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# Inferential stat handout 2013 2014 part 2

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• 1. GRADE 8 RESEARCH INFERENTIAL STATISTICS HANDOUT PART 2 Statistical Procedures Transforming a raw score into a z-score and plotting it into the normal curve might give us an idea of the difference between means. However, there are more appropriate and practical statistical procedures to use when dealing with research data. The correct statistical procedure to use is primarily dependent on the type of hypothesis, type of data, and number of groups utilized in your research study. The one-sample t-test The one-sample t-test must be used to test the difference between a population mean ( ) with a single sample mean(x-bar ) when both (1) the population standard deviation ( ) is unknown, and (2) the sample size (n) is less than 30 (That is, the z-test is not practical to becomputed under these conditions). Just like z-scores, results of the one-sample t-test should be compared to a critical value to conclude that the means are significantly different from each other (or to reject the null hypothesis).But, the critical values of a t-test are computed differently from the critical values of z-scores. We need a tdistribution table for this. To find the critical value for a t-test, we need to consider the following: Level of significance = α (usually 0.05) Type of test = whether it is one-tailed (usually 0.05) of two-tailed (usually 0.025) Degrees of freedom = computed by getting n-1 The t-test for twoindependent samples Samples are “independent” when subjects are randomly selected and assigned individually to groups. The formula for the t-test for independent samples is: x-bar = mean of first group y-bar = mean of second group nx = no. of subjects in first group ny = no. of subjects in second group sx = st. dev of first group sy = st. dev of 2nd group To find the critical value for a t-test for independent samples, we need to consider the following: Level of significance = α (usually 0.05) Type of test = whether it is one-tailed (usually 0.05) of two-tailed (usually 0.025) Degrees of freedom = computed by getting n1 + n2 - 2
• 2. The t-test for two matched samples The Correlated Samples (or “Matched Samples”) t-test is used when data come from (1) naturally occurring pairs, (2) two measures taken on asingle subject, such as pre- and post-tests, or (3) two subjects that have been paired or “matched” on some variable, such as intelligence, age, or educational level. The formula is: d-bar = difference of the means of the two groups( mean of all post-test minus mean of all pre-test) summation of d = sum of the differences between data (sum of all post-test minus pre-test) n = no. of pairs Example: Pair Number 1 2 3 4 5 6 n=6 Pre-Test Post-Test 10 12 13 14 10 12 Mean = 11.83 d-bar = 5.84 19 20 17 13 19 18 Mean = 17.67 Difference 9 8 4 -1 9 6 Σd = 35 (Σd)2 = 1,225 Difference squared 81 64 16 1 81 36 Σd2= 279 sd2= The critical value for the –test for matched samples is derived by: = 14.97 sd2 sd = sd = 1.58 t= t = 3.69 Level of significance = α (usually 0.05) Type of test = whether it is one-tailed (usually 0.05) of two-tailed (usually 0.025) Degrees of freedom = computed by getting n0. of pairs -1 For all t-tests, compare the critical value to the computed t-value. If the computed value is higher, then there is a significant difference. Your null hypothesis must be rejected.
• 3. Summary: When do we use the t-tests? t-tests are used when comparing TWO groups. No matter what your design is, as long as you are only comparing TWO groups, then the t-test is the most applicable to use. Critical values of all t-tests depend on the type of alternative hypothesis (one- or two-tailed), level of significance (α), and degrees of freedom. Type of t-test Description Critical Value One-sample Used when the n > 30, and when you want to compare your experimental group to an already given mean of the population. Independent Samples Used when comparing two unrelated groups (e.g. experimental vs. control, expt’al A vs. expt’al B) Matched / Paired Used when comparing two related groups specially the pre-test and post-test groups. Level of significance = α (usually 0.05) Type of test = whether it is onetailed (usually 0.05) of two-tailed (usually 0.025) Degrees of freedom = computed by getting n-1 Level of significance = α (usually 0.05) Type of test = whether it is onetailed (usually 0.05) of two-tailed (usually 0.025) Degrees of freedom = computed by getting n1 + n2 - 2 Level of significance = α (usually 0.05) Type of test = whether it is onetailed (usually 0.05) of two-tailed (usually 0.025) Degrees of freedom = computed by getting no. of pairs -1 Sample Problems: Use level of significance 0.05 (α). 1. “Are intellectuals more likely to wear a beard?” wondered a social psychologist, he asked samples of bearded and clean-shaven men about their level of attainment. Using these data on years of schooling, test for the significance (non-directional) of the between means. Beard No Beard 18 23 12 14 18 16 15 17 11 12 16 14 12 12 2. A social psychologist was interested in sex differences in the sociability of teenagers. Using the number of good friends as a measure, he compared the sociability of eight female and seven male teenagers. Test the null hypothesis of no difference (non-directional) with respect to sociability between females and males. What do your results indicate?
• 4. Males 8 3 1 7 7 6 8 5 Females 1 5 8 3 2 1 2 3. In a test of the hypothesis that females smile at others more than males do, females and males were videotaped while interacting and the number of smiles emitted by each sex was noted. Using the following number of smiles in the five-minute interaction, test the non-directional hypothesis that there is a difference between males and females. Males 8 11 13 4 2 Females 15 19 13 11 18 4. A criminologist was interested in whether there was any disparity in sentencing based on the race of the defendant. She selected at random 18 burglary convictions and compared the prison terms given to the 10 whites and 8 blacks sampled. The sentence lengths (in years) are shown for the white and black offenders. Using these data, test the null hypothesis that whites and blacks convicted of burglary in this jurisdiction do not differ with respect to prison sentence length.For this problem, test both non-directional and directional hypotheses. Whites 3 5 4 7 4 5 6 4 3 2 Blacks 4 8 7 3 5 4 5 4 5. A personnel consultant was hired to study the influence of sick-pay benefits on absenteeism. She randomly selected samples of hourly employees who do not get paid when out sick and salaried employees who receive sick pay. Using the following data on the number of days absent during a one-year period, test the directional hypothesis that salaried employees incur less absences than hourly employees. Hourly Salaried 1 1 2 3 3 Salaried 2 2 4 2 2