### Proportions and Inference - 1 and 2 prop z-tests

```AP Statistics
A coin that is balanced should come
up heads half the time in the long
run. The French naturalist Count
Buffon (1707-1788) tossed a coin
4040 times. He got 2048 heads. Is
this evidence that Buffon’s coin was
not balanced?
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Parameters (listed as variables AND in context)
Hypotheses
Assumptions and Conditions
Name of the test
 Formula without the numbers plugged in
 Formula with the numbers plugged in
 Value (check this with your calculator)
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Obtained P-value
Make decision
Statement of conclusion in context
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The data are an SRS
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This takes care of randomization
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The population is at least 10 times the sample
size
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The sample size is large enough that np0 ≥ 10
and n(1-p0) ≥ 10
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Parameters (listed as variables AND in
context)
Hypotheses
Assumptions and Conditions
Name of the test
Test statistic – formula and value
Obtain P-value
Make decision
Statement of conclusion in context
Some boxes of a certain brand of breakfast cereal include a voucher
for a free video rental inside the box. The company that makes the
ceral claims that voucher can be found in 20 percent of the boxes.
However, based on their experiences eating this ceral at home, a
group of students believes that the proportion of boxes with
vouchers is less than 0.2. This group of student purchased 65
boxes of the cereal to investigate the company’s claim. The
students found a total of 11 vouchers for free video rentals in the
65 boxes.
Suppose it is reasonable to assume that the 65 boxes purchased by
the students are a random sample of all boxes of this cereal.
Based on this sample, is there support for the students’ belief that
the proportion of boxes with vouchers is less than 0.2? Provide
2
4
3
To study the long-term effects of preschool programs for poor
children, the High/Scope Educational Research Foundation has
followed two groups of Michigan children since early childhood.
One group of 62 attended preschool as 3 and 4 year olds. This is
a sample from population 2, poor children who attended
preschool. A control group of 61 children from the same area
and similar backgrounds represents population 1, poor children
with no preschool. Thus the sample sizes are n1 = 61 and n2 = 62.
One response variable of interest is the need for social services as
adults. In the past 10 years, 38 of the preschool sample and 49 of
the control sample have needed social services (mainly welfare).
Is there significant evidence that preschool reduces the latter need
for social services?
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We don’t know either population proportion,
only the two sample proportions.
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We have two n’s and two p-hats
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We want to turn this into a single hypothesis
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Look at some combination of the two
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What are we trying to test?
What are our parameters / statistics?
Hypotheses
 H0: p1 = p2
▪ There’s no difference between the two proportions
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Ha: p1 < p2
• The first proportion is less than the second
We’re trying to find if there is a difference.
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If there is no difference then p1 – p2 =0
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If p1 is greater than p2 then p1 – p2 >0
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We’ll standardize this difference.
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This is standard error (SE)
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The data from each sample are an SRS
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This takes care of randomization
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Independent samples
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The population is at least 10 times the sample
size for both samples and populations
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The sample size is large enough that np1 ≥ 10
and n(1-p1) ≥ 10 AND np2 ≥ 10 and n(1-p2) ≥ 10
P – Parameters
H – Hypotheses
A – Assumptions / Conditions
N – Name the test
T – Test statistic
O – Obtain P-value
M – Make decision
S – State result in context
```