Learning Objectives of this session
At the end of the lecture the student should :Understand what is meant by hypothesis testing (test of significance)Understand when , and be able , to carry out (Z test) for a parameter (µ or Π) of single population using information from one sample. Understand when , and be able , to carry out (Z test) for the equality of two population means or proportions using information from two independent samples.What Is a Test of Significance?
A test of significance is a formal procedure for comparing observed data with a hypothesis whose truth is to be assessed.The results of a test are expressed in terms of a probability that measures how well the data and hypothesis agree.
Stating hypotheses
Significance testing involves two competing hypotheses 1.Null hypothesis 2. Alternative hypothesis*
The Null- Hypothesis (H0 )
The null hypothesis (H0) is the statement that the effect we want is not present. Typically, H0 is a statement of “no difference” or “no effect”.The Alternative Hypothesis (Ha )
The alternative hypothesis (Ha) is the statement of what we hope or suspect is true. (i.e what we are trying to prove or the effect we are hoping to see). Ha is a statement of difference or relationship It can be One Tailed either > or <( e.g. ) Or Two Tailed <>(e.g. )Note: A hypothesis is a statement about the parameters in a population.
Hypothesis Forms for Means
Hypothesis Forms for Proportions
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Test of SignificanceA statistical tests begins by stating the null hypothesis (H0) Then we try to find evidence against this hypothesis . We also want to assess the strength of the evidence against the null hypothesis
Independent Samples
Paired SamplesIndependent Samples
* Two-samples t test * Z test
* Paired t test
*Chi-Squared *Mann-Whitney U Test
Chi-Squared McNemar Wilcoxon signed-rank test
A Classification of Hypothesis TestingHypothesis Tests
One Sample
Two Samples
One Sample
Two Samples
* t test * Z test
* Chi-Squared
Parametric Tests
Non-parametric Tests
Paired Samples
Parametric v. Nonparametric
Parametric tests (e.g., t-test) make assumptions about characteristics of data Scores must be on an interval or ratio scale Assume normal distribution*
Nonparametric tests
Used if parametric assumptions are not satisfied Usually examine ranks (order) of scores, rather than scores Test differences in Median, rather than mean*
The samples are independent if they are drawn randomly from different populations. If data pertaining to different groups of respondents even within the same sample e.g., males and females, are generally treated as independent samples. The samples are paired when the data for the two samples relate to the same group of respondents. E.g., pretest posttest, control group design
Independent and paired samples
State the Null Hypothesis State the Alternative Hypothesis State the level of significance. Choose the correct statistical test. Compute the test statistic Determine the critical value of a statistics (needed to reject the null hypothesis) from a table of sampling distribution values) Compare computed to critical value Accept or reject the Null Hypothesis
Steps in Hypothesis Testing:
Significance level
Significance level ( represented as ) is the value of probability below which we start consider significant differences. Typical levels used are 0.1, 0.05,0.01 and 0.001.a =0.05 is widely accepted as criterion for statistical significance.It is the statement of the chance of making a wrong decision about rejecting a true null hypothesis (it tells how many times out of a hundred or thousand that rejecting the null hypothesis would be wrong)How do we define the critical values
The reasoning of statistical tests, like that of confidence intervals, is based on the sampling distribution of the means.Trying to prove the parameter is “less than”
Trying to prove the parameter is “more than”
Trying to prove the parameter is “not equal to” or “different from”Hypothesis testing (one sample tests) I. One sample mean
One sample mean with s is known or large sample size- Normal test(Z test) Steps for Testing one sample Mean 1. State the null hypothesis: 2. State the alternative hypothesis: 3. State the level of significance (for example, a = 0.05). 4. Calculate the test statistic5. Find the Critical ValueFor a one-Tailed test: z=0.05 = 1.64For a two-Tailed test:1.96Decision : Reject H0 if Test Statistic > Critical Value i.e P-value < (Significance level) Accept H0if Test Statistic < Critical Value i.e P-value > (Significance level)
7. State your conclusion.If H0 is rejected, “There is significant statistical evidence that the population mean is different from µ. The results are said to be statistically significant.If H0 is accepted , results are said to be not significant. “There is no significant statistical evidence that the population mean is different from µ.
The P-value
If the P-value is as small or smaller than , we say that the data are statistically significant at level .Not likely to happen by chanceThe lower the level of significance, the more confidence there is in correctly rejecting the null hypothesis.II. One sample proportion- z test
Steps for Testing one sample Proportion 1. State the null hypothesis: 2. State the alternative hypothesis: 3. State the level of significance (for example, a = 0.05). 4. Calculate the test statistic)
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Hypothesis testing ( Two sample Means)
Independent Samples and known population variances or large samples (both more than 30)- Normal test ( Z test)1. Set up hypotheses2. Test Statistics (Z)If is unknown, but the sample is large then replace with S3. Critical Value: For two-tailed test: Z/2 ; For one-tailed test: Z 4. Decision Rule: Reject H0 if Test Statistic > Critical Value i.e.if P-value < (Significance level)5. ConclusionHypothesis testing (Two sample Proportions)- Z test
1. Set up hypotheses1. Set up hypotheses2. Test Statistics (Z*)3. Critical Value: For two-tailed test: Z/2 ; For one-tailed test: Z 4. Decision Rule: Reject H0 if Test Statistic > Critical Value orif P-value < (Significance level) 5. Conclusion P = pooled proportion (pooled p) P=or
Errors in Hypothesis Testing
Conclusions from significance tests can be wrongType I error: No effect, but we find an effectProbability of Type I error alpha (a)The researcher chooses the alpha level Type II error: True effect, but we don’t find itProbability of Type II error beta (b) Statistical power = 1 – betaBeta is inversely related to alphaBeta (and power) is also related to sample size *Truth
outcomeH0 is True
Accept H0
H0 is False
Reject H0
Correct Decision
Correct Decision
Type I Error (alpha)
Type II Error (beta)
Errors in Hypothesis Testing