inferential statistics confidence intervals and hypothesis testing
Post on 19-Dec-2015
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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
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)
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:
n
ssX
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?
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(.025,.975), df=90)– qt(c(.167,.833), df=90)
# Distributionsx <- 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 Areasplot(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 functionconf <- function(x, conf) {
conf <- ifelse(conf>1, conf/100, conf)tail <- (1-conf)/2mean(x)+qt(c(tail, 1-tail), df=length(x)-1)*sd(x)/sqrt(length(x))
}
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
# 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))
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
Hypothesis Testing
• Assumptions and Null Hypothesis• Test Statistic (method)• Significance Level• Observe Data• Compute Test Statistic• Make Decision
Assumptions
• Data are a random sample– Every combination is equally likely
• Appropriate sampling distribution
Null Hypothesis
• Represented by H0
• 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)
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
Errors in Hypothesis Testing
Null Hypothesis (H0) is
True False
Research Decision Reject H0
ErrorType I, α
Correct Decision
Accept H0 (fail to reject)
Correct Decision
ErrorType II, β
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
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()
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()
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
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?