Question

Respond to the following in a minimum of 175 words, please type response:

Describe statistical inferences about two populations: How can comparisons be made for independent and dependent samples.

Respond to the following in a minimum of 175 words, please type response:

Describe analysis of variance & design of experiments: How does ANOVA extend the hypothesis testing analysis for two sample & more.

Answer 1

STATISTICAL INFERENCES

Statistical inference mainly deals with estimation and testing of hypothesis.

Statistical hypothesis may be defined as statement about one or more population.Since populations are characterized by their parameters , as such statistical hypothesis becomes a statement about one or more parameters. One of the major steps in the process of statistical statistical hypothesis testing involves the choice of statistical test.The most appropriate test is chosen according to the desired power of the study,the nature of the population from which the observations are taken and the nature of measurement of the variable.

Parametric and Nonparametric Analysis

Statistical tests may be classified into two categories,known as parametric and nonparametric tests.

The following assumptions should hold before a parametric statistical method is used,

1. The samples should be from a normally distributed population.
2. The samples should be independent.
3. The variances of the population under examination should be similar(Homoscedasticity).
4. The variable under examination must be scale, discrete or continuous.

Therefore, parametric tests utilized for the metric or scale data while nominal and ordinal data is analysed by nonparametric tests.

Independent V/S Dependent samples

Dependent samples are measurements of one set of items.That is,these samples are characterized by a measurement followed by an intervention of some kind and then another measurement.This could be called a ‘before’ an ‘after’ study. Dependent and samples are characterized by matching or pairing observations.

Independent samples are measurements made on two different sets of items. That is ,if the values in one sample reveal no information about those of the other sample, then the samples are independent.

Type	Parametric Tests	Non-Parametric tests
Independent samples	Z-test (Sample size>30) t-test (Sample size<30)	Chi-square test(Sample size > 20) Fisher Exact test(Samle size <20 Mann-Whiteney U test(Metric)
Dependent/Paired samples	Paired t-test	Mc Nemars test(Categorical) Wilcoxon Signed Rank test(Metric)

Analysis of Variance (ANOVA) and Design of Experiments

Literally , an experiment is a test. That is, we can define an experiment as a test or series of tests in which purposeful changes are made to the input variables of a process or system so that we may observe and identify the reasons for changes that may be observed in the output response.

Analysis of Variance (ANOVA) is a tool to study whether the means of more than two populations are equal or not. Though t –test of significance is also designed to test the same, but the limitation of t-test is that it relates to only two samples. If the number of samples is greater than two ,the t-test formula is not equipped to test the significance. For such situations , ANOVA has been designed, originally developed for use in agricultural sciences, ANOVA has found widespread application in almost every field.

Analysis of Variance helps in understanding the sources of total variation, through studies known as ‘treatment’ effects on sample. Treatment refers to the factor that the sample is being subjected and the factor is controlled. That is, researcher knows it and is controlling it.

Assumptions of ANOVA

1. Normality: The samples are drawn from normal population
2. Homogeneity: The population from which the samples are drawn have the same variance.
3. Independence: The observations are independent.
4. Additive: The effects of different treatments as well as environmental factors on the observations are additive in nature

The ANOVA involves determining if the observed values belong to the same population, regardless of the group, or whether the observations in at least one of these groups come from a different population.

To obtain a F value we need two estimates of the population variance. It is necessary to examine the variability (analysis of variance) of observations within groups as well as between groups. The F statistic is computed using a simplified ratio similar to the t-test,

To calculate the F statistic for the decision rule either the definitional or computational formulas may be used.