You work for the Quality Assurance Department of a manufacturing company. Your c

Question

You work for the Quality Assurance Department of a manufacturing company. Your company manufactures pencils. Machine A, Pencil Maker, is set to produce pencils of mean length 190mm with a standard deviation of 0.1mm. Machine B, Pencil Giant, produces pencils of mean length 190mm with a standard deviation of 0.2mm. Your boss believes that the machines are not producing pencils of correct length. As the Quality Assurance Officer, you must select a sample of pencils, measure the length of each pencil and use the data to determine whether your boss’ claim is true.

Machine A Machine B
1 190.2 190.1
2 190.0 190.2
3 190.1 190.2
4 190.2 190.2
5 190.2 190.1
6 190.2 190.3
7 190.1 190.2
8 190.0 190.4
9 190.3 190.4
10 190.3 190.4
11 190.0 190.2
12 190.0 190.0
13 190.0 190.1
14 190.2 190.4
15 190.1 190.3
16 190.1 190.5
17 190.1 190.4
18 190.4 190.0
19 190.4 190.2
20 190.3 190.4
21 190.3 190.2
22 190.2 190.1
23 190.0 189.8
24 190.3 189.7
25 190.1 190.0
26 190.3 189.5
27 189.7 189.5
28 189.8 189.9
29 189.7 189.6
30 189.9 189.7
31 189.9 189.8
32 189.7 189.9
33 189.7 189.5
34 189.9 189.7
35 189.7 189.7
36 189.8 189.7
37 189.7 189.5
38 189.9 189.8
39 189.8 189.6
40 189.6 189.

Complete the contingency table below:

Machine A

(f) Use the contingency table to find the probability that a randomly selected pencil from the sample: (i) is from Machine A and exactly 190mm; (ii) is from Machine A, given that it is 190mm; (iii) is from Machine A or is exactly 190mm.

(g) Test the claim that the mean length of a pencil from Machine A is not 190mm. Use = 0.10 and assume that the population is normally distributed.

Machine Exactly 190 mm in length Within 1 standard deviation of 190mm More than 1 stndard deviations of the 190 mm Total

Machine A

Machine B Total

Explanation / Answer

(f) (i) Is from Machina A and exactly 190mm

Pr(Machin A and exactly 190 mm) = 6/80 = 0.075

(ii) Pr(from machine A, given that it is 190mm) = 6/ (6 + 3) = 2/3

(iii) Pr(Machine A or is exactly 190 mm) = (9 + 40 -6)/ 80 = 0.5375

(g) Here sample mean x = 190.03

sample standard deviation s = 0.22555

standard error of the mean se0 = s/ sqrt(n) = 0.22555/ sqrt(40) = 0.03566

Test statistic

t = ( x - H )/ se0 = (190.03 - 190.0)/ 0.03566 = 0.84

so here dF = 39; alpha = 0.10

tcritical = 1.6848

so t > tcritical so we shall not reject the null hypothesis and can conclude that mean lengthof a pencil is not 190mm.

Machine Exactly 190 mm in length Within 1 standard deviation of 190mm More than 1 stndard deviations of the 190 mm Total Machine A 6 16 18 40 Machine B 3 23 14 40 Total 9 39 32 80

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