Unit S3: Statistics 3

Distribution of linear combinations If X ∼ N(μ, σ 2) and Y∼ N(μ, σ 2) independently, then x x y y of independent Normal random aX± bY ∼ N(aμ ± bμ, a2σ 2 + b2σ 2). variables. x y x y No proofs required.

Use linear combinations of normal variables to connect the rule to the data and decision in the question.

This matters because linear combinations of normal variables determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply linear combinations of normal variables to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Linear combinations of Normal variables is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Methods for collecting data.; Simple random sampling.; Use of random numbers for sampling.

Use methods for collecting data to connect the rule to the data and decision in the question.

This matters because methods for collecting data determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply methods for collecting data to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Methods for collecting data is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Other methods of sampling: The circumstances in which they might be used.; Their stratified, systematic, quota. advantages and disadvantages.

Use other methods of sampling to connect the rule to the data and decision in the question.

This matters because other methods of sampling determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply other methods of sampling to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Other methods of sampling is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Concepts of standard error, The sample mean, x, and the sample variance, estimator, bias. 1 n s2 = ∑(x − x)2, as unbiased estimates of the n −1 i i=1 corresponding population parameters.

Use concepts of standard error, to connect the rule to the data and decision in the question.

This matters because concepts of standard error, determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply concepts of standard error, to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Concepts of standard error, is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Know that the sample mean X̄ has mean μ and variance σ²/n; if X is normal, then X̄ ~ N(μ, σ²/n). Proofs are not required.

Use distribution of the sample mean to connect the rule to the data and decision in the question.

This matters because distribution of the sample mean determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply distribution of the sample mean to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Distribution of the sample mean is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Concept of a confidence interval and its interpretation.; Link confidence intervals with hypothesis tests.

Use concept of a confidence interval to connect the rule to the data and decision in the question.

This matters because concept of a confidence interval determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply concept of a confidence interval to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Concept of a confidence interval is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Confidence limits for a Normal Students will be expected to know how to apply the Normal mean, with variance known. distribution and use the standard error and obtain confidence intervals for the mean, rather than be concerned with any theoretical derivations.

Use confidence limits for a normal to connect the rule to the data and decision in the question.

This matters because confidence limits for a normal determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply confidence limits for a normal to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Confidence limits for a Normal is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Hypothesis tests for the mean of a X −µ Use of ∼ N(0, 1).; Normal distribution with variance σ/ n known.

Use hypothesis tests for the mean of a normal distribution to connect the rule to the data and decision in the question.

This matters because hypothesis tests for the mean of a normal distribution determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply hypothesis tests for the mean of a normal distribution to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Hypothesis tests for the mean of a Normal distribution is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Use of Central Limit theorem to X −µ can be treated as N(0, 1) when n is large. extend hypothesis tests and S / n confidence intervals to samples from non-Normal distributions.; Use A knowledge of the t-distribution is not required. of large sample results to extend to the case in which the variance is unknown.

Use central limit theorem for sample means to connect the rule to the data and decision in the question.

This matters because central limit theorem for sample means determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply central limit theorem for sample means to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Central Limit theorem for sample means is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Hypothesis test for the difference (X −Y)−(µ −µ) between the means of two Normal x y Use of ∼ N(0, 1). distributions with variances known. σ2 σ2 x + y n n x y.

Use hypothesis test for the difference to connect the rule to the data and decision in the question.

This matters because hypothesis test for the difference determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply hypothesis test for the difference to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Hypothesis test for the difference is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Use of large sample results to (X −Y)−(µ −µ) Use of x y ∼ N(0, 1). extend to the case in which the S2 S2 population variances are unknown. x + y n n x y A knowledge of the t-distribution is not required.

Use use of large sample results to connect the rule to the data and decision in the question.

This matters because use of large sample results determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply use of large sample results to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Use of large sample results is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

The null and alternative hypotheses.; Applications to include the discrete uniform, binomial, n (O − E)2 Normal, Poisson and continuous uniform (rectangular) The use of ∑ i i as an distributions.; Lengthy calculations will not be required.; E i=1 i approximate χ2 statistic.

Use null and alternative hypotheses to connect the rule to the data and decision in the question.

This matters because null and alternative hypotheses determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply null and alternative hypotheses to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: null and alternative hypotheses is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Degrees of freedom.; Students will be expected to determine the degrees of freedom when one or more parameters are estimated from the data.; Cells should be combined when E < 5.; Yates’ i correction is not required.

Use degrees of freedom to connect the rule to the data and decision in the question.

This matters because degrees of freedom determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply degrees of freedom to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Degrees of freedom is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Spearman’s rank correlation Numerical questions involving ties will not be set.; Some coefficient, its use, interpretation understanding of how to deal with ties will be expected. and limitations.

Use spearman’s rank correlation to connect the rule to the data and decision in the question.

This matters because spearman’s rank correlation determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply spearman’s rank correlation to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Spearman’s rank correlation is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Testing the hypothesis that a Use of tables for Spearman’s and product moment correlation is zero. correlation coefficients.

Use hypothesis tests for zero correlation to connect the rule to the data and decision in the question.

This matters because hypothesis tests for zero correlation determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply hypothesis tests for zero correlation to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: Hypothesis tests for zero correlation is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.