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t Tests Hypothesis Testing The most basic and commonly used procedures One Sample t Test Two Sample t Test Compare a single mean to a fixed number Compare two population means based on independent samples from the two populations or groups Paired t Test Compare two means based on samples that are paired in some way One Sample t Test or z test Compare sample results with a known value Mean weight of male Geography majors vs mean weight male UA students The advertised tension strength of garbage bags vs your sample SAT scores for a random sample vs the mean SAT value from 3 years ago Government specification on the percentage of fruit juice that must be in a drink before it can be sold as fruit juice Considerations for 1 sample t Test If the sample size is small (< 15), t test should only be used if there is minimal skewness and outlier impact If the sample size is moderate (15 – 40), t test can be used in most cases unless there are extreme outliers If the sample size is large (> 40), t test may be used safely 2 tailed or 1 tailed test 1 tailed If you are interested in rejecting the null hypothesis if the population mean differs from the hypothesized value in a direction of interest 2 tailed Ho: µ = µo (the pop mean is = to the hypothesized value Ha: µ > µo (the pop mean is > µo) Ho: µ = µo (the pop mean is = to the hypothesized value Ha: µ ≠ µo (the pop mean is ≠ to µo) SPSS reports 2 tailed p value Divide by 2 for 1 tailed test 2 Sample t Test Determine whether the unknown means of 2 populations are different from each other based on independent samples from each population Equal Variance Unequal Variance 2 Sample Considerations Comparison of means Data should not be categorical even if it is recoded Independent Samples 2 samples from same population 2 samples from different populations Check for normality with smaller samples Equal variances Levene’s test Sample sizes from both groups are similar paired t Test Before and after Two measurements on the same subject Control group vs treatment group Ho: µd = 0 (population meqan of the differences is 0) Ha: µd ≠ 0 (population mean of the difference is not zero) Data analyzed are differences within pairs Paired t Test considerations Pairing observations may increase the ability to detect differences Normality of difference scores One Sample Test Practice Calculate Test Statistic Calculation of the test statistic requires four components: The average of the sample (observed average) The population average or other known value (expected average) The standard deviation of the average The number of observations. T stat- compare to table Having calculated the t-statistic, compare the t-value with a standard table of t-values to determine whether the t-statistic reaches the threshold of statistical significance. Practice SE = s / sqrt( n ) t = (x - μ) / SE We are testing to see if a sample mean score of freshman GY101 students is less than the class mean of 70 on exam 1. 65 67 52 58 46 85 44 41 43 50 83 53 49 52 74 39 72 76 Freshman test scores your list of numbers: 1, 3, 4, 6, 9, 19 SD calculation example mean: (1+3+4+6+9+19) / 6 = 42 / 6 = 7 list of deviations: -6, -4, -3, -1, 2, 12 squares of deviations: 36, 16, 9, 1, 4, 144 sum of deviations: 36+16+9+1+4+144 = 210 divided by one less than the number of items in the list: 210 / 5 = 42 square root of this number: square root (42) = about 6.48

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