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Chapter 6: Inferences Comparing Two Popula data=c(10,3,2,1,4) > mean(data)  4
> median(data)  3 > median(c(3,2,1,4))  2.5
Comparing Two Populaμ2, or H1:μ10)
Population of females with diabetes μ2 = mean glucose level σ2 = sd glucose level
Recall, that when both populations are normally distributed,
Using R: Welch Two-‐Sample T-‐test • Example 6.1 (p. 294): Company officials were concerned about the length of time
a particular drug product retained its potency. A random sample of 10 bottles of the product was drawn from the production line and analyzed for potency. A second sample 10 bottles was obtained and stored in a regulated environment for a period of 1 year. The readings obtained from each sample are given below:
> fresh=c(10.2,10.5,10.3,10.8,9.8,10.6,10.7,10.2,10,10.6)! > stored=c(9.8,9.6,10.1,10.2,10.1,9.7,9.5,9.6,9.8,9.9)! > t.test(fresh,stored)!
!Welch Two Sample t-test! data: fresh and stored! t = 4.2368, df = 16.628, p-value = 0.000581! alternative hypothesis: true difference in means is not equal to 0! 95 percent confidence interval:! 0.2706369 0.8093631! sample estimates:! mean of x mean of y ! 10.37 9.83 !
Hence, we are 95% confident that the reduction in potency is between 0.27 and 0.81.
Since the p-value is less than α=0.05, we reject H0:µ1=µ2 and conclude that mean potency is statistically lower after 1 year.
Using R: Pooled T-‐test • If it is reasonable to assume that the two unknown standard deviations are equal,
(Rule of Thumb: If the larger sample standard deviation is not more than twice the smaller, it is reasonable to make this assumption. In the next chapter, we will learn a formal way to test this assumption.) we should use the Pooled T-Test.
> sd(fresh) ! ! !# 0.3233505! > sd(stored) ! ! !# 0.2406011! > t.test(fresh,stored,var.equal=T)!
!Two Sample t-test! data: fresh and stored! t = 4.2368, df = 18, p-value = 0.0004959! alternative hypothesis: true difference in means is not equal to 0! 95 percent confidence interval:! 0.2722297 0.8077703! sample estimates:! mean of x mean of y ! 10.37 9.83!
Although we get the same conclusion, it is worth noting that the p-value is a little smaller and the confidence interval is a little shorter using the pooled t-test compared to the Welch.
Verifying Assump plot(fresh,type=‘b’)!
Shows Independence Shows Serial Correlation
Verifying Assump qqnorm(fresh); qqline(fresh)! > qqnorm(stored); qqline(stored)! !
Fresh Data Set Stored Data Set
Both qq-plots show no strong evidence against the normality assumption as the points are close to the reference lines.
Shapiro-‐Wilk Test • You can also perform a formal test to assess the normality of the
data using the Shapiro-Wilk Test. • Test H0:Data are normal vs. H1:Data are not normal
> shapiro.test(fresh)! !Shapiro-Wilk normality test!
data: fresh! W = 0.9528, p-value = 0.7013! > shapiro.test(stored)! !Shapiro-Wilk normality test!
data: stored! W = 0.9346, p-value = 0.4951! ! Word of caution: The Shapiro-Wilk Test was found to be sensitive to minor deviations from normality. So when the sample size is big, this test might give a significant result even when the data are reasonably normal. !
Since both p-values are larger than 0.05, it is reasonable to assume that both data come from normal populations.
Using R: Pooled T-‐test • Suppose we want to test H0:μ1-μ2= 0.5 vs. H1:μ1-μ2 > 0.5.
> t.test(fresh,stored,alternative="greater",mu=.5,var.equal=T)! !
!Two Sample t-test! ! data: fresh and stored! t = 0.3138, df = 18, p-value = 0.3786! alternative hypothesis: true difference in means is greater than 0.5! 95 percent confidence interval:! 0.3189872 Inf! sample estimates:! mean of x mean of y ! 10.37 9.83 !
Since the p-value is not less than α=0.05, we don’t reject the null hypothesis. Hence, we didn’t find sufficient evidence that reduction in mean potency is more than 0.5.
Wilcoxon Rank Sum Test • What do we do if at least one of the two samples is not normal?
– We could try to transform the data to make them normal. We will talk more about this strategy later on.
– We could use a nonparametric test called Wilcoxon Rank Sum Test (also know as the Mann-Whitney Test).
!• The assumptions for this test are: 1. The two random samples are
independent. 2. The samples come from identical
distributions with the exception that one distribution may be shifted to the right.
Using R: Wilcoxon Rank Sum Test • H0: Δ=0 (The 2 populations are identical) vs. H1:Δ≠0.
> ?wilcox.test! > wilcox.test(fresh,stored)!
!Wilcoxon rank sum test with continuity correction! data: fresh and stored! W = 91, p-value = 0.00211! alternative hypothesis: true location shift is not equal to 0! ! > median(fresh) # 10.4! > median(stored) # 9.8!
Since the p-value is less than α=0.05, we reject the null hypothesis. Hence, we found sufficient evidence that the median potency is statistically lower after 1 year.
Note: By default, R will compute an exact p-value if the samples contain less than 50 values and there are no ties. Otherwise, a normal approximation is used.
This indicates that a normal approximation was used to compute the p-value.
Wilcoxon Rank Sum Test: Example 2
Example 2: Assump qqnorm(placebo);qqline(placebo) > qqnorm(alcohol);qqline(alcohol) > shapiro.test(placebo) W = 0.8634, p-‐value = 0.08367
> shapiro.test(alcohol) W = 0.8149, p-‐value = 0.02201
Both qq-plots show deviations from the reference line and the Shapiro-Wilk test yielded a p-value for the alcohol data that is less than 0.05. Hence, the normality assumption is violated.
Wilcoxon Rank Sum Test: Example 2 • H0: Δ=0 (The distributions of reaction times for the placebo and alcohol
populations are identical). • H1: Δ placebo=c(.9,.37,1.63,.83,.95,.78,.86,.61,.38,1.97)! > alcohol=c(1.46,1.45,1.76,1.44,1.11,3.07,.98,1.27,2.56,1.32)! > wilcox.test(placebo,alcohol,alternative="less",conf.int=T)!
!Wilcoxon rank sum test! data: placebo and alcohol! W = 15, p-value = 0.003421! alternative hypothesis: true location shift is less than 0! 95 percent confidence interval:! -Inf -0.37! sample estimates:! difference in location ! -0.61 ! > median(placebo) # 0.845! > median(alcohol) # 1.645!
Since the p-value is less than α=0.05, we reject the null hypothesis. Hence, we found sufficient evidence that the median reaction time for the placebo population is statistically lower than that of the alcohol population.
Wilcoxon Rank Sum Test Sta
> data=read.csv("Example6_7.csv",header=T) > head(data) Car Garage1 Garage2 1 1 17.6 17.3 2 2 20.2 19.1 3 3 19.5 18.4 4 4 11.3 11.5 5 5 13.0 12.7 6 6 16.3 15.8 > attach(data)
Since the estimates made by Garages I and II are for the same set of cars, these values are not independent.
Paired Data > plot(Garage1,Garage2,pch=19) > cor.test(Garage1,Garage2)
Pearson's product-moment correlation data: Garage1 and Garage2 t = 37.5052, df = 13, p-value = 1.243e-14 alternative hypothesis: true correlation is not equal to 0 95 percent confidence interval: 0.9858393 0.9985176 sample estimates: cor 0.9954108
Both the linear pattern in the above scatterplot and the close- to-1 correlation coefficient indicate that the two set of values are not independent.
r= sample correlation coefficient The small p-value indicates that correlation ρ is not 0.
Paired T-‐test • H0: There is no difference in the estimates given by the two garages. • H1: Garage I estimates are higher than those of Garage II .
> t.test(Garage1,Garage2,alternative=”greater”,paired=T)! !Paired t-test!
data: Garage1 and Garage2! t = 6.0234, df = 14, p-value = 1.563e-05! alternative hypothesis: true difference in means is greater than 0! 95 percent confidence interval:! 0.4339886 Inf! sample estimates:! mean of the differences ! 0.6133333 !
Since the p-value is extremely small, we reject the null hypothesis. Hence, we found sufficient evidence that, on average, Garage I estimates are higher than Garage II.