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Unit S3: Statistics 3

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  1. S3.1 - Combinations of random variables

    1. S3.1.1

      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.

  2. S3.2 - Sampling

    1. S3.2.1Methods for collecting data

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

    2. S3.2.2Other methods of sampling

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

  3. S3.3 - Estimation, confidence intervals and tests

    1. S3.3.1Concepts of standard error,

      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.

    2. S3.3.2Distribution of the sample mean

      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.

    3. S3.3.3Concept of a confidence interval

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

    4. S3.3.4Confidence limits for a Normal

      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.

    5. S3.3.5Hypothesis tests for the mean of a Normal distribution

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

    6. S3.3.6Central Limit theorem for sample means

      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.

    7. S3.3.7Hypothesis test for the difference

      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.

    8. S3.3.8Use of large sample results

      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.

  4. S3.4 - Goodness of fit and contingency tables

    1. S3.4.1null and alternative hypotheses

      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.

    2. S3.4.2Degrees of freedom

      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.

  5. S3.5 - Regression and correlation

    1. S3.5.1Spearman’s rank correlation

      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.

    2. S3.5.2Hypothesis tests for zero correlation

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