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Inference for Categorical & Quantitative Data: Chi-Square &…
Inference for
Categorical & Quantitative
Data:
Chi-Square & Slopes
Chi-square Test
Goodness of Fit(1 way table)
Observed Count
- The actual observed counts from a sample/study
Expected Count
- The counts expected if the null hypothesis is true
One-way table
- displays counts for categories of a single categorical variable
Chi-square Statistics:
df
:
As df increases the chi-sqaure dist. becomes less right skewed
Conditions for contructing GOF test
Independence
: Independent treatments/N>=n*10
Large Counts
: All expected counts are >= 5
Randomness
: Random sampling/assignemnt
Ho
: The distribution of the variable is the SAME as the claim
Ha
: The distribution of the variable is DIFFERENT than the claim
p-value
= X²cdf
Chi-square test for Homogeniety (1 categorical varibale)
Same Conditions: Randomness Independence, Large Counts
Ho
: The distribution of var. A is the same as var. B
Ha
: The distribution of var. A is the not the same as var. B
Expected Counts
= (row total * colum total) / (table total)
2- way table
Chi-square test for Independence (2 categorical vairbales)
Same Conditions: Randomness Independence, Large Counts
Ho
: There is no association between var A & var. B
Ha
: There is an association between var A & var. B (They are dependent)
Expected Counts
= (row total * colum total) / (table total)
2-way table
Confidence Interval for Slopes
Conditions
N
ormality - no strong skew or outliers on the resdual plot
E
qual Variance - No fanning on the residual plot / same distribution around residual=0
I
ndependene - N>=10n or treatments are indpt from each other
R
andomness - random assignment/random selection
L
inear - The relationship between x and y is linear (NO pattern in the residual plot)
If all conditions met perform a linear regression t-intreval for slope
Standard Deivation
:
Margin of Error
: (t*)(SEb) =
Confidence Interval
: b±(t*)(SEb) =
Interpretations
Constant(α)
: when (Y) is 0, the predicted (X) is ___
Slope(β)
: for every 1 increase in (Y), their predicted (X) will increase by ___
SD(σ)
: On average, the predicted (X) typically vary by ___ from the actual (X)
Standard error of slope(SEb)
: the slope of the sample regression line typically vary by ___ from the true regression line for predicting (X) from (Y)
Conclsion for 4-step plan
: We are (confidence level) confident that the interval
to
captures the slope of the true regression line for predicting (X) from (Y)
Mean
:
df
= n-2
Test for Slopes
Test Statistics:
Hypothesized slope = 0 when checking for linear association
Ho
β = 0
(if sample suggests linear relationship)
β = hypothesized slope
(if given)
Ha
β ≠ 0
(There is an linear relationship)
β > 0
(if there is a positve linear relationship)
β < 0
(if there is a negative linear relationship)
df
= n-2
if testing for β ≠ 0 remember to times tcdf by 2 to get the p-value
Conditions
LINER
if all conditions met, perform a linear regression t-test for slope
if p-value>α, we fail to reject the Ho. Therefore we do not have convincing evidence that (Y) increases (X) increases
if p-value<α, we reject the Ho. Therefore we do have convincing evidence that (Y) increases (X) increases