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Inferences About Two Means:
Dependent Samples
Inferences About Two Means:
Dependent Samples
Definition
• Two samples are independent if the sample values
from one population are not related to or
somehow naturally paired with the sample values
from the other population.
• Two samples are dependent if the sample values
are paired.
Inferences About Two Means:
Dependent Samples
Notation:
𝑑 = individual difference between the two values in
a single matched pair.
𝑢𝑑 = mean value of the difference d for the
population of all pairs of data.
𝑑 =mean value of the differences d for the paired
sample data.
𝑠𝑑 = Standard deviation of the difference d for the
paired sample data.
𝑛 = number of pairs of sample data.
Inferences About Two Means:
Dependent Samples
Requirements:
1. The two samples are Dependent.
2. Both samples are simple random samples.
3. Both populations have sample size 𝑛 > 30 or
both samples are normal.
Inferences About Two Means:
Dependent Samples
Hypothesis Test Statistic
𝑑 − 𝜇𝑑
𝑡= 𝑠
𝑑
𝑛
• with df = 𝑛 − 1
Inferences About Two Means:
Dependent Samples
Hypothesis Test
1. Identify claim and state Hypotheses. (𝐻0 : 𝜇𝑑 = 0,
𝐻𝑎 is that 𝜇𝑑 is different in some way).
2. Find the Test Statistic (here we find a 𝑡 stat)
In calc 𝐿1 − 𝐿2 → 𝐿3 then Ttest on 𝐿3
3. Draw a picture and find the p-value.
4. Make a decision (compare p-value and 𝛼)
5. Summarize
Inferences About Two Means:
Dependent Samples
The data below is a small sample of measure weights
of college students in September and April of their
freshman year. Using a 0.05 significance level test
the claim that for the population of students, the
mean change in weight from September to April is
equal to 0 kg.
April Weight
66
52
68
69
71
September Weight
67
53
64
71
70
Difference
-1
-1
4
-2
1
Inferences About Two Means:
Dependent Samples
Researchers collected data on the numbers of hospital
admissions resulting from motor vehicle crashes, and results
are given below for Fridays on the 6th of the month and
Fridays on the following 13th of the month. Use a 0.05
significance level to test the claim that when the 13th day of a
month falls on a Friday, the numbers of hospital admissions
from motor vehicle crashes are not affected.
Friday the 6th
9
6
11
11
3
5
Friday the 13th
13
12
14
10
4
12
Inferences About Two Means:
Dependent Samples
Homework:
9-4: 9-19 odd