Question

Q1. Consider the population of adult females resident in Melbourne. Our focus is on the population mean height. Assume we know o (population standard deviation) is 25, but we do not know the population mean, u. We take a sample of adult females resident in Melbourne (n=100) and calculate the sample mean height as 70 cm. i) Write down the distribution of X, female height. (2 marks) ii) Write down the distribution of Are you relying on any theorems in your answer? If yes explain. (3 marks) iii) Derive the 95% confidence interval around the sample mean. (3 marks) iv) In your answer to iii) will the specific confidence interval you have derived contain the population mean? Explain carefully. (2 marks)

Answer 1

ANSWER

(I)

The distribution of X, female height is unknown. Only statistic we know is that the distribution of X, female height has mean = 70 cm.

(II)$\bar{x}$

The distribution of $\bar{x}$ , the sample mean is Normal Distribution with mean equal to population mean and standard deviation given by:

SE = $\sigma$ /$\sqrt{n}$ = 25/$\sqrt{100}$ = 2.5

We are relying on Central Limit Theorem, according to which the sampling distribution of sample statistic is Normal distribution forlarge sample size irresptive of the shape of the population distribution.

(III)

$\bar{x}$ = 70

$\sigma$ = 25

n = 100

$\alpha$ = 0.05

From Table, critical values of Z = $\pm$ 1.96

Confidence Interval:

ZXσ , Ξ 1.96 x 25 = = 70 +4.9 = (65.1, 74.9) V100 vos 704

So,

Answer is:

(65.1, 74.9)

(IV)

The 95% Confidence Interval (65.1, 74.9) is a range of values we are 95% confident will contain the true unknown population mean height of adult females resident in Melbourne.

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