Page 73 - Applied Statistics with R

5.2. HYPOTHESIS TESTS IN R 73

c(mean(capt_crisp$weight) - qt(0.975, df = 8) * sd(capt_crisp$weight) / sqrt(9),
mean(capt_crisp$weight) + qt(0.975, df = 8) * sd(capt_crisp$weight) / sqrt(9))

## [1] 15.70783 16.09217

5.2.3 Two Sample t-Test: Review

2
2
Suppose ∼ N( , ) and ∼ N( , ).

Want to test ∶ − = versus ∶ − ≠ .

0
0

0

1
Assuming is unknown, use the two-sample Student’s test statistic:

( ̄ − ̄) − 0
= ∼ + −2 ,
1
1
√ +

2
∑ ∑ ( − 1) + ( − 1) 2
2
where ̄ = =1 , ̄ = =1 , and = .

+ − 2
A 100(1 − )% CI for − is given by

1
1
( ̄ − ̄) ± + −2 ( /2) ( √ + ) ,

where + −2 ( /2) is the critical value such that ( > + −2 ( /2)) = /2.
5.2.4 Two Sample t-Test: Example
2
2
Assume that the distributions of and are N( , ) and N( , ), respec-
2
1
tively. Given the = 6 observations of ,
x = c(70, 82, 78, 74, 94, 82)
n = length(x)
and the = 8 observations of ,
y = c(64, 72, 60, 76, 72, 80, 84, 68)
m = length(y)
we will test ∶ = versus ∶ > .
0
1
1
1
2
2
First, note that we can calculate the sample means and standard deviations.