1. Inferential Statistics Confidence Intervals and Hypothesis Testing

2. Samples vs. Populations Population –All of the objects that belong to a class (e.g. all Darl projectile points, all Americans, all pollen grains) –A theoretical distribution Sample –Some of the objects in a class –Observations drawn from a distribution

3. Two Distributions The sample distribution is the distribution of the values of a sample – exactly what we get plotting a histogram or a kernel density plot The sampling distribution is the distribution of a statistic that we have computed from the sample (e.g. a mean)

5. Confidence Intervals Given a sample statistic estimating a population parameter, what is the parameter’s actual value? Standard Error of the Estimate provides the standard deviation for the sample statistic:

6. Example 1 Snodgrass house size. Mean area is 236.8 with a standard deviation of 94.25 based on 91 houses. Area is slightly asymmetrical Can we use these data to predict house sizes at other Mississippian sites?

8. Example 1 (cont) The confidence interval is based on the mean, sd, and sample size Mean ± t(p<confidence)*sd/sqrt(n) For 95%, 90%, 67% confidence –qt(c(.025,.975), df=90) –qt(c(.167,.833), df=90)

9. # Distributions x <- seq(10, 40, length.out=200) y1 <- dnorm(x, mean=25, sd=4) y2 <- dnorm(x, mean=25, sd=1) max(y2) plot(x, y1, type="l", ylim=c(0,.4), col="red") lines(x, y2, col="blue") text(c(28, 26.3), c(.08,.30), c("Sample Distribution\n mean=25, sd=4", "Sampling Distribution\n m=25, sd=1, n=16)"), col=c("red", "blue"), pos=4) # Snodgrass House Areas plot(density(Snodgrass$Area), main="Snodgrass House Areas") lines(seq(0, 475, length.out=100), dnorm(seq(0, 475, length.out=100), mean=236.8, sd=94.2), lty=2) abline(v=mean(Snodgrass$Area)) legend("topright", c("Kernel Density", "Normal Distribution"), lty=c(1, 2)) # Confidence interval function conf <- function(x, conf) { conf 1, conf/100, conf) tail <- (1-conf)/2 mean(x)+qt(c(tail, 1-tail), df=length(x)-1)*sd(x)/sqrt(length(x)) }

10. Bootstrapping Confidence intervals depend on a normal sampling distribution This will generally be a reasonable assumption if the sample size is moderately large We can draw multiple samples of house areas to get some idea

12. # Draw 100 samples of size 50 samples <- sapply(1:100, function(x) mean(sample(Snodgrass$Area, 50, replace=TRUE))) range(samples) quantile(samples, probs=c(.025,.975)) conf(Snodgrass$Area, 95) plot(density(samples), main="Sample Size = 50") x <- seq(175, 300, 1) lines(x, dnorm(x, mean=mean(samples), sd=sd(samples)), lty=2) legend("topright", c("Kernel Density", "Normal Distribution"), lty=c(1, 2)) # Draw 1000 samples of size 91 samples <- sapply(1:100, function(x) mean(sample(Snodgrass$Area, 91, replace=TRUE))) range(samples) quantile(samples, probs=c(.025,.975)) conf(Snodgrass$Area, 95) plot(density(samples), main="Sample Size = 91") x <- seq(175, 300, 1) lines(x, dnorm(x, mean=mean(samples), sd=sd(samples)), lty=2) legend("topright", c("Kernel Density", "Normal Distribution"), lty=c(1, 2))

13. Example 2 Radiocarbon Ages are presented as an age estimate and a standard error: 2810 ± 110 B.P. The probability that the true age is between 2700 and 2920 B.P. is.6826 or.3174 that it is outside that range The probability that the true age is between 2590 and 3030 B.P. is.9546 or.0545 that it is outside that range

14. Hypothesis Testing Assumptions and Null Hypothesis Test Statistic (method) Significance Level Observe Data Compute Test Statistic Make Decision

15. Assumptions Data are a random sample –Every combination is equally likely Appropriate sampling distribution

16. Null Hypothesis Represented by H 0 Must be specific, e.g. S1-S2 = 0 The difference between two sample statistics is zero, e.g. they are drawn from the same population (two tailed test) Or S1-S2>0 (one tailed)

17. Test Statistic Measurement Levels Number of groups Dependent vs. Independent Power

18. Significance Level Nothing is absolute in probability Select probability of making certain kinds of errors Cannot minimize both kinds of errors Social scientists often use p ≤ 0.05 Consider how many tests

19. Errors in Hypothesis Testing Null Hypothesis (H 0 ) is TrueFalse Research Decision Reject H 0 Error Type I, α Correct Decision Accept H 0 (fail to reject) Correct Decision Error Type II, β

20. Difference of Means (t-test) Independent random samples of normally distributed variates Samples: 1, 2 independent, 2 related If 2 independent – variances equal or unequal Sample statistics follow the t- distribution

21. Example Snodgrass site is a Mississippian site in Missouri that was occupied about A.D. 1164

23. Using Rcmdr Snodgrass Site – House sizes inside and outside are the same Check normality - shapiro.test() Check equal variances – var.test() or bartlett.test() Compute statistic and make decision – t.test()

24. Wilcoxon Test If data do not follow a normal distribution or are ranks not interval/ratio scale Nonparametric test that is similar to the t-test but not as powerful Tests for equality of medians –wilcox.test()

25. Difference of Proportions Uses the normal distribution to approximate the binomial distribution to test differences between proportions (probabilities) This approximation is accurate as long as N x (min(p,(1-p))>5 where N is the sample size, p is the proportion, and min() is the minimum

26. Using Rcmdr Must have two or more variables defined as factors, eg, –Create ProjPts to be equal to as.factor(ifelse(Points>0, 1, 0)) using Data | Manage variables... | Compute new variable –Statistics | Proportions | Two sample... –prop.test() –Are the % Absent equal inside and outside the wall?