BN121, BN422

Mathematics

Worksheet 3 solutions

Question 1

A company is manufacturing components with 3 separate production

processes. During the production process, 4 important faults in the

components are identified and the management seek to identify whether

there is a link between production method and identified fault. The company

recorded each incidence of a component with one fault produced in a given

time-period, sorted by production method and fault. The values are given in

the table shown

Factory 1 Factory 2 Factory 3

Scheme A 126 154 175

Scheme B 152 151 142

Scheme C 169 157 145

Scheme D 178 153 125

Determine, using an appropriate statistical test, whether or not the 4 faults

are arising equally across the three production methods. The steps involved

are:

1. Frame the null and alternative hypothesis.

2. Decide the level of significance, e.g. 0.05 or 0.01.

3. Look up critical values for 0.05.

4. Calculate the value of 2, from the data.

5. The null hypothesis is rejected if this exceeds the critical value.

Solution

The test proceeds as follows:

Framing the Hypotheses: The null hypothesis is that there is no

link between the faults and the method. The alternative

hypothesis is that there is a link.

We will take the level of significance as 0.01 (or 0.05).

The number of degrees of freedom is (4 1)(3 1) = 6.

Using the tables, the critical value for these parameters is 16.81

(or 12.59 for 0.05); this means

P[2 > 16.81] = 0.01 (or P[2 > 12.59] = 0.05).

The value found is 18.277, higher than the critical value of

16.81. The null hypothesis is therefore rejected, and we

conclude there is a connection between the faults and the

methods.

Question 2

As part of a study into the effects of smoking, a group of 70 students counted

the number of incidents of respiratory related illness during each of the four

seasons, among non-smokers (NS), light smokers (LS) and heavy smokers

(HS). Determine, using an appropriate statistical test, whether there is a link

between smoking and respiratory health across the four seasons. The results

are shown in the following table.

NS LS HS

Winter 8 14 16

Spring 4 11 10

Summer 0 4 6

Autumn 1 6 6

Question 3

A Study was carried out into what sports were played by residents of three

environments; Dublin city, the Commuter Belt towns and Rural areas. A

random sample of 320 people were classified by where they live, and asked

what sport they play on a regular basis. Determine, using an appropriate test,

whether there is a link between type of sport played and the urban/rural

environment. The results are shown in the following table.

GAA Soccer Rugby Other

Dublin 39 31 23 12

Commuter 48 45 26 34

Rural 18 22 18 4

Question 4

A pharmaceutical company is researching the effects of cigarette

smoking and urban pollution levels on the occurrence of respiratory

infections. A list of 1600 subjects were selected at random from 5

cities. Each subject was classified as non-smoker (NS), occasional

smoker (OS) or heavy smoker (HS). The cities were listed in order of

the recorded levels of pollution, City 1 with the lowest pollution and

City 5 the highest. The table below records the number of incidents of

respiratory infection in the subjects, counted for each city across the

three categories of subject. Carry out an appropriate statistical test to

determine whether the data suggests the smokers are suffering more

illnesses because of pollution).

City 1 City 2 City 3 City 4 City 5

HS 131 241 252 222 303

OS 52 54 68 63 85

NS 23 18 25 29 34