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Hypothesis Testing-I SUBJECT FORENSIC SCIENCE SUBJECT FORENSIC SCIENCE Paper No. and Title PAPER No.15: Forensic Psychology Module No. and Title MODULE No.31: Hypothesis Testing-1 Module Tag FSC_P15_M31 FORENSIC SCIENCE PAPER No.15 : Forensic Psychology MODULE No.31: Hypothesis Testing-I TABLE OF CONTENTS 1. Learning Outcomes 2. Introduction 3. Single Sample Parametric Test 3.1 Single sample Z test 3.2 Single sample t-test 4. Single Sample Non-Parametric Test 4.1 Wilcoxon Signed-Rank Test 5. Summary FORENSIC SCIENCE PAPER No.15 : Forensic Psychology MODULE No.31: Hypothesis Testing-I 1. Learning Outcomes After studying this module, you will be able to- Know about parametric and non-parametric method. Understand one sample Z and t test. Understand Wilcoxon Signed-Rank Test 2. Introduction Parametric Methods These methods are categorized on the basis the population we are studying. Parametric methods are those for which the population is almost normal, or we can estimate using a normal distribution after we invoke the central limit theorem. Eventually the categorization of a method as non-parametric depends upon the conventions that are made about a population. A few parametric methods include: Confidence interval for a population mean, with known standard deviation. Confidence interval for a population mean with unknown standard deviation Confidence interval for a population variance. Confidence interval for the difference of two means with unknown standard deviation. Non-Parametric Methods Non-parametric methods are statistical techniques for which any assumption of normality for the population is not required unlike parametric methods. Certainly the methods do not have any dependency on the population of interest. Non-parametric methods are also known as distribution free methods. These are emerging in reputation and impact for many reasons. The main reason is that we are not forced to making as many assumptions about the population that we are working with as what we have to make with a parametric method. Many of these non-parametric methods are easy to apply and to understand. FORENSIC SCIENCE PAPER No.15 : Forensic Psychology MODULE No.31: Hypothesis Testing-I A few non-parametric methods include: Sign test for population mean Bootstrapping techniques U test for two independent means Spearman correlation test Comparison between Parametric and Non-Parametric Method There are multiple ways to use statistics to find a confidence interval about a mean. Many times parametric methods are more efficient than the corresponding non-parametric methods. Although this difference in proficiency is usually not that much of a concern, there are cases where we need to consider which method is more proficient. Non-parametric statistical processes are less potent because they use less information in their calculation. For example, a parametric correlation uses information about the mean and deviation from the mean while a non-parametric correlation will use only the ordinal position of pairs of scores. The basic dissimilarity between parametric vs non-parametric is: If your measurement scale is nominal or ordinal then you use non- parametric statistics. If you are using interval or ratio scales, you use parametric statistics. There are other considerations which have to be taken into account. You have to look at the distribution of your data. If your information is supposed to take parametric statistics, you should check that the distributions are nearly normal. The best way to do this is to check the skew and kurtosis measures from the frequency output. FORENSIC SCIENCE PAPER No.15 : Forensic Psychology MODULE No.31: Hypothesis Testing-I Analysis type Parametric Non parametric procedure procedure Compare means between two Two sample t test Wilcoxon rank-sum test distinct/independent groups Compare two quantitative Paired t-test Wilcoxon-signed rand test measurements taken from the same individual Compare means between three or ANOVA Kruskal-Wallis Test more distinct/independent groups Estimate the degree of association Pearson coefficient of Spearman’s rank correlation between two quantitative variables correlation 3. Single Sample Parametric Tests 3.1 One-Sample Z Test To determine whether the sample belongs to a particular population, one sample Z test is used. For example, if we are doing research on data collected from various groups of students of a school studying English. We may want to know if this particular sample of students is identical to or different from school students in general. This test is used only for the test of the sample mean. The main hypothesis is to determine whether the average of our sample (M) suggests that the students selected belong to the population with known mean (μ) or it comes from different population. Depending upon the type of hypothesis i.e. directional or non-directional, following forms of hypothesis can be used. In the equations, μ1 is the population from which the sample for study was chosen and μ is replaced by the actual value of the population mean. FORENSIC SCIENCE PAPER No.15 : Forensic Psychology MODULE No.31: Hypothesis Testing-I H0 : µ1 = µ HA : µ1 ≠ µ H0 : µ1 ≤ µ HA : µ1 > µ H0 : µ1 ≥ µ HA : µ1 < µ In one sample Z test, one group of subject is chosen, data is composed for this subject and then the comparison between sample statistic (M) and population parameter (μ) is made. The population parameter helps us determine what to expect if our sample was found to be from that particular sample. In case the sample statistic is very diverse then we can determine that it is from a dissimilar population. In this test, we are comparing the mean calculated on a single set of score to a known population mean. This test associates a sample to a defined population in which defined population stands for various parameters of population which are known. Population distribution is defined in terms of central tendency and variability. For one sample Z-test, known population is μ and σ. Without known μ and σ, one sample Z test cannot be performed. A set of assumptions are there for all parametric statistics which needs to be met in order to properly use the statistics to test hypothesis. These assumptions used in one sample Z test are: random sampling form a defined population, interval or ratio scale of measurement and population is normally distributed. Since random sampling is based on probability, it is required for all statistical inference. The psychologist may use Z test for dependent means on approximately interval scales even though the tests require interval or ratio data. Central limit theorem helps us to understand that even if the population distribution is unknown, the sampling distribution of the mean will be approximately normally distributed in case sample size is large. FORENSIC SCIENCE PAPER No.15 : Forensic Psychology MODULE No.31: Hypothesis Testing-I For example: A psychologist claims that thе childrеn who havе nеvеr facеd a crimе havе morе IQ. A random samplе of thirty childrеn IQ scorеs havе a mеan scorе of 112. Is thеrе sufficiеnt еvidеncе to support thе psychologist’s claim? Thе mеan population IQ is 100 with a standard dеviation of 15. Solution: 1. 1. Statе thе null hypothеsis. Thе accеptеd fact is that thе population mеan is 100, so, H0: μ=100. 2. Now statе thе altеrnatе hypothеsis. Thе claim is that thе childrеn who havе nеvеr facеd a crimе havе morе IQ scorе, so. H1: μ > 100.Sincе wе arе looking for scorеs grеatеr than thе cеrtain point mеans that this is a onе tailеd tеst. 3. Statе thе alpha lеvеl and if thе alpha lеvеl is not givеn thеn usе 5% (0.05). 4. Find thе rеjеction rеgion arеa givеn by thе alpha lеvеl from thе Z-tablе. An arеa of 0.05 is еqual to a Z scorе of 1.645. 5. Find thе tеst statistic using thе formula: 푥̅ −휇0 Z = 휎/√푛 6. For this sеt of data: Z= (112-100) / (15/√30) = 4.56. If Stеp 6 is grеatеr than Stеp 4, rеjеct thе null hypothеsis. If it’s lеss than Stеp 4, you cannot rеjеct thе null hypothеsis. In this casе, it is grеatеr, so you can rеjеct thе null. 3.2 One-Sample T-Test When information about full populations is not available, then one sample t test is used to know whether the sample comes from a particular population. For example, if we want to know if a particular sample of cloth is similar to or different from other cloth pieces in general. This test is used only for tests of the sample mean. Therefore, our hypothesis tests whether the mean of our sample (M) suggests that our cloth piece come from a population with a known mean (μ) or whether it comes from a different population. On the basis of the fact that whether the research hypothesis is directional or non-directional, the hypothesis for one sample t test can take one the following forms. In the following, μ1 is the population from which the sample to be studied was chosen whereas μ is replaced by the actual value of the population mean. FORENSIC SCIENCE PAPER No.15 : Forensic Psychology MODULE No.31: Hypothesis Testing-I The hypothesis are similar to those which are used for one sample Z tests. H0 : µ1 = µ HA : µ1 ≠ µ H0 : µ1 ≤ µ HA : µ1 > µ H0 : µ1 ≥ µ HA : µ1 < µ In one sample t test, one group of subject is selected, data is collected for this subject and then the sample statistic (M) is compared to the population parameter (μ). The population parameter helps us determine what to expect if our sample was found to be from that particular sample. In case the sample statistic is very different then we can conclude that it is from a different population.
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