tTest for OneSample

1
Chapter 18

• t-Test for One-Sample

2
Problem with z-test

• When can the z-test be used?
• Single Sample Comparisons
• Hypothesis test about a single population.
• Comparing a single sample to constant value.
• s (or s?) is known
• Requirement in order to use the z-test
• n is large enough to satisfy the central limit
  theorem

3
Problem with the z-test
s, s?, and Significant Differences
Large Standard Error
Small Standard Error
4
Problem with the z-test
s, s?, and Significant Differences
5
Problem with the z-test

• Hypothesis test dependent on
• Must know s, to know s?, so we can assign
  significance.
• or
• have values to approximate s s?
• Estimated population standard deviation s
• Estimated standard error s?

6
Estimating the Population Standard Deviation

• will allow us to complete the hypothesis test!

7
Coffee and Drunkenness (no s)
There has been the suggestion that drinking
coffee makes a person less drunk. There is also a
good reason to suspect that it might make a
person less drunk since caffeine is a stimulant
(it wont make them more drunk). This issue is
important because the legislature is debating a
policy that would allow people to drive home from
bars if they have a few cups of coffee. To test
whether people who drink coffee have the same
drunkenness score or a lower drunkenness score, a
researcher decided to give caffeine pills to a
group of 9 randomly selected people who had had 3
drinks. They gave the following ratings What
will this researcher conclude about the
drunkenness of people who drink coffee (level of
significance is .05).
8
Coffee and Drunkenness (no s)
There has been the suggestion that drinking
coffee makes a person less drunk. There is also a
good reason to suspect that it might make a
person less drunk since caffeine is a stimulant
(it wont make them more drunk). This issue is
important because the legislature is debating a
policy that would allow people to drive home from
bars if they have a few cups of coffee. To test
whether people who drink coffee have the same
drunkenness score or a lower drunkenness score, a
researcher decided to give caffeine pills to a
group of 9 randomly selected people who had had 3
drinks. They gave the following ratings What
will this researcher conclude about the
drunkenness of people who drink coffee (level of
significance is .05).
m3drinks30 s3drinks??? mhyp-caf30 scaf??? n
9 a .05
9
Coffee and Drunkenness (no s)
There has been the suggestion that drinking
coffee makes a person less drunk. There is also a
good reason to suspect that it might make a
person less drunk since caffeine is a stimulant
(it wont make them more drunk). This issue is
important because the legislature is debating a
policy that would allow people to drive home from
bars if they have a few cups of coffee. To test
whether people who drink coffee have the same
drunkenness score or a lower drunkenness score, a
researcher decided to give caffeine pills to a
group of 9 randomly selected people who had had 3
drinks. They gave the following ratings What
will this researcher conclude about the
drunkenness of people who drink coffee (level of
significance is .05).
m3drinks30 s3drinks??? mhyp-caf30 scaf??? n
9 a .05
10
Using Samples to Estimate the Population

• ? as m
• S(descriptive) as s

11
Sample Sampling Distributions
5
2
3
4
Shape Rectangular m 3.5 s 1.12
12
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
4
5
? 4.5
Sample
Sample
Sample
Sample
13
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
2
5
? 3.5
Sample
Sample
Sample
Sample
14
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
5
4
? 3.5
Sample
Sample
Sample
Sample
15
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
2
3
? 2.5
Sample
Sample
Sample
Sample
16
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
2
4
? 3
Sample
Sample
Sample
Sample
17
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
2
2
? 2
Sample
Sample
Sample
Sample
18
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
3
2
? 2.5
Sample
Sample
Sample
Sample
19
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
3
3
? 3
Sample
Sample
Sample
Sample
20
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
3
4
? 3.5
Sample
Sample
Sample
Sample
21
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
3
5
? 4
Sample
Sample
Sample
Sample
22
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
4
2
? 2
Sample
Sample
Sample
Sample
23
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
4
3
? 3.5
Sample
Sample
Sample
Sample
24
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
4
4
? 4
Sample
Sample
Sample
Sample
25
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
5
2
? 3.5
Sample
Sample
Sample
Sample
26
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
5
3
? 4
Sample
Sample
Sample
Sample
27
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
5
5
? 5
Sample
Sample
Sample
Sample
28
Sample Sampling Distribution
3
5
2
4
The Sampling Distribution of the (Sample)
Mean(s) n2
Sample
Sample
Sample
Sample
29
Sample Sampling Distribution
Shape Normal m? 3.5 s? .79
30
Sample Sampling Distributions
Sampling Distribution of the Sample
Means Parameters
Population Parameters
Shape Rectangular m 3.5 s 1.12
Shape Normal m? 3.5 s? .79
? is an unbiased estimate of m
31
Sample Sampling Distributions
5
2
3
4
Shape Rectangular m 3.5 s 1.12
32
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
4
5
S .5
Sample
Sample
Sample
Sample
33
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
2
5
S 1.18
Sample
Sample
Sample
Sample
34
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
5
4
S .5
Sample
Sample
Sample
Sample
35
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
2
3
S .5
Sample
Sample
Sample
Sample
Sample
Sample
Sample
Sample
36
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
2
4
S 1
Sample
Sample
Sample
Sample
Sample
Sample
Sample
Sample
37
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
2
2
S 0
Sample
Sample
Sample
Sample
Sample
Sample
Sample
Sample
38
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
3
2
S .5
Sample
Sample
Sample
Sample
Sample
Sample
Sample
Sample
39
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
3
3
S 0
Sample
Sample
Sample
Sample
Sample
Sample
Sample
Sample
40
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
3
4
S .5
Sample
Sample
Sample
Sample
Sample
Sample
Sample
Sample
41
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
3
5
S 1
Sample
Sample
Sample
Sample
Sample
Sample
Sample
Sample
42
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
4
2
S 1
Sample
Sample
Sample
Sample
Sample
Sample
Sample
Sample
43
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
4
3
S .5
Sample
Sample
Sample
Sample
Sample
Sample
Sample
Sample
44
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
4
4
S 0
Sample
Sample
Sample
Sample
Sample
Sample
Sample
Sample
45
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
5
2
S 1.18
Sample
Sample
Sample
Sample
Sample
Sample
Sample
Sample
46
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
5
3
S 1
Sample
Sample
Sample
Sample
Sample
Sample
Sample
Sample
47
Sample Sampling Distribution
3
5
Lets Go Sampling n2
2
4
5
5
S 0
Sample
Sample
Sample
Sample
Sample
Sample
Sample
Sample
48
Sample Sampling Distribution
3
5
2
4
The Sampling Distribution of the (Sample)
Standard Deviations (Descriptive) n2
Sample
Sample
Sample
Sample
49
Sample Sampling Distribution
Shape Positively Skewed mS .66
50
Sample Sampling Distributions
Sampling Distribution of the Sample
Standard Deviations (Descriptive) Parameters
Population Parameters
s 1.12
mS .66
S is an biased estimate of s (consistently
underestimates)
51
Why Biased? Degrees of Freedom

• S divides by too many.
• Not all sample values actually vary.

52
Degrees of Freedom

• Attempt to estimate s
• how much the scores freely vary

53
Degrees of Freedom

• Attempt to estimate s
• how much the scores freely vary
• by observing how much the scores in the sample
  freely vary.

Do all of the scores in the sample freely vary?
54
Degrees of Freedom

• Attempt to estimate s
• how much the scores freely vary
• by observing how much the scores in the sample
  freely vary.

Do all of the scores in the sample freely vary?
55
Degrees of Freedom

• Attempt to estimate s
• how much the scores freely vary
• by observing how much the scores in the sample
  freely vary.

Do all of the scores in the sample freely vary?
56
Degrees of Freedom

• Attempt to estimate s
• how much the scores freely vary
• by observing how much the scores in the sample
  freely vary.

Do all of the scores in the sample freely vary?
57
Degrees of Freedom

• Attempt to estimate s
• how much the scores freely vary
• by observing how much the scores in the sample
  freely vary.

Do all of the scores in the sample freely vary?
58
Degrees of Freedom

• Attempt to estimate s
• how much the scores freely vary
• by observing how much the scores in the sample
  freely vary.

Do all of the scores in the sample freely vary?
59
Degrees of Freedom

• Attempt to estimate s
• how much the scores freely vary
• by observing how much the scores in the sample
  freely vary.

Do all of the scores in the sample freely vary?
60
Degrees of Freedom

• Attempt to estimate s
• how much the scores freely vary
• by observing how much the scores in the sample
  freely vary.

Do all of the scores in the sample freely vary?
61
Degrees of Freedom

• Attempt to estimate s
• how much the scores freely vary
• by observing how much the scores in the sample
  freely vary.

Do all of the scores in the sample freely vary?
Must be
62
Degrees of Freedom

• Attempt to estimate s
• how much the scores freely vary
• by observing how much the scores in the sample
  freely vary.

Do all of the scores in the sample freely vary?
Must be
Must be
63
Degrees of Freedom

• Attempt to estimate s
• how much the scores freely vary
• by observing how much the scores in the sample
  freely vary.

3 scores freely vary?
Degrees of Freedom df (n - 1)
Must be
1 did not (had to make up slack)
64
Sample Sampling Distribution
3
5
2
4
The Sampling Distribution of the (Sample)
Standard Deviations (Inferential) n2
Sample
Sample
Sample
Sample
65
Sample Sampling Distribution
Shape Positively Skewed mS .66
66
Sample Sampling Distributions
Sampling Distribution of the Sample
Standard Deviations (Descriptive) Parameters
Population Parameters
s 1.12
mS .66
S is an biased estimate of s (consistently
underestimates)
67
Better Estimate of s

• Sample standard deviation (inferential) s
  (little)

68
Better Estimate of s

• Attempt to estimate s
• how much the scores freely vary

Divide by true degrees of freedom.
69
Better Estimate of s
Sample Standard Deviation (Inferential)
Computational Formula
Definition Formula
70
The new hypothesis test

• t-test for One-Sample
• (What Changes?)

71
t-test for One Sample

• Steps of a one sample t-test
• Step 0) Rewrite every number from the story
  problem next to its symbol.
• Step 1) Rewrite the research problem
• Same as z
• Same basic question
• Step 2) Write Statistical Hypotheses
• Same as z
• Same basic question

72
t-test for One Sample

• Step 3) Write Decision Rule
• Shading
• same as z
• Same logic
• Determine critical score
• Use a t-table to determine tcrit
• Takes into account that the estimated standard
  deviation changes from sample to sample
• Write down conditions under which you will reject
  H0 or fail to reject H0
• Same as z

73
t-test for One Sample

• Step 4) Calculate test statistic
• Obtained Score
• tobt
• Based on samplig distribution with mhyp s?
• Step 5) Make Decision
• Compare tobt to tcrit
• Step 6) Interpret Decision
• Write in terms of the problem
• How you would tell your boss.

74
Hypothesis Test 7

• Coffee and Drunkenness (no s)

75
Coffee and Drunkenness (no s)
There has been the suggestion that drinking
coffee makes a person less drunk. There is also a
good reason to suspect that it might make a
person less drunk since caffeine is a stimulant
(it wont make them more drunk). This issue is
important because the legislature is debating a
policy that would allow people to drive home from
bars if they have a few cups of coffee. To test
whether people who drink coffee have the same
drunkenness score or a lower drunkenness score, a
researcher decided to give caffeine pills to a
group of 9 randomly selected people who had had 3
drinks. They gave the following ratings What
will this researcher conclude about the
drunkenness of people who drink coffee (level of
significance is .05).
76
Coffee and Drunkenness (no s)
There has been the suggestion that drinking
coffee makes a person less drunk. There is also a
good reason to suspect that it might make a
person less drunk since caffeine is a stimulant
(it wont make them more drunk). This issue is
important because the legislature is debating a
policy that would allow people to drive home from
bars if they have a few cups of coffee. To test
whether people who drink coffee have the same
drunkenness score or a lower drunkenness score, a
researcher decided to give caffeine pills to a
group of 9 randomly selected people who had had 3
drinks. They gave the following ratings What
will this researcher conclude about the
drunkenness of people who drink coffee (level of
significance is .05).
m3drinks30 s3drinks??? mhyp-caf30 scaf??? n
9 a .05
77
Coffee and Drunkenness (no s)

• Step 1) Rewrite Research Question
• Is the mean of the caffeine population 30?

mhyp-caf30 scaf??? n 9 ? ??? a .05
? 256/9 28.44
(256)2
7392 -
9
9 -1
3.71
78
Coffee and Drunkenness (no s)
Step 2) Write the statistical hypotheses
H0 m 30
H1 m lt 30
lt mhyp mhyp gt mhyp
79
Coffee and Drunkenness (no s)

• Step 3) Form Decision Rule
• Draw Normal Curve
• Shade in a
• Mark Rejection Region(s)
• Determine Critical Scores
• Write conditions for rejection H0

• Hypothesis
• H0 mconv ? 30
• H1 mconvlt 30

t(8)crit -1.86
80
Coffee and Drunkenness (no s)
m? m
mhyp
30

• Hypothesis
• H0 mconv ? 30
• H1 mconvlt 30

• Decision Rule
• Reject H0
• tobt lt -1.86

1.24
81
Coffee and Drunkenness (no s)
Step 4) Calculate Test Statistic
zobt (? - mhyp) / s?
tobt (? - mhyp) / s?
tobt (28.44 - 30) / 1.24
tobt -1.56 / 1.24

• Hypothesis
• H0 mconv ? 30
• H1 mconvlt 30

tobt -1.26

• Decision Rule
• Reject H0
• tobt lt -1.86

Based upon a sampling distribution with m? 30
s? 1.24
.3
82
Coffee and Drunkenness (no s)
Step 5) Make Decision
Step 6) Interpret Decision

• We have no reason to believe that caffeine makes
  a person less drunk.

• Hypothesis
• H0 mconv ? 30
• H1 mconvlt 30

• Decision Rule
• Reject H0
• tobt lt -1.86

Test Statistic tobt -1.26
Based upon a sampling distribution with m? 30
s? 1.24
.3