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Crim 320 week 10.docx

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CRIM 320
Patrick Lussier

Crim 320 Week 10 March 12, 2012 Introduction to t test - Context - Main goal - Assumptions - Procedures - Independent-sample t-test - Summary Difference between two population means - Type of research questions - Two variables are under investigations o One nominal variable with only two attributes as the IV o One interval/ratio variable considered the DV Of fingers, toes and penises - Study published in Nature, by Kondo et a. (1997) - Identification of the Hox genes: Hox_a and Hox_d - Responsible for the development of both fingers and the urogenital system - Removal of the genes in mice leads to foetus characterized y the absence of digits and genitalia - Hox gene also found in humans - 2 to 4 digit ratio increasingly used as a proxy measure of exposure to fetal testosterone (in utero) th - The relative 2D:4D finger lenth is established as early as the 14 week after birth - 2D:4D is lower in males than females T-test - Research question o Research seems to suggest a link between gender digit ratio - Research hypothesis o H1: Males have a lower 2D:4D digit ratio than females - Null hypothesis o H0: There are no significant differences in 2D:4D digit ratio between males and females Difference between two-sample means Population 1 “males” Population 2 “females” Pop meanPop st. dev Sample mean Sample st. dev Sample size n1 n2 Main goal of the t-test - Inferential statistical procedure o Allow to test hypotheses about the differences between two populations means (u1-u2) o The statistical procedures involve testing a hypothesis using two sample means (x1-x2) Difference between two sample means - Descriptive analysis of the variables under investigation o First, the nominal variable o The two “samples” or groups being compared o Gender of the participant o Second, the interval variable o The variable reflecting the 2D:4D ratio o Length of the 2 digit divided by the length of the 4 digit o Third, inspection of group means o Analysis can only be done on cases for which we have information obn both variables, i.e., gender and 2D:4D ratio o Inspection of group means reveal differences:  1-Mean of .98 (SD=.04) for women  2-Mean of ..95 (SD=.04) for men  Notice that the st. dev is very similar for men and women  Notice also that the range or “response” is the same for men and women - Differnece between two-saple means o Preliminary observations o Based on our sample statistics, we can conclude that:  Xwomen > Xmen  A mean difference of .03 is observed  Two scenarios might explain this result - Scenario #1: no difference in the population means, if you have data for the whole province then we won’t be finding any difference between males and females. (graph looks symmetrical) - Scenario #2: a true difference in the population means. The difference we’re getting using sample statistics might actually be indicative of TRUE differences in general population (there is two graphs because of the different ratios) - Because we have sample data and not population data, mean differences observed may be due to: o Real differences in the popn (u2>u1)  We could conclude that there is a statistically significant relationship between gender and digit ratio o Chance or random variation (u1=u2)  When we draw random samples from the popn, the two sample means will sometimes differ  We could conclude that the observed difference between men and women is no greater that what would be expected by chance - Need to determine whether the mean differences are due to “true” differences in the popn or “random” variation due to sampling? o Need to rely on probability theory o Testing the null hypothesis o “what is the probability of making a mistake by rejecting the null hypothesis? o When comparing the difference between two popn means, we use the t probability distribution - Distribution of t stats allows us to make conclusion about null hypothesis t statistic - Calculation of the t statistic in order to evaluate whether the mean difference is statistically significant - Mean difference between group 1 and 2 DIVIDED BY variance difference for group 1 and 2 - T = x1-x2/ S*(x1-x2) t distribution and t statistic x1-x2 u2 - Then, more likely to sample X1>X2 Determining the critical region - Example: for a df of 40 and a desired level of confidence of 5%, the t statistic is 2.02 - Means we split the 5% on both sides of the distribution, a=.025 - If the t statistic of calculated from our two sample means is 1.96, then we cannot reject the null hypothesis - If the t statistic of calculated from our two sample means is 2.25, then we can reject the null hypothesis Assumptions - T-test is a parametric test assuming: o The samples were randomly selected o The samples are drawn from normally distributed populations o The dependent variable is interval/ratio and normally distributed o The variance of the two samples is homogenous Procedures for a sample t-test State rese
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