(18 marks) Hypertension (high blood pressure) is a common health problem amongst the Australian population. It is the greatest contributor to the burden of cardiovascular disease (CVD). According to Australian Heart Foundation data, 25% Australians of aged between 45- 54 years suffer from hypertension. To investigate how Australians are affected by the incidence of hypertension, a researcher selected a random sample of 12 Australians of the above age group. Use this information to answer the following questions: (a) (2 marks) Suggest an appropriate model to estimate the probabilities of the number of Australians of the above age group (45-54 years) who have a hypertension. What are the parameters of this model? An appropriate model to estimate the probabilities of the number of Australians of the above age group (45-54 years) who have a hypertension is a binomial distribution model. The parameters of this model are the sample size (number of experiments or occurrences. Therefore, letting the number of Australian aged between 45-54 who suffer from hypertension be denoted by X, �~��������(�, �) �ℎ��� � �� �ℎ� ������ ���� ��� � �� �ℎ� ����������� �� �������. �ℎ��, �~�������� (12, 0.25) (b) (4 marks) Explain why this model is appropriate by discussing the important distinguishing features of the model, focusing on the underlying conditions for the validity of the model. The appropriate model is Binomial distribution Model because the outcomes are independent, there are two possible occurrences (with or without hypertension and the probability of having hypertension (success) is constant. (c) (2 marks) Using the model from part (a), estimate the probability that, in the randomly selected sample of 12 Australians of the above age group, at most 3 will have hypertension. �(�) = E � �F �G�IJG; �ℎ��� � �� �ℎ� ����������� �� �������. ����������� �ℎ�� �� ���� 3 ���� ℎ��� ℎ����������� = �(� ≤ 3) �(� ≤ 3) = E 12 0 F ∗ 0.25Q0.75ST + E 12 1 F ∗ 0.25S0.75SS + E 12 2 F ∗ 0.25T ∗ 0.75SQ + E 12 3 F 0.25V ∗ 0.75W = 0.391 Thus, there is a 39.1% probability that randomly selected sample of 12 Australians of the above age group will contain at most 3 people who have hypertension. (d) (4 marks) A random sample of 120 Australians of the above age group have been selected to have their hypertension checked. For this sample of 120, estimate the mean and standard deviation of the number of Australians with hypertension. With a larger sample size, normal distribution could be used to model binomial occurrences. ���� = � ∗ � = 120 ∗ 0.25 STA2300 Data Analysis, Semester 3 2019, 0061032987 �������� ��������� = ��� = 120 ∗ 0.25 ∗ 0.75 = 22.5 (e) (4 marks) Determine the probability that, in the random sample of 120 Australians of the age group (selected in part (c)), 20 or more will have hypertension. ����� �ℎ� ������� �� ������ ������������ �� �������� ������������ ��� �ℎ� parameters �� ���� �, �(� > 20) = 1 − �(� < 20) = 1 − � E� < hJi j F = 1 − � E� < TQJVQ TT.k F = 1 − ∅(−0.44) = 1 − 0.3284 = 0.6716 �ℎ�������, �ℎ� ����������� �ℎ�� �� � ������ ������ �� 120, 20 �� ���� ���� ℎ��� ℎ����������� �� 67.16% (f) (2 marks) State and check any assumptions, conditions or rules of thumb that should be considered for the validity of the result obtain in part (e). i) �ℎ� ������ �������� �ℎ���� �� ��������� � �������� ������������.�� �ℎ�� ����, �~���(120, 0.25) ℎ���� �ℎ� ��������� �� ���. ii) The sample size is large and/or probability of success is close to ½. In this case, a sample size of 120 can be considered large. Accordingly, the two conditions are met.

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