AP Statistics Chapter 12: Chi Squared and Inferences for Slope

Prerequisite

AP Statistics Chapter 9 : Hypothesis Testing For Proportions

12.1 Chi Squared Test and Distribution

Chi Square

Chi squared is used to test categorical variables using their expected outcomes.

The expected outcome is as below.

\[\tag{1} E(X_i) = \sum_{i}np_i\]

where

• $n$ is number of samples
• $p_i$ is probability of getting outcome of $i$th category

and Chi Squared test statistics is as follows

\[\tag{2} \chi^2 = \sum _i \frac{(O_i-E(X_i))^2}{E(X)}\]

where

• $O_i$ is actual outcome of $i$th category
• $E(X_i)$ is expected outcome of $i$th category (see equation $(1)$)

Chi Square Distribution

$n$ categorical variables follows chi squared distribution of $n-1$ degrees of freedom

This, we can get P-value of chi square test statistics using $\chi$cdf in calculator.

Chi square is always greater or equal to 0 because it is analysis of variances.

Chi Square Test

We can carry out the test as we did in Ch.9 and Ch.11

Conditions

1. Random (10% condition)
2. Large Counts: Every $E(X_i)$ is greater than 5

12.2 Two way Chi Squared Test

Test for independence and homogeneity is same except hypothesis.

Test for Independence

Typically, Test for Independence is used when

• Single sample
• 2 categorical variables

Here, we are looking for association (or independence) between two categorical variable in a single population

Performing the test

1. State:
   • $H_0$: Two variable __ and __ is independent in population in interest
   • $H_a$: Two variable __ and __ is independent is associated in population in interest
2. Plan: Check Conditions
3. Do: Perform Calculations
4. Conclude: Make Conclusion

Test for Homogeneity

Test for Homogeneity is used when

• More than one samples or different treatments
• One Categorical Variable

Here, we are testing if the distribution is same across different populations.

Performing the test

1. State:
   • $H_0$: The distribution of __ is same across __ and __ in population in interest
   • $H_a$: The distribution of __ is different __ in population in interest
2. Plan: Check Conditions
3. Do: Perform Calculations
4. Conclude: Make Conclusion

Comparison with 2 sample z test (proportions)

1. 2 sample z test is powerful since we can set directions

Conditions for Inference

1. Random: Random sample or assignment
2. Independence: 10% condition or independent treatment
3. Large counts: All expected counts are 5 or more ($\forall E(X_i) \geq 5$)

12.3 Inference for Slope

Population Regression Line

Population regression line is regression line for the entire population.

The Least Square Regression line for the population will be

\[\mu_y = \alpha + \beta x\]

This assumes that the $y$ variable will linearly depend on $x$

Sample Regression Line

If we sample some portion of the population, the slope $\beta$ of regression line will also have sampling distribution.

The Least Square Regression line for a sample will be

\[\hat y = a + bx\]

Sampling distribution of Slope

Since we are randomly selecting sample from the population, the regression line will also vary depending on the sample we got.

This is similar to sampling distribution for means.

1. Center : Mean if sampling distribution of slopes is unbiased estimator for population slope.

   \[\mu_b = \beta\]
2. Variability : \(\sigma_b = \frac{\sigma}{\sigma_x \sqrt{n}}\) Where $\sigma$ is standard deviation of residuals and $\sigma_x$ is standard deviation of $x$
3. Shape : Approximately Normal when conditions are met

Since we dont know the parameter, we estimate the variability with standard error

\[SE_b = \frac{s}{s_x \sqrt{n-1}}\]

Conditions for Regression Inference

1. Linear : y and x should have linear relationship
2. Independent : 10% conditions, independent observations or treatments
3. Normal : y should vary apporoximately normal for any x value in range
4. Equal SD : $\sigma$ (residuals) are equal throughout the range of x
   This also means that $\sigma_y$ is equal for any given x in range
5. Random : random sample or random assignment

Confidence Interval

Since it is similar to sampling distribution for means, we can think of constructing confidence interval for slopes too.

The t distribution follows n-2 degrees of freedom because we lost one degrees of freedom when constructing the regression line.

\[b \pm t^*_{df=n-2} \cdot SE_b\]

T-test

we can also perform t test with slopes.

The t distribution here also follows n-2 degrees of freedom.

\[t = \frac{b - \beta_0}{SE_b}\]