Section 6.12
Test for a Difference
in Means
Statistics: Unlocking the Power of Data
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Outline
 Test for a difference in means
Statistics: Unlocking the Power of Data
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T-Test for a Difference in Means
H0:μ1=μ2
𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐 − 𝑛𝑢𝑙𝑙
𝑆𝐸
𝑡=
𝑥1 − 𝑥2
𝑠1 2 𝑠2 2
+
𝑛1
𝑛2
• If the population is approximately normal or if
sample sizes are large (n1 ≥ 30, n2 ≥ 30), the pvalue can be computed as the area in the tail(s)
beyond t of a t-distribution with degrees of
freedom equal to the smaller of n1 – 1 and n2 – 1
Statistics: Unlocking the Power of Data
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The Pygmalion Effect
• Pygmalion Effect: the greater the expectation
placed upon people, the better they perform
• Teachers were told that certain children
(chosen randomly) were expected to be
“growth spurters,” based on the Harvard Test
of Inflected Acquisition (a test that didn’t
actually exist).
• The response variable is change in IQ over the
course of one year.
Source: Rosenthal, R. and Jacobsen, L. (1968). “Pygmalion in the Classroom: Teacher
Expectation and Pupils’ Intellectual Development.” Holt, Rinehart and Winston, Inc.
Statistics: Unlocking the Power of Data
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The Pygmalion Effect
Control Students
“Growth Spurters”
n
255
65
𝒙
8.42
12.22
s
12.0
13.3
Does this provide evidence that the Pygmalion
Effect exists? (that merely expecting a child to
do better actually causes the child to do better?)
(a) Yes
(b) No
*s1 and s2 were not given, so I set them to give the correct p-value
Statistics: Unlocking the Power of Data
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Pygmalion Effect
1. State hypotheses: H0: 1 = 2 1 = average change in IQ for “growth spurters”
Ha: 1 > 2
1 = average change in IQ for control
2. Check conditions: n1 = 65 ≥ 30, n2 = 255 ≥ 30
3. Calculate statistic:
𝑥1 − 𝑥2 =12.22 – 8.42 = 3.8
4. Calculate standard error: SE =
𝑠1 2
𝑛1
+
𝑠2 2
𝑛2
=
13.32
65
12.02
+
255
=
1.813
𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐 − 𝑛𝑢𝑙𝑙
3.8
=
= 2.096
𝑆𝐸
1.813
5. Calculate test statistic: t =
6. Compute p-value: t with 65 – 1 = 64 df, upper tail
p-value = 0.02
7. Interpret in context:
We have evidence that positive teacher expectations significantly
increase IQ scores, on average, in elementary school children.
Statistics: Unlocking the Power of Data
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Pygmalion Effect
From the paper:
Statistics: Unlocking the Power of Data
“The difference in gains could
be ascribed to chance about 2
in 100 times”
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