POLI 443 Applied Political Research Session 3 – Statistical Significance & Tests of Hypotheses Lecturer: Prof. A. Essuman-Johnson , Dept. of Political Science Contact Information: aessuman-johnson@ug.edu.gh College of Education School of Continuing and Distance Education 2014/2015 – 2016/2017
Introduction • This session introduces you to statistical significance and tests of hypothesis. It will deal with how you conduct tests of hypothesis. • Objectives • By the end of this session, you should be able to understand what sampling distributions are and do tests for the normal distribution Slide 2
Type and Type Errors • There are three main types of problems in statistical inference. They are concerned with making decisions and deriving broad generalizations about social problems. The first of such problems is determining whether a sample has been selected from a specified population. The second problem may be about determining if two different samples have been drawn from a specified population i.e. the test is about the relationship between statistics from two samples. Slide 3
Type and Type Errors • In the third problem we might be given a statistic from a sample and we have to determine between what limits the parameter lies i.e. we try to find out the confidence limits within which the parameter would lie. To arrive at a proper decision we make use of a null and alternative hypotheses. Hypotheses are assumptions or guesses about the populations concerned. A statistical hypothesis can be formulated for the sole purpose of rejecting or nullifying it: Slide 4
Type and Type Errors • e.g. if we want to decide whether one procedure is better than another, we formulate the hypothesis that there is no difference between the two procedures i.e. any observed differences are due merely to fluctuation in sampling from the same population. Such populations are usually called the null hypotheses and denoted by H O. Any hypothesis that differs from a given hypothesis is called an alternative hypothesis and denoted by H 1 . If after computations it is found that the difference is not significant then we accept the H O , otherwise we Slide 5
Type and Type Errors • reject the H O and accept H 1 i.e. if on the supposition that a particular H O is true, we find that the results observed in a random sample differ markedly from those expected under the hypothesis on the basis of pure chance using sampling theory, we say the observed differences are significant and would be inclined to reject the hypothesis or at least not accept it. In carrying out a test of significance we make decisions about accepting or rejecting the hypothesis. If we reject the hypothesis when Slide 6
Type and Type Errors • it should be accepted, we say that a Type I error has been made. If on the other hand we accept a hypothesis when it should be rejected, we say that a Type II error has been made. For any tests of hypothesis or rules of decision to be good, they must be designed such as to minimize such errors of decision. The best way of reducing such errors is to increase the sample size which is not easy due to issues relating to funding etc. Slide 7
Level of Significance • In hypothesis testing, the maximum probability with which the researcher is willing to risk a type error is called the level of significance of the test. In the social sciences we usually choose a significance level of 0.05 (5%) or 0.01 (1%). If a 5% level of significance is chosen for test, then we are saying that we are about 95% confident that we have made the right decision. In this case the null hypothesis (H o ) is rejected at the 0.05 level of significance i.e. there is a 0.05 probability that we can be wrong. Slide 8
Summary • In this section we have learned about what type and type errors are in tests of significance as well as the relevance of the level of significance in scientific decision making. Slide 9
Sampling Distributions • The drawing of successive samples of the same size from a population produces a frequency distribution of measures computed from the samples. Frequency distributions of statistics from a large number of samples of a specified size are known as sampling distributions. Sampling distributions can be derived for various kinds of statistics such as means and proportions. The standard deviations of such statistics i.e. sampling distributions are known as their standard errors. Slide 10
Standard Error of Means • The standard error of means is the deviation of the sample means around the population mean. It is found by dividing the standard deviation of the cases of the universe (population) by the square root of the number of cases of the sample. • SE ( x̅ ) = SD/ = ( x̅ ) = Slide 11
• Since for most of the time we do not know the standard deviation (SD) of the population we are dealing with, we must estimate it from the SD of the sample and this is denoted by s. The s is usually less than the SD of the population ( ). When we take the SD around the sample mean which gives a minimum result we have lost one degree of freedom (the degree of freedom refers to the number of observations that are free to vary and it is usually (N-1). Slide 12
The Degree of Freedom • In order to compute a statistic like the standard error of means ( x̅ ), it is necessary to use observations obtained from a sample as well as certain population parameters i.e. population quantities like population mean (µ) variance opposed to sample quantities like sample mean ( x̅ ) variance (called statistics). If the population parameters are unknown, they must be estimated from the sample statistics given. The number of degrees of freedom of a statistic, which is generally denoted by v. Slide 13
• is defined as the number #(N) of independent observations in the sample i.e. the sample size, minus the number (k) of population parameters which must be estimated from the sample observations. This is symbolized as (N-K). Where in ( x̅ ) = we have to estimate from the sample, then the number (#) of independent observations in the sample is N and since we have to estimate then k = 1 so v = (N-1). This provides a basis for computing the SE of the mean when the sampled SD i.e. s must be used as estimate of i.e. population standard deviation. The formula ( x̅ ) = becomes ( x̅ ) = - 1 Slide 14
Summary • In this section, you have learned about what sampling distributions are and the relevance of such distributions in tests of significance. Slide 15
THANK YOU Slide 16
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