Question 3 [19 marks]

Table 3a shows the number of defective bolts produced by two different types of machines (I and II) on 9 consecutive days. It can be assumed that both machines have the same total output per day.

Day	1	2	3	4	5	6	7	8	9
Machine I	47	54	55	49	32	48	51	61	49
Machine II	71	63	45	49	55	50	42	53	57
Difference	-24	-9	10	0	-23	-2	9	8	-8

A quality control engineer wishes to conduct a paired-sample sign test to determine whether both machines are the same in terms of the numbers of defective bolts produced.The null and alternative hypotheses are stated as follows:

Null Hypothesis: H₀: = 0 (both machines are the same)

Alternative Hypothesis: H₁: 0 (both machines are different),

where denotes the population median of differences between the numbers of defective bolts for both machines.

Answer all the following questions. Give your final numerical answers correct to 2

decimal places, unless otherwise stated.

1. Is the paired-sample sign test a parametric or non-parametric hypothesis test? State threeadvantages of using the sign test in hypothesis testing. [4 marks]

1. Is this test lower-tail, upper-tail or two-tail? [1 mark]

1. i) State the total number of signs, and the number of positive signs, . [2 marks]

ii) Calculate the p-value for this test. [6 marks]

1. If the level of significance is 5%, can the null hypothesis be rejected? Explain. [2 marks]
2. Write down a formal conclusion in the context of the problem. [2 marks]
3. Explain the meaning of "5% level of significance" in the problem context. [2 marks]