Question

Probability with samples?

Answer whichever part (a-e) you can/want to :)

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Answer 1

a) `P(x\lt40)\approx0.189`

b) `P(\bar{x}\lt40),n=9\rArr\approx0.0041`

c) `P(\bar{x}\lt40),n=12\rArr\approx0.0011`

d) not sure see explanation for more

e) unusual as it exceeds 2 z-scores (or 2 standard deviations)

Explanation:

Given information
`\mu=43.7` cm, `\sigma=4.2` cm

Working it out

a) `P(X\lt40)=P(z\lt(40-43.7)/4.2)=P(z\lt-3.7/4.2)` `=P(z\lt-0.8810)\approx0.189`

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b) `n=9\rarr\sigma_{\bar{x}}=4.2/\sqrt{9}=4.2\div3=1.4\rArrP(\bar{x}\lt40)` `=P(z\lt-3.7\div1.4)=P(z\lt-2.6429)\approx0.0041`

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c) `n=12\rarr\sigma_{\bar{x}}=4.2/\sqrt{12}\approx1.2124\rArrP(\bar{x}\lt40)` `=P(z\lt-3.7\div1.2124)=P(z\lt-3.0517)\approx0.0011`

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d) Increasing sample size appears to lower probability. This is likely because --

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e) `n=15\rarr\sigma_{\bar{x}}=4.2/\sqrt{15}\approx1.0844` `\bar{x}=46;z=(46-43.7)/(1.0844)\approx2.1209`
This result is unusual as it exceeds a z-score of 2.