Question

STAT10060 – Statistical Modelling

Exercise 1

To find out if four flavours of Dairy Milk bars are equally liked, members of the public were asked to vote in an online survey for their favourite flavour from four options: Caramello, Golden Crisp, Mint Crisp or Oreo. Labelling these categories from 1 to 4 in the order listed, the results were:

n1 =249 n2 =399 n3 =401 n4 =346

1. Denote symbols clearly and state your hypothesis.

2. What are the expected counts for each flavour, assuming that each are equally liked?

3. Compute the χ2 test statistic using the assumption from Part (i) stating clearly the degrees of freedom associated with it.

4. Carry out the χ2 test to find out if there is a favourite a flavour amongst the four or if they are all equally liked. Use α = 0.05.

   [Total: 25 marks]

Exercise 2

Empathy is defined as the ability to understand and share the feelings of another individual. One researcher is interested in testing whether or not there is an association between being named John, Conor or Joseph and being empathetic. She collected a sample of n1 = 346 individuals named John and a sample of n2 = 217 individuals named Conor and a sample of n3 = 313 individuals named Joseph in Ireland and asked them to fill out a short survey testing their empathy. She claims that 297 Johns, 206 Conors and 301 Josephs showed empathy. What is an appropriate statistical test in this scenario. What are the results of such an analysis? Make clear all steps involved and state your conclusions. Take α = 0.01.

[Total: 25 marks]

Answer 1

Exercise 1

Denote symbols clearly and state your hypothesis.

The hypothesis being tested is:

H0: They are all equally liked

Ha: There is a favourite flavour amongst the four

What are the expected counts for each flavour, assuming that each are equally liked?

expected

348.750

348.750

348.750

348.750

Compute the χ2 test statistic using the assumption from Part (i) stating clearly the degrees of freedom associated with it.

degrees of freedom = 3

χ2 test statistic = 43.62

Carry out the χ2 test to find out if there is a favourite a flavour amongst the four or if they are all equally liked. Use α = 0.05.

observed	expected	O - E	(O - E)² / E
249	348.750	-99.750	28.531
399	348.750	50.250	7.240
401	348.750	52.250	7.828
346	348.750	-2.750	0.022
1395	1395.000	0.000	43.621
43.62	chi-square
3	df
1.82E-09	p-value

The p-value is 0.0000.

Since the p-value (0.0000) is less than the significance level (0.05), we can reject the null hypothesis.

Therefore, we can conclude that there is a favourite flavour amongst the four.

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