Statistics 2 (S2): Hypothesis testsEdexcel International A Level Further Maths: Topic test

1. 1
   A school has $1250$ students, all listed by name on its register. The head teacher wants to find the mean time students spend travelling to school, and plans to select $80$ students for a survey.

   (a)

   What is the sampling frame for this survey?

   [1 mark]

   • AAll $1250$ students at the school
   • BThe $80$ students who are selected
   • CThe school register of $1250$ names
   • DThe mean travel time of the students

   (b)

   What is the sampling unit?

   [1 mark]

   • AA single student
   • BThe register of students
   • CThe travel time of a student
   • DThe group of $80$ students selected

   (c)

   State one advantage and one disadvantage of using a sample rather than a census in this survey.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A population consists of the values $2$ and $6$, which occur with equal probability. Two independent observations $X_1$ and $X_2$ are made, and the statistic $M$ is the larger of the two values (if they are equal, $M$ takes that value).

   (a)

   Find $\mathrm{P}(M=6)$.

   [1 mark]

   • A$\frac14$
   • B$\frac34$
   • C$\frac12$
   • D$\frac13$

   (b)

   Find $\mathrm{E}(M)$.

   [1 mark]

   • A$4$
   • B$3$
   • C$6$
   • D$5$

   (c)

   Write down the sampling distribution of $M$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A pharmaceutical company claims that its new treatment cures $60\%$ of patients. A doctor suspects that the cure rate is different from this. The doctor gives the treatment to a random sample of $12$ patients and records the number $X$ who are cured. The significance level is $10\%$.

   (a)

   State suitable hypotheses for the test, defining the parameter, and explain why a two-tailed test is appropriate.

   [3 marks]

   (b)

   Given that $10$ of the $12$ patients are cured, carry out the test.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A university has $4500$ students, listed on its enrolment database. The students' union believes that more than $30\%$ of students cycle to campus. It will test this using a sample of $25$ students at the $5\%$ significance level.

   (a)

   Identify the population, the sampling unit and the sampling frame. Describe how a simple random sample of $25$ students could be selected using random numbers, and give one reason why a sample is used rather than a census.

   [6 marks]

   (b)

   In the sample, $12$ of the $25$ students cycle to campus. Test, at the $5\%$ significance level, whether the proportion of students who cycle is more than $30\%$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A hospital with $300$ beds wants to find out how satisfied its patients are. Each patient could give a satisfaction score from $1$ to $10$. The hospital asks a random sample of $40$ patients for their scores.

   (a)

   Which is an advantage of a census over a sample in this situation?

   [1 mark]

   • AIt would be quicker to carry out
   • BIt would be cheaper to carry out
   • CIt would need fewer questionnaires to be processed
   • DIt would give the true mean score for all patients, with no sampling error

   (b)

   Which of these is a statistic?

   [1 mark]

   • AThe mean score of all patients the hospital has ever treated
   • BThe mean score of the $40$ patients in the sample
   • CThe number of beds in the hospital, $300$
   • DThe mean score the hospital hopes to achieve

   (c)

   Explain what is meant by the sampling distribution of the sample mean score.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A supplier claims that $8\%$ of its eggs are cracked on delivery. A restaurant owner believes that the proportion is higher. The owner checks a random sample of $25$ eggs, and $X$ is the number that are cracked.

   (a)

   Which pair of hypotheses should the owner test?

   [1 mark]

   • A$\mathrm{H}_0:p=0.08$, $\mathrm{H}_1:p>0.08$
   • B$\mathrm{H}_0:p=0.08$, $\mathrm{H}_1:p<0.08$
   • C$\mathrm{H}_0:p>0.08$, $\mathrm{H}_1:p=0.08$
   • D$\mathrm{H}_0:p=0.08$, $\mathrm{H}_1:p\ne0.08$

   (b)

   For the test at the $5\%$ significance level, the critical region for $X$ is

   [1 mark]

   • A$X\ge4$
   • B$X\ge6$
   • C$X\ge5$
   • D$X\ge3$

   (c)

   The owner finds that $4$ of the $25$ eggs are cracked. State the conclusion of the test at the $5\%$ significance level, showing your working.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Potholes appear on a rural road at random, at a mean rate of $1.5$ per kilometre. After the road is resurfaced, an engineer believes that the rate has decreased. In a randomly chosen $4$ km stretch of the resurfaced road, $X$ potholes are found. The significance level is $5\%$.

   (a)

   State suitable hypotheses for the test, and the distribution of $X$ under the null hypothesis.

   [3 marks]

   (b)

   Given that $2$ potholes are found in the $4$ km stretch, carry out the test.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A taxi firm receives phone bookings at random, at a mean rate of $4$ per ten-minute period. A manager believes that this rate has changed and will test the belief at the $10\%$ significance level using the number of bookings $X$ in one randomly chosen ten-minute period. The firm also takes bookings through an app, and claims that $35\%$ of all its bookings are made through the app.

   (a)

   Find the critical region for the manager's test, and state the actual significance level of the test.

   [6 marks]

   (b)

   A customer group believes that the proportion of bookings made through the app is higher than $35\%$. In a random sample of $150$ bookings, $63$ were made through the app. Use a Normal approximation to carry out a test at the $5\%$ significance level.

   [6 marks]

   Total for question 8: 12 marks

End of questions