark scheme for question 4
Question	Scheme	Marks
4	Two-tailed Acceptance region: $3\leq X\leq13$	2
Notes
There is a critical region in each tail, so the test is two-tailed. The integer values not in either critical region run from $3$ to $13$ inclusive.

(2 marks)

Tier 2 · Standard

Under

H_0

X\sim\operatorname{B}(18,0.45)

. For a lower-tailed test at the

5\%

level,

P(X\leq4)=0.0411

and

P(X\leq5)=0.1077

. State the critical region, critical value, acceptance region and actual significance level.

(4)

(Total for Question 5 is 4 marks)

Mark scheme

Mark scheme for question 5
Question	Scheme	Marks
5	Critical region: $X\leq4$ Critical value: $4$ Acceptance region: $X\geq5$ Actual significance level: $0.0411$ (or $4.11\%$ )	4
Notes
The boundary $5$ would make the lower-tail probability exceed $0.05$ , while the probability up to $4$ is below $0.05$ . Hence the critical region is $X\leq4$ , its largest value is the critical value, and the complement is $X\geq5$ . The null probability of the chosen region is $0.0411$ .

(4 marks)

Tier 3 · Hard

For a particular sample size, the

5\%

critical value for a one-tailed positive-correlation test is

0.497

, while the two-tailed

5\%

critical values are

\pm0.576

. A pre-registered test of

H_0:\rho=0

against

H_1:\rho>0

gives

r=0.532

. Carry out this test. A second researcher chose a positive alternative only after seeing the scatter diagram; explain how this changes the appropriate conclusion.

(5)

(Total for Question 6 is 5 marks)

Mark scheme

Mark scheme for question 6
Question	Scheme	Marks
6	Reject $H_0$ for the pre-registered one-tailed test because $0.532>0.497$ ; there is sufficient evidence of positive population correlation. Choosing the direction after seeing the data is not a valid pre-specified one-tailed test. The second researcher should use the two-tailed comparison, for which $\|0.532\|<0.576$ , so there is insufficient evidence of population correlation at the $5\%$ level. Neither test establishes causation.	5
Notes
The planned directional alternative places the rejection region only in the positive tail, and $r$ exceeds its supplied critical value. A direction selected after inspection gives an unfair second opportunity to choose the favourable tail, so the non-directional two-tailed threshold is appropriate. The observed magnitude does not reach that threshold.

(5 marks)

Tier 2 · Standard

Under

H_0

, the number

X

of successes in a sample of

20

has distribution

\operatorname{B}(20,0.5)

. For a two-tailed test at the

5\%

level,

P(X\leq5)=0.0207

and

P(X\leq6)=0.0577

. State the critical region, acceptance region and actual significance level, using equal tail allocations.

(4)

(Total for Question 7 is 4 marks)

Mark scheme

Mark scheme for question 7
Question	Scheme	Marks
7	Critical region: $X\leq5$ or $X\geq15$ Acceptance region: $6\leq X\leq14$ Actual significance level $=2(0.0207)=0.0414$ , or $4.14\%$	4
Notes
Each tail may contain at most $0.025$ . The boundary $6$ is too large because its lower-tail probability is $0.0577$ , while $X\leq5$ is admissible. Symmetry about $10$ makes the corresponding upper region $X\geq15$ . The complement is $6\leq X\leq14$ , and the total null probability of the two critical regions is $2(0.0207)=0.0414$ .

(4 marks)

Tier 3 · Hard

Under

H_0:p=0.30

, the number

X

of successes in a sample of

12

has distribution

\operatorname{B}(12,0.30)

. A two-tailed test of

H_0:p=0.30

against

H_1:p\neq0.30

is to have no more than

2.5\%

in either tail. The following probabilities are rounded to

4

decimal places:

P(X\leq0)=0.0138

P(X\leq1)=0.0850

P(X\geq7)=0.0386

and

P(X\geq8)=0.0095

. Find the critical region and actual significance level. Conduct the test when

X=8

(6)

(Total for Question 8 is 6 marks)

Mark scheme

Mark scheme for question 8
Question	Scheme	Marks
8	Critical region: $X=0$ or $X\geq8$ Actual significance level $=0.0138+0.0095=0.0233$ , or approximately $2.33\%$ . Since $8$ is in the critical region, reject $H_0$ . There is sufficient evidence that the population success probability differs from $0.30$ .	6
Notes
In the lower tail, adding $X=1$ would raise the probability above $0.025$ , so only $X=0$ is included. In the upper tail, $X\geq7$ is too large but $X\geq8$ is admissible. Using the supplied probabilities, the regions have total null probability $0.0138+0.0095=0.0233$ . The observation $8$ lies on the upper critical boundary, so reject the null and state the conclusion about the population parameter $p$ .

(6 marks)

Tier 3 · Hard

Two independent studies both obtain a sample product-moment correlation coefficient

r=0.460

. Study A has sample size

18

and two-tailed

5\%

critical values

\pm0.468

. Study B has sample size

40

and two-tailed

5\%

critical values

\pm0.312

. For each study, test

H_0:\rho=0

against

H_1:\rho\neq0

. Explain why the decisions differ and state what neither result establishes.

(5)

(Total for Question 9 is 5 marks)

Mark scheme

Mark scheme for question 9
Question	Scheme	Marks
9	Study A: do not reject $H_0$ because $\|0.460\|<0.468$ ; there is insufficient evidence of population correlation. Study B: reject $H_0$ because $\|0.460\|>0.312$ ; there is sufficient evidence of a correlation. The larger sample has a smaller critical magnitude, so the same sample correlation supplies stronger evidence against $H_0$ . Neither result establishes that one variable causes the other.	5
Notes
Use the population parameter $\rho$ in the hypotheses and compare $\|r\|$ with the supplied two-tailed critical magnitude. The coefficient misses A's boundary by $0.008$ but exceeds B's by $0.148$ . Increased sample size reduces the magnitude needed for significance, while a correlation test still addresses association rather than causation.

(5 marks)

Q10

Tier 3 · Hard

10.

Under

H_0:p=0.25

, the number

X

of successes in a sample of

16

has distribution

\operatorname{B}(16,0.25)

. For an upper-tailed test,

P(X\geq7)=0.07956

P(X\geq8)=0.02713

and

P(X\geq9)=0.00747

. Find the critical region and actual significance level for a nominal

5\%

test. Conduct the test when

X=7

. State how the critical region and conclusion change at the

10\%

level.

(6)

(Total for Question 10 is 6 marks)

Mark scheme

Mark scheme for question 10
Question	Scheme	Marks
10	At the $5\%$ level, the critical region is $X\geq8$ and the actual significance level is $0.02713$ . With $X=7$ , do not reject $H_0$ ; there is insufficient evidence that the population success probability exceeds $0.25$ . At the $10\%$ level, the critical region is $X\geq7$ and its actual significance level is $0.07956$ . The observation $X=7$ is then in the critical region, so reject $H_0$ and conclude that there is sufficient evidence that $p>0.25$ .	6
Notes
For $5\%$ , including $7$ would make the null tail probability exceed $0.05$ , while $X\geq8$ is admissible. At $10\%$ , $X\geq7$ is admissible and is larger than the region starting at $8$ , so it is used. The observed boundary value is excluded from the first critical region but included in the second, producing different evidence statements at the two pre-specified levels.

(6 marks)

Verified exam appearances

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Other points in S5 Statistical hypothesis testing

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