According to the Australian Bureau of Statistics, 70% of all Australian men aged

Question

According to the Australian Bureau of Statistics, 70% of all Australian men aged 18 and over are overweight or obese. A former STA2300 student takes a random sample of 20 men aged 18 years and over in Toowoomba. A particular variable of interest is the 'number of men aged 18 and over who are overweight or obese'. Based on the above information answer the following questions: (a) What is an appropriate model to represent the variable of interest? Write down the parameters of the model, if any. (b) Discuss how the conditions of the appropriate model are satisfied in the current study. (c) Find the mean and standard deviation of the number of men aged 18 years and over who are overweight or obese using the parameters of the model. (d) Find the probability that at least 15 of the men aged 18 years and over are overweight or obese. (e) Determine the probability that, in a random sample of 100 men aged 18 years or more in Toowoomba, 75 or more men are overweight or obese. State and check any assumptions, conditions or rules of thumb that should be considered before performing the calculations to determine this probability.

Explanation / Answer

Let X = Number of men aged 18 and over who are overweight or obese in a sample of 20 men randomly selected from the population of Toowoomba.

Part (a)

Model is: X ~ B(n, p), where n = sample size and p = probability that a man aged 18 and over randomly selected from the population of Toowoomba is overweight or obese.

Parameters are: n {= 20 (given)} and p (= 0.7) given 70% of all Australian men aged 18 and over are overweight or obese.

Part (b)

Binomial distribution is appropriate because the population infinitely large and the probability p remains constant so far as sample is drawn from Australian population of men aged 18 and over.

Part (c)

Mean = np = 20x0.7 = 14.

Standard Deviation = sq.rt{np(1 - p)} = sqrt(4.2) = 2.05

Part (d)

Probability at least 15 of the men are overweight or obese = P(X 15) = 1 - P(X 14)

= 1 – 0.5836 = 0.4164 ANSWER [probability is obtained using Excel Function]

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