Further Statistics 2

What further statistics 2 covers

One of the eight optional papers of 9FM0, sat as Paper 4B. It is an Option 2 paper, so it may be taken only in a matching pair with Further Statistics 1. It fits lines by least squares, treats continuous densities properly, and then builds the intervals and tests that apply when a variance is estimated rather than known.

The main ideas

• Least squares regression, residuals, the reading of a residual plot, and the residual sum of squares.
• Continuous random variables: the density function and the cumulative distribution function, moved between in both directions.
• Mean, variance, mode, median, percentiles and skewness by integration, then the continuous uniform distribution and its derived results.
• The product moment and Spearman rank correlation coefficients, the effect of coding, and testing each against a tabulated critical value.
• Linear combinations of independent normal variables, where the variances add for a difference as well as for a sum.
• Estimators, bias, standard error and confidence intervals for a normal mean, then two-sample tests on a difference of means.
• Chi-squared tests and intervals for a variance, the F test for two variances, and the t-distribution for one-sample, paired and pooled tests.

The results it turns on

b = S(xy)/S(xx), with the line passing through the mean point

least squares regression

residual = observed − predicted, and RSS = S(yy) − S(xy)²/S(xx)

what a fitted line leaves unexplained

F(x) = ∫f from the lower end, and F'(x) = f(x)

density and cumulative distribution function

U(a, b) has mean (a + b)/2 and variance (b − a)²/12

the continuous uniform distribution

aX ± bY is normal, with variance a²σx² + b²σy²

a combination of independent normal variables

(n − 1)S²/σ² is χ² on n − 1 degrees of freedom

a test or an interval for a variance

Where it usually goes wrong

• A residual is the observed value minus the predicted one. The other order is the commonest sign error on the topic.
• Variances add for a difference of independent normal variables just as they do for a sum, with each coefficient squared before it is used.
• The chi-squared distribution is skewed, so a two-tailed test on a variance reads two different critical values rather than one used twice.
• A confidence interval is a statement about the procedure that produced it, not about the probability that the parameter lies inside this particular interval.

Where to start

Regression first, then the three continuous-variable lessons as one block, since everything later assumes them. Correlation and its test go together. Combinations of normal variables before estimators and the two-sample work. Variance tests and the t-distribution close the paper and are the most table-heavy lessons in it.