S3.3 - Estimation, confidence intervals and tests

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.