Week 6 - Introduction to statistics

tools for describing and measuring the properties of populations from samples.

To estimate the values of important parameters, including: means, proportions, variances, effects To test hypotheses about those parameters - are estimates more extreme than we expect simply by chance? If yes, results are significant

a specific statement about a population parameter made for the purposes of argument. A good null hypothesis is a statement that would be interesting to reject. Tested Rejected in support of the alternative hypothesis

includes all other feasible values for the population parameter besides the value stated in the null hypothesis.

State the hypotheses Compute the test statistic with the data Determine the p-value Draw appropriate conclusions

Alternative hypothesis is two-sided (includes parameter values on both sides of the parameter value specified by the null hypothesis)

A number calculated from the data that is used to evaluate how compatible the data are with the result expected under the null hypothesis Evaluates how well the observed result matches the result claimed by the null hypothesis Some mismatch is expected by chance so is our result too extreme for chance to explain? To decide we need a suitable test statistic Then we need its p-value

Signal - value of interest Noise - uncertainty of value

The sampling distribution of outcomes for a test statistic under the assumption that the null hypothesis is true Distribution of test statistic when the null is true Shape varies for different test statistics and sample size but: All possible values is on x-axis The densities of values are on the y axis Densities sum to 1, giving probabilities

The probability of obtaining the data (or data showing as great or greater difference from the null hypothesis) if the null hypothesis were true The refers to the probability of a specific event when sampling data under the null hypothesis: it is the probability of obtaining a result as extreme as or more extreme than the observed

The significance level is a probability used as a criterion for rejecting the null hypothesis. If the p value is less than or equal to a then the null hypothesis is rejected. If the p-value is greater than a the null hypothesis is not rejected The rejection threshold is called the significance level (a) Usually a = 0.05

Summary of statistical test: The value of the test statistic The sample size The p-value

rejecting the true null hypothesis. The significance level sets the probability of committing a Type I error. False positive

failing to reject a false null hypothesis. False negative

of a test is the probability that a random sample will lead to rejection of a false null hypothesis.

the alternative hypothesis includes parameter values only one side of the value specified by the null hypothesis. Rejected only if the data depart from it in the direction stated by p-value.

both treatments are applied to every sampled unit. Similar conditions in each block - can control for some noise Usually more powerful

each treatment group is composed of an independent, random sample of units. Every plot is independent

Paired measurements are converted to a single measurement by taking the difference between them

The sampling units are randomly sampled from the population The paired differences have a normal distribution in the population

the average of the variances of the samples weighted by their degrees of freedom.

Each of the two samples is a random sample from its population The numerical value is normally distributed in each population The standard deviation of the numerical variable is the same in both populations

compares the means of two groups and can be used even when the standard deviations of the two groups are not equal.

estimates and tests for an association between two or more categorical variables.