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Unit 8: Chi-Square; Unit 9: Inference for Quantitative Data - Slopes 📝…
Unit 8: Chi-Square;
Unit 9: Inference for Quantitative Data - Slopes 📝
Chi-Square: Goodness of Fit (GOF)
About chi-square
A statistical value that measures how expected counts compare to observed counts w/
multiple
categorical data values involved
Always
right
skewed +
no negative
values!
Larger
chi-square value gives
more
convincing evidence for the alternative hypothesis
As degrees of freedom
increase
, the graph becomes
less right-skewed
Vocabulary
One-way table:
A table that displays counts for categories of a single categorical variable
Expected counts:
The counts expected if the null hypothesis is true
Observed counts:
The actual observed counts from a sample/study
State
Null hypothesis: the
distribution
of the variable is the SAME as the claim
Alt. hypothesis: the
distribution
of the variable is DIFFERENT from the claim
Plan
Random - random sampling/random assignment
Independence - 10% condition
3. Large Counts (NOT normality!)
- all expected counts have to be greater than 5
By calc: got to 2nd -> matrix -> insert observed values for [A] -> run test -> go to matrix [B] for expected values
"All conditions met, perform a chi-square goodness-of-fit test."
Do
Components
add up to form the chi-square test statistics:
using test statistics? x^2 cdf (lower: test statistics, upper: positive infinity, df = n-1)
df = # of categories - 1
Calculator: X^2 GOF-Test
Conclude
(compare the p-value with the significance level and interpret based on the null and alt. hypotheses)
Chi-Square: Two Way Tests
Chi-square test for homogeneity
- a test to see if a categorical variable has a
different distribution
for several populations or treatments
4-step plan same as chi-square GOF except:
Null: The distribution of the variable is the SAME for the groups or treatments
Alt.: The distribution of the variable is DIFFERENT from the groups or treatments
df = (number of rows-1)(number of columns - 1)
"All conditions met, perform a chi-square test for homogeneity."
Calculator: X^2 - Test
Chi-square test for independence
- a test to see if there is an
association
between two categorical variables.
4-step plan same as chi-square GOF except:
Null: There is NO association between the two variables
Alt.: There IS an association between the two variables
"All conditions met, perform a chi-square test for independence."
Calculator: X^2 - Test
Confidence Interval for Slopes
Plan: LINER
1. Linear
- the residual plot shows a random scattering of residuals
2. Independence
- assume that the response variable for each x is independent
3. Normality
- the residuals show an approximately normal distribution
4. Equal Variance
- the residuals have about the same scattering around 0
5. Random
— random sample/random assignment
"All conditions met, construct a confidence interval for slope."
Calculator: LinRegTInt
Interpreting the standard error of slope:
If we repeat the random assignment/sample many times, the slop of the sample regression line would typically vary by about
(SE of b)
from the slop of the true regression line for predicting
(response var.)
from
(explanatory var.)
.
Interpreting the standard deviation:
The actual y-value typically varies by about
(SD)
from the amounts predicted with the least-squares regression line (LSRL) using x = __(state x-context).
Do
Remember context for LSRL!
Test for Slopes
State
Null
beta = 0 -> no linear relationship
Alternative
beta greater than (positive linear)
beta less than (negative linear)
beta not equal to 0 (linear relationship or not)
Define beta (population slope) in context
Plan: LINER
"All conditions met, perform a linear regression t-test for slope."
Calculator: LinRegTTest
Do
t = [(stats - parameter)^2/standard error of slope]
df = # of categories - 2
tcdf if by test stats
Conclude
(same as before)
