Download Proportions and Confidence Intervals: Understanding p-hat and Its Distribution and more Lecture notes Mathematical Statistics in PDF only on Docsity! Chapter 19 – Proportions P = population proportion (parameter) p-hat = sample proportion (statistic) TESTING HYPOTHESIS
Test Statistic for Proportion
We want to test Ho: p = po where pp is the hypothesized value for p.
Recall the general formula for a test statistic:
Hypothesized
__stimate ~ valueforp
Standard deviation expected under Ho
p =N(p, JPUT=P)) whennp > 10 and n(1—p) > 10,
p — Po
} p(1—p)
n
Z=
Test Statistic for Proportion 9
B= Po
z=
Po(1— po) .
n
Question: What is the problem with this formula f ‘
in practice?
pis unknown!
Question: What should we do?
Use po to compute the standard deviation
since we assume Ho to be true!
Question: When can the standard normal table be used to find z*?
the P-value corresponding to this test statistic?
Whenever data are collected with an SRS and
po > 10 and n(1 - po) > 10.
Careful with notation - Notice: this proportion p is a population parameter NOT A P-VALUE, NOT A PROBABILITY ON A STATISTIC Categorical Data and Proportions
Summary
p Population Parameter (0<p<1)
proportion
n Sample Statistic (0<p<1)
p proportion P
Definitions of Sampling Distribution of p
Theoretical Sampling Distribution of B: The distribution of all 6's
from all possible samples of the same size from the same population.
Approximate Sampling Distribution of p: The distribution of p's
obtained from taking repeated SRS's of the same size from the same
population.
The simulation is done with a problem in which we know p in order to investigate the behavior of p-hat.
How Likely?
Let p = proportion of births that are boys.
Assume p = 0.5 in usual birthing conditions.
Officials in India are concerned about selective
abortions.
One large hospital reported 100 births
in 24 hours: 80 boys and 20 girls,
Is the event of getting 80 or more boys in
100 births unlikely if p = 0.5?
Does this provide evidence of selective abortions?
Let's compute the probability and find out.
How Likely?
Assuming p = 0.5 is true, what is the probability of getting 80 or
more boys in 100 births? (i.e., what is the probability that p = 0.8)?
Needed: the sampling distribution of p assuming p = 0.5 is true.
Mean = p=0.5 The standard deviation of pi:
, | pl =p) ,, | 0.5 (1 — 0.5)
n - 100
= 0.05
Is the shape approximately normal?
Checking Assumptions:
SRS? Assume ok
Is np >10? Yes 100(0.5) = 50 > 10
Isn(1—p)>10? Yes 100(0.5) = 50 > 10
Implications? Shape is approximately normal.
This is an indication that selective abortion may be taking place
Three Different Distributions
Population Sample _ Sampling
Categorical Data Categorical Data Distribution of p
Proportion p B Doesn't apply
Categorical data | Categorical data
Center don't have center | don't have center p
Categorical data | Categorical data | 4fo7,_,\,,
Spread don't have spread | don't have spread p(1-p)/n
Bar graphs Bar graphs Approx. normal
Shape don't have shape | don't have shape inns is lar eo
np=10, n(1-
Probability Only if comping
using
Normal Table
Never
Never
distribution is
approx. normal
