Question

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Let X be the live weight of a randomly selected 4-month old male lamb. Suppose X follows a normal distribution with a mean of 38 kg and a standard deviation of 2.5 kg. In this question, we consider a sample of sixteen 4-month old male lambs. Denote by X¯ the mean live weight of these 16 lambs.

(a) State the distribution, with the corresponding parameters, of X¯ .

(b) Manually calculate the probability that the mean live weight of these 16 lambs is greater than 39 kg.

(c) An important assumption of the Central Limit Theorem [which is used to derive the distribution of X¯ in (a)] is independence. In the context of this question, suggest a situation where the independence assumption may be violated.

Answer 1

Where X is the weights of 4 month old male lamb.

n = 16

a) We have the weights as normally distributed and they are independent. So we can use the central limit theorem for the distribution of the means.

b.

z-score = =

P(> 39) = P(Z > 1.6)

= 1 - P(Z < 1.6)

= 1 -0.9452 .................using normal distribution tables

P( > 39) = 0.0548

c. The important assumption is the sample being independent. It means that the sample is selected at random and therefore there is no bias. It can be violated if the sample is not randomly selected and that there is some kind of bias (eg: only healthy males being selected or only unhealthy ones selected.)