Paired t- test - Example
• OISE sponsored summer institute to improve the skills of high
school teachers of foreign languages. One such institute hosted 20
French teachers for four weeks. At the beginning of the period the
teachers were given the Modern Language Association’s listening
test of understanding spoken French. After 4 weeks of immersion in
French, the listening test was given again. The data is given in the
following slide. Dose the data provide evidence that the course
improves French-spoken language skills?
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Data Display
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Teacher
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Pretest
30
28
31
26
20
30
34
15
28
20
30
29
31
29
34
20
26
25
31
29
Posttest
29
30
32
30
16
25
31
18
33
25
32
28
34
32
32
27
28
29
32
32
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improvement
-1
2
1
4
-4
-5
-3
3
5
5
2
-1
3
3
-2
7
2
4
1
3
2
• One sample t-test for the improvement
T-Test of the Mean
Test of mu = 0.000 vs mu > 0.000
Variable
N
Mean
StDev
SE Mean
improvem 20
1.450
3.203
0.716
T
2.02
P
0.029
• MINITAB commands for the paired t-test
Stat > Basic Statistics > Paired t
Paired T-Test and Confidence Interval
Paired T for Posttest – Pretest
N
Mean
StDev
SE Mean
Posttest
20
28.75
4.74
1.06
Pretest
20
27.30
5.04
1.13
Difference
20
1.450
3.203
0.716
95% CI for mean difference: (-0.049, 2.949)
T-Test of mean difference=0 (vs > 0):
T-Value = 2.02 P-Value = 0.029
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6
Frequency
5
4
3
2
1
0
-4
-2
0
2
4
6
8
improvement
Character Stem-and-Leaf Display
Stem-and-leaf of improvement
Leaf Unit = 1.0
2
-0 54
4
-0 32
6
-0 11
8
0 11
(7)
0 2223333
5
0 4455
1
0 7
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N
= 20
4
Goodness of Fit Tests
• The goal of χ2 goodness of fit tests is to test is the data comes from a
certain distribution.
• There are various situations to which these tests apply.
• The first situation we will explore is when we observe count data in
k different categories.
• The aim is to test the null hypothesis that the probabilities of the k
categories are p1, p2,…,pk.
• We distinguish between two cases.
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Case 1
• The null hypothesis completely specifies the probabilities of each of
the k categories.
• For each category we calculate the expected count Ei = npi.
• The test statistic and its distribution are…
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Example
• The statistic department at U of T offers introductory courses for
students from other disciplines. The department believes that 40% of
the students are math major, 30% are computer science, 20%
biology and 10% chemistry. A random sample of 120 students
revealed 52, 38, 21, and 9 from the four majors above. Does this
data support the department claim?
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Case 2
• The null hypothesis does not fully specify the probabilities.
• In this case the probabilities of the different categories may be
functions of other parameters.
• First use the sample data to estimate r unknown parameters.
• Then use the estimated parameters to estimate the k probabilities.
• For each category, calculate the estimated expected count.
• The test statistic is…
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Example
• A farmer believes that the number of eggs a chicken will give per
day has a Poisson(λ) distribution. He observed the following data….
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Remark
• In many cases we will observe data that are not categorized and we
would want to test is the data comes from a certain distribution.
• If the distribution we are testing is discrete the values of the variable
will be the actual categories.
• However, if the variable takes infinite possible values, the grouping
should be done so that the expected frequency in each category is at
least 5.
• If the distribution we are testing is continuous we need to group the
measurement of the random variable of interest into k intervals.
Very often the choice of cells is done arbitrarily.
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Contingency Tables
• The goal is to test if two categorical variables are independent.
• The row variable has r categories while the column variable has c
categories.
• The data is the count of observations in the rxc table…
• The null hypothesis states that the row variable and the column
variable are independent. The alternative states that the variables are
dependent.
• To conduct the test, we calculate the expected count for each cell…
• The test statistic and its distribution is….
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Example
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