Statistics: Statistical hypothesis testingEdexcel A-Level Maths: Topic test

1. 1
   A cafe owner believes that the proportion $p$ of her customers who order a vegetarian meal is different from the national figure of $0.2$. She takes a random sample of 30 customers and carries out a two-tailed test at the $10\%$ significance level. Let $X$ be the number in the sample who order a vegetarian meal.

   (a)

   Which pair of hypotheses should she use?

   [1 mark]

   • A$H_0:p=0.2$, $H_1:p>0.2$
   • B$H_0:p=0.2$, $H_1:p\neq0.2$
   • C$H_0:p\neq0.2$, $H_1:p=0.2$
   • D$H_0:\bar{x}=0.2$, $H_1:\bar{x}\neq0.2$

   (b)

   What is the largest probability allowed in each tail of the critical region?

   [1 mark]

   • A$0.10$
   • B$0.025$
   • C$0.2$
   • D$0.05$

   (c)

   Under $H_0$, find the largest critical value for the lower tail of the critical region.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Nationally, $45\%$ of candidates pass a practical test. A teacher believes that her new method of coaching increases the pass rate. In a random sample of 30 of her candidates, 18 pass. She tests at the $5\%$ significance level, with $H_0:p=0.45$ and $H_1:p>0.45$, where $p$ is the proportion of her candidates who pass.

   (a)

   What is the $p$-value for her result?

   [1 mark]

   • A$0.0714$
   • B$0.0334$
   • C$0.9286$
   • D$0.1356$

   (b)

   What is the critical region for the test?

   [1 mark]

   • A$X\geqslant18$
   • B$X\geqslant17$
   • C$X\geqslant19$
   • D$X\leqslant18$

   (c)

   State the conclusion of the test, in context.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The daily commute time of employees at a company has a Normal distribution with standard deviation $8$ minutes and, until now, a mean of $35$ minutes. After a new bus route is introduced, a manager believes that the mean commute time has decreased. A random sample of 16 employees has a mean commute time of $31.2$ minutes. She tests at the $5\%$ significance level.

   (a)

   State suitable hypotheses and the distribution of the sample mean $\bar{X}$ under the null hypothesis.

   [3 marks]

   (b)

   Carry out the test and state your conclusion in context.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   For a random sample of 15 cyclists, a researcher records the mean distance each cycles per week and the resting heart rate of each cyclist. The product moment correlation coefficient is $r=-0.52$. The critical values for 15 pairs of data are: for a one-tailed test, $0.4409$ at the $5\%$ level and $0.5923$ at the $1\%$ level; for a two-tailed test at the $5\%$ level, $0.5140$.

   (a)

   Test, at the $5\%$ significance level, whether there is evidence of correlation between weekly distance and resting heart rate. Interpret your result in context and comment on the strength of the evidence.

   [6 marks]

   (b)

   The researcher had good reason to expect a negative correlation. Carry out a one-tailed test at the $5\%$ level, and again at the $1\%$ level. Explain when it is acceptable to use a one-tailed test and state what the significance level means.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   An online retailer states that $40\%$ of its orders are delivered the next day. After changing courier, the retailer believes the proportion has increased. In a random sample of 25 orders, $X$ are delivered the next day. Let $p$ be the proportion of all orders now delivered the next day. The retailer tests at the $5\%$ significance level.

   (a)

   Which pair of hypotheses should the retailer use?

   [1 mark]

   • A$H_0:p=0.4$, $H_1:p\neq0.4$
   • B$H_0:p>0.4$, $H_1:p=0.4$
   • C$H_0:p=0.4$, $H_1:p>0.4$
   • D$H_0:p=0.4$, $H_1:p<0.4$

   (b)

   What is the critical region for $X$?

   [1 mark]

   • A$X\geqslant15$
   • B$X\geqslant14$
   • C$X\geqslant13$
   • D$X\leqslant14$

   (c)

   In the sample, 14 orders are delivered the next day. State the conclusion of the test, in context.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A bike-share scheme finds that $18\%$ of hires last longer than one hour. After a change to its prices, the operator believes the proportion has changed. In a random sample of 50 hires, 4 last longer than one hour. The operator carries out a two-tailed test at the $5\%$ significance level, with $H_0:p=0.18$ and $H_1:p\neq0.18$.

   (a)

   With which value should the probability $P(X\leqslant4)$ be compared?

   [1 mark]

   • A$0.05$
   • B$0.18$
   • C$0.95$
   • D$0.025$

   (b)

   Given that $P(X\leqslant4)=0.0399$ under $H_0$, which conclusion is correct?

   [1 mark]

   • AReject $H_0$, because $0.0399<0.05$.
   • BDo not reject $H_0$, because $0.0399>0.025$.
   • CAccept that $p=0.18$ is definitely true, because $H_0$ is not rejected.
   • DReject $H_0$, because only $4$ hires is less than $18\%$ of $50$.

   (c)

   Find the critical region for the lower tail of this test.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The fuel economy of a model of car, in km per litre, is Normally distributed with standard deviation $1.2$. The manufacturer states that the mean fuel economy is $18.5$. An independent tester believes that the mean is different and tests a random sample of 9 cars at the $10\%$ significance level. The sample mean is $17.9$.

   (a)

   Find the critical region for the sample mean $\bar{X}$ for this two-tailed test.

   [3 marks]

   (b)

   Use a $p$-value to test the manufacturer's claim and state your conclusion in context.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The marks on an end-of-year mathematics paper are Normally distributed with standard deviation $12$ and, in previous years, a mean of $58$. A head of department believes that a new teaching programme has raised the mean mark. A random sample of 36 students who followed the programme has a mean mark of $62.5$. Let $\mu$ be the mean mark for students who follow the programme.

   (a)

   Test the head of department's belief at the $1\%$ significance level. State your conclusion in context.

   [6 marks]

   (b)

   The head of department then tests at the $5\%$ significance level instead. Find the critical value of $\bar{X}$ for this test, state the conclusion, and explain why the two conclusions differ. State one assumption needed for the test.

   [6 marks]

   Total for question 8: 12 marks

End of questions