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VICSpecialist MathematicsQuick questions

Unit 4: Data analysis, probability and statistics

Quick questions on Linear combinations and statistical inference: VCE Specialist Mathematics Unit 4

10short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What are hypotheses?
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H0:μ=500H_0: \mu = 500 versus H1:μ>500H_1: \mu > 500 (one-sided).
What is standard error?
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σn=816=84=2\dfrac{\sigma}{\sqrt{n}} = \dfrac{8}{\sqrt{16}} = \dfrac{8}{4} = 2 mL.
What is test statistic?
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z=xˉ−μ0σ/n=504−5002=42=2z = \dfrac{\bar{x} - \mu_0}{\sigma/\sqrt{n}} = \dfrac{504 - 500}{2} = \dfrac{4}{2} = 2.
What is pp value?
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For a one-sided upper test, p=P(Z>2)p = P(Z > 2). From the standard normal, P(Z>2)≈0.0228P(Z > 2) \approx 0.0228.
What is decision?
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Since p≈0.0228<0.05p \approx 0.0228 < 0.05, we reject H0H_0 at the 5% level. There is evidence that the machine overfills.
What is sense check?
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The observed mean is two standard errors above the target, which is reasonably extreme, so rejecting H0H_0 is consistent with z=2z = 2 sitting beyond the one-sided critical value z=1.645z = 1.645.
What is wrong tail for the pp value?
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A one-sided test uses one tail; a two-sided test doubles the tail probability. Match the pp value to the alternative hypothesis.
What is q1?
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XX and YY are independent with Var(X)=5\mathrm{Var}(X) = 5, Var(Y)=3\mathrm{Var}(Y) = 3. Find Var(X−Y)\mathrm{Var}(X - Y). [2 marks]
What is q2?
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A sample of n=36n = 36 from a population with σ=12\sigma = 12 has mean xˉ=70\bar{x} = 70. State the standard error of the mean. [1 mark]
What is q3?
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For a test of H0:μ=100H_0: \mu = 100 against H1:μ≠100H_1: \mu \neq 100, the test statistic is z=2z = 2. State the pp value. [2 marks]

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