Mathematics 243
t-methods
Day 28 - March 26
1. The relationship between a hypothesis test with significance level α and a 100(1 − α)% confidence interval.
2. What does a confidence interval mean?
3. Inference for µ – review of the t-test
(a) Preliminary analysis: (data in vector x)
> boxplot(x)
> stripchart(x,pch=1)
> summary(x)
(b) Assumptions: simple random sample (independent trials of a process), population close to normal or sample
size large
x̄ − µ0
√
(c) Test statistic:
s/ n
s
(d) Confidence interval: x̄ ± t∗ √
n
(e) t-test and additional functions from package mosaic
> x=sample(sr$GPA,30)
> t.test(x,mu=3.2)
One Sample t-test
data: x
t = -1.5053, df = 29, p-value = 0.1431
alternative hypothesis: true mean is not equal to 3.2
95 percent confidence interval:
2.862712 3.251288
sample estimates:
mean of x
3.057
> interval(t.test(x))
mean of x
lower
upper
3.057000 2.862712 3.251288
> pval(t.test(x,mu=3.2))
p.value
0.143054
> stat(t.test(x,mu=3.2))
t
-1.505332
4. Simulating confidence intervals using mosaic
> mu=mean(sr$GPA)
> ints = do(1000) * interval(t.test(sample(sr$GPA,4)))
> str(ints)
'data.frame':
1000 obs. of 3 variables:
$ mean of x: num 3.48 2.76 3.32 2.97 3.31 ...
$ lower
: num 2.36 2.14 2.5 2.16 2.73 ...
$ upper
: num 4.59 3.37 4.14 3.77 3.89 ...
> misshigh=sum(ints$lower>mu)
> misslow=sum(ints$upper<mu)
> misslow;misshigh; (misslow+misshigh)/1000
[1] 10
[1] 47
[1] 0.057
5. Inference for µ1 − µ2
(a) Preliminary analysis: (data in dataframe d, quantitative variable x and factor variable f)
> boxplot(x~f,data=d)
> stripchart(x~f,data=d,pch=1)
> aggregate(x~f,data=d,FUN=summary)
(b) Assumptions: two simple random samples, independent of each other, population close to normal or sample
size large or randomized assignment to two treatments
(x̄1 − x̄2 ) − (µ1 − µ2 )
p
(c) Test statistic:
s21 /n1 + s22 /n2
(d) Confidence interval: (x̄1 − x̄2 ) ± t∗ SE
(e) t-test
> t.test(length~sex,data=KidsFeet)
Welch Two Sample t-test
data: length by sex
t = 1.9174, df = 36.275, p-value = 0.06308
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
-0.04502067 1.61291541
sample estimates:
mean in group B mean in group G
25.10500
24.32105
> interval(t.test(length~sex,data=KidsFeet))
mean in group B mean in group G
lower
upper
25.10500000
24.32105263
-0.04502067
1.61291541
> pval(t.test(length~sex,data=KidsFeet))
p.value
0.06308223
> stat(t.test(length~sex,data=KidsFeet))
t
1.917445
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