Inferential Statistics 4 Maarten Buis 18/01/2006

Outline interpretation of confidence interval confidence interval and testing Analysis of Variance

Interpreting confidence intervals If you draw a hundred samples and compute a 95% confidence interval of the mean in each of these samples than the population mean will be inside the interval in 95 samples If you draw one sample and compute the confidence interval, than the population mean is either within that interval or it is not. So you are not 95% sure that the population mean is in that interval.

Confidence vs. Probability The procedure will give the correct conclusion in 95% of the times it is used. You have no way of knowing if you are one of the 95% ‘lucky ones’ or the 5% ‘unlucky ones’ when you have drawn one sample and computed a confidence interval. All you can say is that you have used a high quality method to construct the interval.

confidence interval and the sampling distribution If we have an estimate of the sampling distribution, than the 2.5 th and the 97.5 th percentiles will form the 95% confidence interval. These percentiles are the critical values and can be looked up in the appropriate table. In 5% of the samples the true parameter will be outside that interval Notice that the true parameter remains fixed and the estimates of the lower and upper bound change between samples.

Best estimate of the sampling distribution of a mean Our best estimate of the mean in the population is the mean in the sample So, our best estimate of the mean of the sampling distribution is the mean of the sample Our best estimate of the standard error is the standard deviation divided by the square root of N So our best estimate of the sampling distribution of the mean is a t-distribution with mean equal to the sample mean, a standard deviation of the standard error, and N-1 degrees of freedom

confidence interval for mean rent N=19, so df =18 look up the two sided critical t-value in Appendix B, table 2: mean is 258, s = 99, so se = lb = *2.101 = 210 ub = *2.101 = 306

Comparing means of more than two groups Until now we have compared the means of two groups, and not –compared means of more than two groups or, –compared means for a continuous x-variable (regression) In these cases we use analysis of variance (ANOVA) and the F-test

The Null Hypothesis The null hypothesis is that the means of all groups are equal:        k We observe the means of group 1 till k: M 1, M 2, M 3,..., M k, and these differ due to sampling error Are these deviations large enough to reject H 0

Decomposition of Sum of Squares McCall p. 358 Y i, M k, M (Y i -M) = (Y i -M k ) + (M k -M) Deviation of a score from the overall mean consists of a deviation of the score to the group mean plus a deviation of the group mean to the overall mean. Square and sum: SS total =SS within + SS between

Mean Sum of Squares Estimates of the Mean Sum of Squares (variance) are obtained by dividing the Sum of Squares by the number of degrees of freedom: –MS total = SS total /(N-1) –MS within = SS within /(N-k) –MS between = SS between /(k-1) N is the sample size and k is the number of groups

old friends MS total = variance MS within = (standard error of the estimate) 2 MS between /MS total = R 2 or proportion of variance explained, so: MS between = variance explained

F-test The F statistic is just an estimate like the mean, or the correlation, so it has a sampling distribution: the F-distribution, appendix 2, table E. The F-distribution has two types of degrees of freedom: –for the numerator, MS between ; k-1) and –for the denominator, MS within ; n-k

F-test If H 0 is true (all group means are equal) than MS within = MS between Otherwise MS between > MS within F = MS between / MS within So H 0 can be rewritten as: F = 1 And H A : F > 1 This is not a directional hypothesis since F>1 implies:        k

To do before Monday read chapter 14, pay special attention to pp Skip: –pp computational procedure –pp Use SPSS when making sums with example data