chapter 12 Flashcards
When testing 3 or more population proportions, what is the null hypothesis
Ho - all proportions are equal
When testing 3 or more population proportions what is the alternative hypothesis
Ha - not all population proportions are equal
what does k represent
3 or more population proportions
what is the degrees of freedom for testing 3 or more population proportions
k - 1 = df
ie.
3 pops
3 - 1 = 2 degrees of freedom
what distribution do we use when testing 3 or more population proportions
chi-squared
what does Pi stand for
is the proportion of the population I, 1 <= I <=k
if the chi-square test computations indicate Ho can be rejected, we have what
statistical evidence to conclude that not all k population proportions are equal
further analysis can be done to conclude which pop proportion or proportions are significantly different from others
What do you need in order to see if 3 or more pop proporitons are equal
- observed frequencies
- expected frequencies
- Chi-squared test statistic for the test of equal pop proportions
What does each pop sample provide
categorical data
How are the expected frequencies determined with what assumption
that Ho is true
What is the chi-squared test statistic used for
to determine whether there is a significant difference between the observed and the expected frequencies
How do you calculate the expected frequencies
by using the total column and calculating “likely” total for all populations / the total number of all populations
- then apply this % to each of the populations to determine the expected frequencies
What is the degrees of freedom for a chi-square test statistic
k-1
what does the expected frequency need to be at least
5 or more for each cell
How do you calculate the test stat for tes of equal pop proportions?
- find the differences between observed and expected frequencies
- sqaure the difference
- divided the squared difference by expected frequency
- add the last column up gives you your chi-square total
- go to the chi-square distribution chart and with the correct degrees of frequency, find what the p-value will be between then compare to the level of significance
where is the chi-square test for equal population proportions will always be (what kind of tail test)
an upper tail test with rejection of Ho occurring when the test statistic is in the upper tail of the chi-square distribution
when using the critical value approach, what numbers are you comparing?
the value in the squared difference divided by expected frequency and the where the level of significance is with the correct degrees of freedom on teh chi-square distribution table
If p-value is less than the level of significance then what
reject HO
if chi-squared value for the squared differences is more than the chi-squared value for the level of significance than what
reject Ho
What is the multiple comparison procedure used for
- used when the chi-squared test concludes that the pop proportions of 3 pop are not all equal
- this will help identify the differences in the pop proportions
Using the Marascuilo Procedures what are the steps
- compute the absolute value of the pairwise difference between sample proportions for each pair of pops in the study
- select a level of significance and compute the corresponding critical value for each pairwise comparison using the formula
- determine if there are significant differences
What is the CV formula for the Marascuilo pairwise comparison procedure
see formula
What is the test of independence
using sample data to test for the independence of two categorical variables
How does the test of independence work
- we take a sample and record the observations for two categorical variables
- we summarize the data by counting the number of responses for each combination of category for variable 1 and a category for variable 2
- the null hyp for this test is that the two categorical variables are independent
What is this an example of
Ho: beer preference is independent of gender
Ha: beer preference is not independent of gender
Test of independence
Goodness of fit test - Definition
A chi-square test
- used to test that a population probability distribution has a specific historical or theoretical probability distribution.
- This test is for both a multinomial probability distribution and a normal probability distribution.
Marascuilo procedure
A multiple comparison procedure that can be used to test for a significant difference between pairs of population proportions. This test can be helpful in identifying differences between pairs of population proportions whenever the hypothesis of equal population proportions has been rejected
Multinomial probability distribution
A probability distribution where each outcome be-longs to one of three or more categories.
- extends the binomial probability from two to three or more outcomes per trial
Test of independence
A chi-square test that can be used to test for the independence between two categorical variables. If the hypothesis of independence is rejected, it can be concluded that the categorical variables are associated or dependent
What is the Chi-square test used for
- test of equality (P1=p2=p3)
- Test for independence
- Goodness of fit test (1. is there a change in pop proportion or 2. test if it has a normal prob distribution)
When you do a test of equality, If you reject HO and conclude that not all pops are equal, what can you do
- use Marascuilo procedure to determine where the differences among the proportions exists
what are the steps when you are conducting a goodness of fit test to see whether a sample appears to come from a normal probability distribution?
- order from smallest to largest
- calculate the mean and standard deviation
- define intervals n/k (k being the number of sections you want/ intervals) ie 25/5 = 5
- find the interval limits - use normal table
1/k x 100 = ie 1/5 = 0.2 - 20th percentile - look on the z table to find 0.2 x= .84 this is for upper limit and lower is -.84
- then find the next one - 40th percentile??why
- Find interval limits
z = -.84 M+ SD/-.84 = 56.72
z = .84 M+SD/.84 = 85.82
z = -.25 M + SD/-.25 = 66.75
z = .25 M+SD/.25 = 75.25 - Find the observed frequencies by putting the total number in each category
- calculate the test statistic x2 with df - k-p-1
- calculate the expected frequency - (n/k for all categories)
What are some applications of the chi-squared test?
- Testing equality of population proportions for 3 or more populations
- Testing the independence of two Categorical variables
- Testing whether a prob distribution for a population follows a historical or theoretical prob distribution
is the Chi-square test restricted to equal sample sizes for each of the k populations?
no
Why are we using the chi-squared test for pops are equal
chi-square test statistic to determine whether there is a significant difference between the observed and expected frequencies.
if the differences between the observed and expected frequencies are low, then what can we say
If this is the case, the value of the chi-square test statistic will be relatively small and H0 cannot be rejected
What type of test is used for testing of equality, proportion and independence
The chi-square test presented in this section is always a one-tailed test with the rejection of H0 occurring in the upper tail of the chi-square distribution
Each of the k populations in this section had two response outcomes, Yes or No. In effect, each population had a ______distribution with parameter p the population pro-portion of Yes responses. The degrees of freedom here is _________.
An extension of the chi-square procedure in this section applies when each of the k populations has 3 or more possible responses. In this case each population is said to have a __________distribution. the degrees of freedom here is ________________
binomial df= k-1
Multinomial df = (r-1) (k-1)
r = # of responses or rows?
This chi-square test for test of independence is also a ________ test with rejection of H0 occurring in the _____ tail of a chi-square distribution with _______ degrees of freedom.
x2
One tailed test
Upper tail
(r-1)(c-1) df
What is a multinomial distribtuion
it is an extension of the binomial distribution
- binomial describes pops whos elements are one of two things success or failure
Multinomial distribution describes pops whose elements are one of many things, but a fixed number of things such as years of experience
What are the attributes of a multinomial distribution
- each element of the pop belongs to one, and only one, of k categories
- The prob that an element form the pop taken at random will be in category i is some unknown but fixed positive number pi, with pi between o and 1
- a random sample from the pop is taken and each element in the sample is classified into one of k categories
The test for goodness of fit is always a _____tailed test with the rejection occurring in the ____tail of the chi-square distribution
one tailed
Upper tail
Goodness of fit for a normal distribution. Because the normal prob distribution is continuous what must we do
we must modify the way the categories are defined and how the expected frequencies are computed
With a continuous
probability distribution, establish intervals such that
each interval has an expected frequency of ____ or more
5 or more
What can be used to determine the boundaries in a normal prob distribution
standard normal probability tables
Chi- sqayre test to determine Normal diatribution steps
- Defibe intervals (each must have a min of 5 per interval) take total and divide by 5 maybe
- Find interval limits using normal table
If intervals are 25/5 = 5 then 1/5 = .2 or 20th percebtike - look on z table for this number
(Upper limit) z= 0.25
Then 2/5= .4 find the z number Z= 0.84
- Limits
Z= 0.84. xbar+Sd/-.84=56.72
Xbar+sd /.84 = 85.28
Z=.25
Xbar+sd / -.25 =66.75
Xbar+sd/.25= 75.25
- Find the oberserved frequencies and our them in their categiries
0-56.72 - has 7
- 73 - 66.75 has 7
- 76-75.25 has 1
- 26 - 85.28 has 1
- 29 plus has 9
- Conpute chi square test
Expected is 5 all the way theough
Df = k-p-1
Marascuillo procedures is used for
testing the equality of 3 or more pop proportions
In a normal distribution with 30 observations what would the expected number be
30/5 = 6
6 indicates intervals
5 is the expected number for each
what is the degrees of freedom for the normal distribution
k-p-1
p = 2 for mean and sd
k is the total number of observations
Using the Marascuillo procedure, when comparing phat 1 - phat 2 to cv12 what is significant
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