• A Level Maths

Demo

• A Level
• Hypothesis Tests (Binomial)

1. It is known that nine in ten children like chocolate. A teacher wants to investigate whether their class of 25 students like chocolate less than average at the 10% significance level and finds that 20 of the class like chocolate.

1. Write down the null and alternate hypotheses for this test.
2. Find the probability that 20 or fewer of the 25 students like chocolate.
3. What conclusion should the teacher draw?

1. $\mathrm{H}_0: p = 0.9$ and $\mathrm{H}_1: p < 0.9$ where $p$ is the probability a student likes chocolate.
2. $X \sim B(25,0.9)$, $\mathrm{P}(X \leqslant 20) = 0.0980$
3. This is less than the significance level so we reject the null hypothesis. There is evidence to suggest that the class likes chocolate less than average.

2. On average, one in five people like pineapple on pizza. A group of 20 friends believe they like pineapple on pizza less than average and wish to test this hypothesis at the 10% significance level.

1. Write down the null and alternate hypotheses for this test.
2. 2 of the group like pineapple on pizza. Complete the test.

1. $\mathrm{H}_0: p = 0.2$ and $\mathrm{H}_1: p < 0.2$ where $p$ is the probability a person likes pineapple on pizza.
2. $X \sim B(20,0.2)$, $\mathrm{P}(X \leqslant 2) = 0.206$
   This is greater than 10%, reject $\mathrm{H}_1$
   There is insufficient evidence to suggest the group likes pineapple on pizza less than average.

3. A teacher believes that girls are better than boys at rock paper scissors and wants to test this belief at the 3% significance level. 200 games of rock paper scissors were played between boys and girls and the teacher recorded the number of girls who won. Find the acceptance region for this test.

$X \leqslant 113$

4. In 2018, 25% of workers like working from home.
In 2020, a survey of twenty people found that only one likes working from home. A student wishes to test whether the proportion people who like working from home has changed at the 1% level of significance.

1. Write down the null and alternate hypotheses for this test.
2. Find the critical value(s) for this test.
3. Carry out the test.

1. $\mathrm{H}_0: p = 0.25$ and $\mathrm{H}_1: p \neq 0.25$ where $p$ is the probability a worker likes working from home
2. $0$ and $11$
3. $1$ is in the acceptance region, so reject $\mathrm{H}_1$
   There is insufficient evidence to suggest the proportion of people who like working from home has changed.

5. After a single dose of a vaccine, 78% of people produce an immune response. A new vaccine claims that their vaccine is better.
20 patients are selected at random and given the new vaccine. Of these, 19 produced the desired immune response.
Carry out a hypothesis test at the 1% significance level to investigate whether the new vaccine is better.

$\mathrm{H}_0: p = 0.78$ and $\mathrm{H}_1: p > 0.25$ where $p$ is the probability a patient produces an immune response.
$X \sim B(20,0.78)$, $\mathrm{P}(X \leqslant 18) = 0.954$ and $\mathrm{P}(X \geqslant 19) = 0.046$
This is greater than $0.01$, so reject $\mathrm{H}_1$
There is insufficient evidence to suggest the new vaccine is better.

6. A teacher claims that one in four people has red as their favourite colour. In a class of 24 students, it was found that only 2 had red as their favourite colour. Test the teacher's belief at the 5% significance level.

$\mathrm{H}_0: p = 0.25$ and $\mathrm{H}_1: p \neq 0.25$ where $p$ is the probability the a student has red as their favourite colour.
$X \sim B(24,0.25)$, $\mathrm{P}(X \leqslant 2) = 0.040$
This is greater than $0.025$, so reject $\mathrm{H}_1$
There is evidence to back the teacher's claim.

7. Last year, a survey showed that 35% of students enjoy doing homework. This year, in a survey of 120 students, it was found that 30 enjoy doing homework.

1. Test, at the 5% significance level, whether the proportion of students who enjoy doing homework has changed.
2. What is the actual significance level of this test?

1. $\mathrm{H}_0: p = 0.35$ and $\mathrm{H}_1: p \neq 0.35$ where $p$ is the probability a student enjoys doing homework.
   $X \sim B(120,0.35)$, $\mathrm{P}(X \leqslant 30) = 0.012$
   This is less than $0.025$ (two tailed test), so reject $\mathrm{H}_0$
   There is evidence to suggest the proportion of students who enjoy doing homework has changed.
2. $0.0439$

8. Over a long period of time, it was found that 1 in 7 students are disliked by teachers. A teacher believes that they dislike more of their students and wishes to test this hypothesis at the 1% level using the 72 students they teach.

1. Find the critical region(s) for the test.
2. The teacher dislikes 19 of their students. Complete the test.
3. What is the p-value of this test?

1. $X \geqslant 19$
2. $\mathrm{H}_0: p = \frac{1}{7}$ and $\mathrm{H}_1: p > \frac{1}{7}$ where $p$ is the probability the teacher dislikes a student
   19 is in the critical region, so reject $\mathrm{H}_0$
   There is sufficient evidence to suggest that the teacher dislikes a higher proportion of their students.
3. $0.00507$

9. Camille believes that, of all seven colours of the rainbow, more people prefer red than any other colour. She tests this belief at the 5% significance level by asking 40 of her friends what their favourite colour of the rainbow is.

1. Find the critical region(s) for this test.
2. 10 people out of 40 chose red as their favourite colour of the rainbow. Complete the test.
3. Explain how Camille choosing to ask only her friends affects this test.

1. $X \geqslant 11$
2. $\mathrm{H}_0: p = \frac{1}{7}$ and $\mathrm{H}_1: p > \frac{1}{7}$ where $p$ is the probability a person selects red as their favourite colour of the rainbow.
   10 is in acceptance region, so reject $\mathrm{H}_1$
   There is insufficient evidence to support Camille's belief.
3. A binomial distribution is not appropriate to model her friends' responses because they are unlikely to be independent of each other.

10. It is claimed 5% of students fall asleep in lessons. A teacher believes their class of $n$ students is better than average at staying awake in lessons. What is the largest value of $n$ for which the teacher will be unable to reject the null hypothesis at the 10% significance level?

$44$

11. It is believed that 30% of students who do Maths get an A*. A teacher wishes to test this claim at the 5% significance level using the 30 students who do Maths this year.

1. Find the critical region(s) for the test.
2. Given that 26 students failed to achieve an A* at the end of the year, carry out the test.
3. What is the actual significance level of this test?

1. $X \leqslant 3$ and $X \geqslant 15$
2. $\mathrm{H}_0: p = 0.3$ and $\mathrm{H}_1: p \neq 0.3$ where $p$ is the probability a student achieves an A*
   4 is not in the critical region, so reject $\mathrm{H}_1$
   There is insufficient evidence to suggest that the claim is false.
3. $0.0263$

12. A teacher claims that their lessons are enjoyed by half their students.

1. It was found that 10 students in a class of 30 enjoyed the teacher's lessons. Test the teacher's claim at the 5% significance level.
2. In a second class of $n$ students, none of them enjoyed the teacher's lessons. Find the smallest value of $n$ such that the teacher's claim is rejected at the 1% significance level.

1. $\mathrm{H}_0: p = 0.5$ and $\mathrm{H}_1: p \neq 0.5$ where $p$ is the probability a student enjoys the teacher's lesson.
   $X \sim B(30,0.5)$, $\mathrm{P}(X \leqslant 10) = 0.0494$
   This is greater than $0.025$, so reject $\mathrm{H}_1$
   There is evidence to back up the teacher's claim.
2. $810$