Download Proportion Tests

Lecturer:
Pauliina Ilmonen
Slides: Ilmonen/Virtanen/Ailus/Kanta
Proportion Test
Two Sample
Proportion Test
Introduction to Statistical Inference
Lecture 4: Proportion Tests
Lecturer: Pauliina Ilmonen
Slides: Ilmonen/Virtanen/Ailus/Kantala
References
Contents
Lecturer:
Pauliina Ilmonen
Slides: Ilmonen/Virtanen/Ailus/Kanta
Proportion Test
Two Sample
Proportion Test
References
Proportion Test
Two Sample Proportion Test
References
Lecturer:
Pauliina Ilmonen
Slides: Ilmonen/Virtanen/Ailus/Kanta
Proportion Test
Two Sample
Proportion Test
References
Proportion Test
Proportion Test
Lecturer:
Pauliina Ilmonen
Slides: Ilmonen/Virtanen/Ailus/Kanta
Proportion Test
Two Sample
Proportion Test
References
Proportion tests can be used for example when testing
proportions of faulty products in a production process.
Proportion Test
Lecturer:
Pauliina Ilmonen
Slides: Ilmonen/Virtanen/Ailus/Kanta
Proportion Test
Two Sample
Proportion Test
Let x1 , x2 , ..., xn be the observed values of a random variable x.
Assume, that the observed values are independent and
identically distributed (i.i.d.) and come from the Bernoulli
distribution with parameter p. (Now P(xi = 1) = p,
P(xi = 0) = 1 − p, E[x] = p and the variance
E[(x − E[x])2 ] = p(1 − p).)
The null hypothesis H0 : p = p0 .
The possible alternative hypotheses: H1 : p > p0 (one tailed),
H1 : p < p0 (one tailed) or H1 : p 6= p0 (two tailed).
References
Proportion Test
Lecturer:
Pauliina Ilmonen
Slides: Ilmonen/Virtanen/Ailus/Kanta
Proportion Test
• The test statistic C =
Pn
i=1
Two Sample
Proportion Test
xi .
• If the null hypothesis H0 is true, then the test statistic
follows binomial distribution with parameters n and p = p0 .
• Under the null hypothesis H0 , the expected value of the
test statistic is np0 (E[C] = np0 ) and the variance of the
test statistic is np0 (1 − p0 ).
• If the value of the test static is large or small (compared to
the expected value np0 ), evidence against the null
hypothesis H0 is found.
• The null hypothesis H0 is rejected, if the p-value is small
enough.
References
Binomial distribution
Lecturer:
Pauliina Ilmonen
Slides: Ilmonen/Virtanen/Ailus/Kanta
Proportion Test
Two Sample
Proportion Test
References
More about binomial distribution in Wikipedia.
Proportion Test, p-value
Lecturer:
Pauliina Ilmonen
Slides: Ilmonen/Virtanen/Ailus/Kanta
Proportion Test
The distribution of the test statistic C is tabulated and statistical
softwares calculate p-values of the test.
Let c denote the observed value of the test statistic C. Then
the p-value of the test is given as follows:
• If alternative hypothesis is H1 : p > p0 , then the p-value is
p = P(C ≥ c).
• If alternative hypothesis is H1 : p < p0 , then the p-value is
p = P(C ≤ c).
• If alternative hypothesis is H1 : p 6= p0 , then the p-value is
p = 2 min{P(C ≥ c), P(C ≤ c)}.
The probabilities P(C ≥ c) and P(C ≤ c) above are calculated
under H0 .
Two Sample
Proportion Test
References
Asymptotic Proportion Test
Lecturer:
Pauliina Ilmonen
Slides: Ilmonen/Virtanen/Ailus/Kanta
Proportion Test
If the sample size is large, then under the null hypothesis H0 ,
the standardized test statistic
Z =p
p̂ − p0
p0 (1 − p0 )/n
Pn
— where p̂ is the unbiased estimator p̂ = n1 i=1 xi of the
parameter p — approximately follows the standard normal
distribution.
The approximation is usually good enough, if np̂ > 10 and
n(1 − p̂) > 10. For smaller samples, the test relies on the exact
distribution of the test statistic C.
Two Sample
Proportion Test
References
Numerical Example
Lecturer:
Pauliina Ilmonen
Slides: Ilmonen/Virtanen/Ailus/Kanta
Proportion Test
Two Sample
Proportion Test
References
Jack’s unpalatable princess cookies are sold in every store.
The cookies are very popular, because some of the cookies
have been made with a different recipe to achieve a horrible
demon cookie taste. It is stated in the package that 10 % of the
cookies are demon cookies. Susan selected 150 cookies
randomly and 21 of the cookies tasted unpalatable. You wish to
know if the package lies, and decide to test, on 5% significance
level, the null hypothesis H0 : p = 0.10.
Numerical Example
Lecturer:
Pauliina Ilmonen
Slides: Ilmonen/Virtanen/Ailus/Kanta
Proportion Test
Two Sample
Proportion Test
In this proportion test, the null hypothesis is H0 : p = 0.10 and
the alternative hypothesis is H1 : p 6= 0.10. Since the sample
size is large, normal
can be used. Estimated
Pn approximation
21
and the test statistic
probability p̂ = n1 i=1 xi = 150
Z =p
p̂ − p0
p0 (1 − p0 )/n
=p
21
150
− 0.1
0.1 · 0.9/150
≈ 1.632.
The p-value is 2 ∗ (1 − 0.9484) = 0.1032 > 0.05. The null
hypothesis is not rejected.
References
Lecturer:
Pauliina Ilmonen
Slides: Ilmonen/Virtanen/Ailus/Kanta
Proportion Test
Two Sample
Proportion Test
References
Two Sample Proportion Test
Two Sample Proportion Test
Lecturer:
Pauliina Ilmonen
Slides: Ilmonen/Virtanen/Ailus/Kanta
Proportion Test
Two Sample
Proportion Test
References
In two sample proportion test, parameters of two independent
Bernoulli distributed samples are compared.
Two Sample Proportion Test
Lecturer:
Pauliina Ilmonen
Slides: Ilmonen/Virtanen/Ailus/Kanta
Proportion Test
Two Sample
Proportion Test
Let x1 , x2 , ..., xn be observed values of a random variable x and
let y1 , y2 , ..., ym be observed values of a random variable y .
Assume, that the observed values x1 , x2 , ..., xn are i.i.d. and
come from the Bernoulli distribution with parameter px and that
the observed values y1 , y2 , ..., ym are i.i.d. and come from the
Bernoulli distribution with parameter py . Assume, that xi and yj
are independent for all i, j.
The null hypothesis H0 : px = py .
The possible alternative hypotheses: H1 : px > py (one tailed),
H1 : px < py (one tailed) or H1 : px 6= py (two tailed).
References
Two Sample Proportion Test
Lecturer:
Pauliina Ilmonen
Slides: Ilmonen/Virtanen/Ailus/Kanta
• Calculate the sample proportions pˆx =
pˆy =
1
m
Pn
i=1
yi , and p̂ =
npˆx +mpˆy
n+m
1
n
Pn
i=1
xi , and
Two Sample
Proportion Test
.
References
• Calculate the test statistic
Z =q
pˆx − pˆy
p̂(1 − p̂)
1
n
+
1
m
.
• If the sample size is large, then under the null hypothesis
H0 , the test statistic Z approximately follows standard
normal distribution. The approximation is usually good
enough, if npˆx > 5, n(1 − pˆx ) > 5, mpˆy > 5 and
m(1 − pˆy ) > 5.
• If the value of the test static has large absolute value,
evidence against the null hypothesis H0 is found.
• The null hypothesis H0 is rejected, if the p-value is small
enough.
Proportion Test
References
Lecturer:
Pauliina Ilmonen
Slides: Ilmonen/Virtanen/Ailus/Kanta
Proportion Test
Two Sample
Proportion Test
J. S. Milton, J. C. Arnold: Introduction to Probability and
Statistics, McGraw-Hill Inc 1995.
J. Crawshaw, J. Chambers: A Concise Course in
Advanced Level Statistics, Nelson Thornes Ltd 2013.
R. V. Hogg, J. W. McKean, A. T. Craig: Introduction to
Mathematical Statistics, Pearson Education 2005.
Pertti Laininen: Todennäköisyys ja sen tilastollinen
soveltaminen, Otatieto 1998, numero 586.
Ilkka Mellin: Tilastolliset menetelmät,
http://math.aalto.fi/opetus/sovtoda/materiaali.html.
References