Week 6 - Introduction to statistics Flashcards
statistics
tools for describing and measuring the properties of populations from samples.
goal of statistics
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
null hypothesis
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
alternative hypothesis
includes all other feasible values for the population parameter besides the value stated in the null hypothesis.
hypothesis testing steps
State the hypotheses
Compute the test statistic with the data
Determine the p-value
Draw appropriate conclusions
stating the hypothesis
Alternative hypothesis is two-sided (includes parameter values on both sides of the parameter value specified by the null hypothesis)
test statistic
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
test statistic = signal/noise
Signal - value of interest
Noise - uncertainty of value
the null distribution
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
quantifying uncertainty: the p value
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
draw the appropriate conclusion
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
reporting the results
Summary of statistical test:
The value of the test statistic
The sample size
The p-value
type I error
rejecting the true null hypothesis. The significance level sets the probability of committing a Type I error. False positive
type II error
failing to reject a false null hypothesis. False negative
the power
of a test is the probability that a random sample will lead to rejection of a false null hypothesis.
one sided test
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.
paired design
both treatments are applied to every sampled unit.
Similar conditions in each block - can control for some noise
Usually more powerful
two sample design
each treatment group is composed of an independent, random sample of units.
Every plot is independent
paired comparisons of means
Paired measurements are converted to a single measurement by taking the difference between them
paired comparisons of means assumptions
The sampling units are randomly sampled from the population
The paired differences have a normal distribution in the population
pooled sample variance
the average of the variances of the samples weighted by their degrees of freedom.
two sided and two sample assumptions
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
welch’s
compares the means of two groups and can be used even when the standard deviations of the two groups are not equal.
contingency analysis
estimates and tests for an association between two or more categorical variables.
Chi squared test H0 and Ha
H0: variables are independent
Ha: variables are not independent
Chi squared test
If two events are independent, then, the probability of both occurring is equal to the probability of one event occurring times the probability of the other event occurring
Use degrees of freedom
Same assumptions as goodness-of-fit tests: samples are independent and no more than 20% of cells have an expected frequency of less than 5 and no cells have an expected frequency of less than 1
