1. Stat 251 (2009, Summer)
Final Lab
TA: Yu, Chi Wai

2. Hypothesis Testing and Confidence intervals
One sample test
Two sample test
(i) Independent samples
1) Equal variance
2) Unequal variance
(ii) Dependent samples (paired samples)

3. One sample t test
Two-sided
One-sided OR

4. Test statistics

5. Based on the test statistic, we have two ways to test the hypothesis.
Compare the test statistic with a critical value;
Get a p-value, and then compare it with a significance level α, say α = 0.05 or 0.1.

6. If p-value is LESS than the significance level, then we conclude that
we have enough evidence
to reject Ho
at the significance level α

7. If p-value is GREATER than the significance level, then we conclude that
we DO NOT have enough evidence
to reject Ho
at the significance level α
We DO NOT say that we ACCEPT H0, because

8. we DO NOT have enough evidence
to reject Ho
at the significance level α
we have enough evidence
to accept Ho
at the significance level α

9. we DO NOT have enough evidence
to reject Ho
at the significance level α
we have enough evidence
to accept Ho
at the significance level α

10. Example Download the “biotest.txt” data file
Read into R using function read.table()
Extract the 1st column and store as ‘x1’
Store the 2nd column as ‘x2’

11. Example x1 = read.table(“biotest.txt”) [ ,1]

12. Example 1 Test Find the p-value of this test.
115
112
107
119
138
126
105
104
Take ‘x1’ as the sample in this case
Test
with the significance level α=0.05
Find the p-value of this test.

13. t.test( , alternative=“ ”, mu= )
[R] command: t.test
t.test( , alternative=“ ”, mu= )
two.sided,
less, or
greater
true value of μ, i.e. μ0
Data set

14. t.test( , alternative=“ ”,mu= )
with the significance level α=0.05
Two sided, baby!
Alternative hypothesis
t.test( , alternative=“ ”,mu= ) x1
115
two.sided
Data set
two.sided,
less, or
greater
true value of μ

15. One Sample t-test
data: x1
t = , df = 9, p-value = 0.858
alternative hypothesis: true mean is not equal to 115
95 percent confidence interval:
sample estimates:
mean of x
115.6

16. One Sample t-test
data: x1
t = , df = 9, p-value = 0.858
alternative hypothesis: true mean is not equal to 115
95 percent confidence interval:
sample estimates:
mean of x
115.6

17. α=0.05 One Sample t-test data: x1 t = 0.1841, df = 9, p-value = 0.858
alternative hypothesis: true mean is not equal to 115
95 percent confidence interval:
sample estimates:
mean of x
115.6
α=0.05
we DO NOT have enough evidence
to reject Ho
at the significance level α=0.05

18. Conclusion with the significance level α=0.05 p-value = 0.858
We DO NOT have enough evidence to say that the mean of x1 is significantly different from 115 at the level of significance α=0.05.

19. 95 % confidence interval:
One Sample t-test
data: x1
t = , df = 9, p-value = 0.858
alternative hypothesis: true mean is not equal to 115
95 % confidence interval:
sample estimates:
mean of x
115.6
μ0=115 is inside this 95% confidence interval for μ

20. Example 2
Test
with the significance level α=0.05

21. t.test( , alternative=“ ”, mu= )
One sided (greater)
with the significance level α=0.05
t.test( , alternative=“ ”, mu= ) x1
greater
108

22. α=0.05 One Sample t-test data: x1
t = , df = 9, p-value =
alternative hypothesis: true mean is greater than 108
95 percent confidence interval:
Inf
sample estimates:
mean of x
115.6
α=0.05
we have enough evidence
to reject Ho
at the significance level α=0.05
(i.e. accept H1)

23. Conclusion with the significance level α=0.05 p-value = 0.02232
We have enough evidence to say that the mean of x1 is significantly greater than 108 at the level of significance α=0.05.

24. By default, the function t.test() includes a 95% confidence interval
How to get the confidence interval with other %, i.e. how to change the confidence level, say 99%?

25. conf.level = 0.99) t.test(x1, alternative=“greater”, mu=108 )
One sided (greater)
with the significance level α=0.05
t.test(x1, alternative=“greater”, mu=108 )
Find the 99 % confidence interval for μ.
t.test(x1, alternative=“greater”, mu=108,
conf.level = 0.99)

26. Two samples a) Independent samples i) Equal variance
ii) Different variances
b) Dependent samples (paired t test)

27. One sample t test
Two-sided
One-sided

28. Two sample t test
Interest: Mean difference
Two-sided
One-sided

29. 1 2 3 x1 = read.table(“biotest.txt”) [,1]
a) Independent samples
x1 = read.table(“biotest.txt”) [,1]
x2 = read.table(“biotest.txt”) [,2]
Want to test if there is a significant difference between the mean of x1 and that of x2. 1 2 3

30. Two samples a) Independent samples i) Equal variance
ii) Different variances
b) Dependent samples (paired t test)

31. t.test(x1, x2, alternative=“two.sided”, mu=0, var.equal = T)
( Two-sided, two samples )
i) Equal variance
t.test(x1, x2, alternative=“two.sided”, mu=0, var.equal = T)

32. t.test(x1, x2, alternative=“two.sided”, mu=0, var.equal = T)
( Two-sided, two samples )
i) Equal variance
t.test(x1, x2, alternative=“two.sided”, mu=0, var.equal = T)
Input the data sets of two samples, x1 and x2

33. t.test(x1, x2, alternative=“two.sided”, mu=0, var.equal = T)
( Two-sided, two samples )
i) Equal variance
t.test(x1, x2, alternative=“two.sided”, mu=0, var.equal = T)
Two sided test, and under H0, assume that the true mean difference is 0

34. t.test(x1, x2, alternative=“two.sided”, mu=0, var.equal = T)
( Two-sided, two samples )
i) Equal variance
t.test(x1, x2, alternative=“two.sided”, mu=0, var.equal = T)
Assume that the two samples have equal variance.

35. Two Sample t-test (equal variance, a pooled variance estimate )
data: x1 and x2
t = , df = 18, p-value =
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
sample estimates:
mean of x mean of y

36. Two Sample t-test
data: x1 and x2
t = , df = 18, p-value =
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
sample estimates:
mean of x mean of y

37. Two Sample t-test
data: x1 and x2
t = , df = 18, p-value =
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
sample estimates:
mean of x mean of y

38. Two Sample t-test
data: x1 and x2
t = , df = 18, p-value =
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
sample estimates:
mean of x mean of y

39. t.test(x1, x2, alternative=“two.sided”, mu=0, var.equal = F)
ii) Different variances
t.test(x1, x2, alternative=“two.sided”, mu=0, var.equal = F)
Welch Two Sample t-test
data: x1 and x2
t = , df = , p-value = 0.378
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
sample estimates:
mean of x mean of y

40. Confidence interval for the mean difference
i) Equal variance
t.test(x1, x2, alternative=“two.sided”, mu=0, var.equal = T, conf.level = 0.90)
ii) Different variances
t.test(x1, x2, alternative=“two.sided”, mu=0, var.equal = F, conf.level = 0.90)

41. t.test(x1, x2, alternative=“two.sided”, mu=0, paired = T)
b) Dependent samples (paired data)
This test is used when the samples are dependent; i.e., when there is only one sample that has been tested twice (repeated measures) or when there are two samples that have been paired.
the data is in form of (x1, x2)
t.test(x1, x2, alternative=“two.sided”, mu=0, paired = T)

42. Paired t-test
data: x1 and x2
t = , df = 9, p-value =
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
sample estimates:
mean of the differences
-4.8

43. Since the paired samples have the same sample size, we can simply use
to be the test statistic for
 can be regarded as the one-sample case
i.e. we consider the one sample di = xi-yi , where i=1,…,n

44. t.test(x1-x2, alternative=“two.sided”, mu=0)
b) Dependent samples (paired data)
t.test(x1, x2, alternative=“two.sided”, mu=0, paired = T)
Alternatively,
t.test(x1-x2, alternative=“two.sided”, mu=0)

45. One Sample t-test
data: x1 - x2
t = , df = 9, p-value =
alternative hypothesis: true mean is not equal to 0
95 percent confidence interval:
sample estimates:
mean of x
-4.8

46. Paired t-test
data: x1 and x2
t = , df = 9, p-value =
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
sample estimates:
mean of the differences
-4.8

47. t.test(x1, x2, alternative=“two.sided”, mu=0, paired = T)
b) Dependent samples (paired data)
t.test(x1, x2, alternative=“two.sided”, mu=0, paired = T)
No need to consider whether the variances of two samples are equal or not!!
Alternatively,
t.test(x1-x2, alternative=“two.sided”, mu=0)

48. Not reject Ho Not reject Ho Reject Ho Remark:
Use the same data sets of x1 and x2
Two Sample t-test (equal variance)
data: x1 and x2
t = , df = 18, p-value =
Not reject Ho
Welch Two Sample t-test (different variance)
data: x1 and x2
t = , df = , p-value = 0.378
Not reject Ho
Paired t-test
data: x1 and x2
t = , df = 9, p-value =
Reject Ho

49. Final Remarks
Notice that the conclusion from the two sample t-test and the paired t-test are different even if we are looking at the same data set.
Should check if the two sample are independent or not.