• Announcements

    - CASA Registration
    - No class next Monday or Tuesday (March 10) 
    - I will record a midterm review and post an online pre-midterm quiz **for points**

What is a hypothesis

A falsifiable statement about what we believe will happen based on the theory we are trying to test.

Falsifiability

• We start from the assumption that our theory is wrong
• Our hypothesis is called the

alternative hypothesis or H1

• because it is the alternative to our assumption of no relationship

null hypothesis or H0

Testing the null hypothesis

• Statistical tests show the degree of certainty that we can reject the null hypothesis
• They show the probability that the alternative hypothesis is due to random chance
• When the probability, p, is below our pre-determined threshold, we reject the null hypothesis
• If we reject the null hypothesis, the alternative hypothesis is true, right?

NO!

• If we reject the null hypothesis, the alternative hypothesis is true, right?

NO!!!!

• If we reject the null, we infer that the alternative hypothesis (our hypothesis) is approximately true within the probability we have chosen.
• If we reject the null, we can say there is strong evidence that the alternative hypothesis (our hypothesis) is true
• We often test multiple hypotheses

How do we get there?

How do we get from a sample with a correlation to talking about testing a hypothesis for a population?

• Tying sample statistics to population parameters (LLN)
• Probability distributions
• Tying our data to the probability distributions (CLT)

Statistics to parameters

• We rely on the Law of Large Numbers and use sample statistics
• With sufficient sample size and proper math, our sample mean is an unbiased estimator of the population mean
• With CLT, \(s^2\) is an unbiased and consistent estimator of \(\sigma^2\)
• For the normal distribution \(s^2\) is also the Minimum Variance Unbiased Estimator for \(\sigma^2\)

Probability distributions

The 68-95-99.7 Rule

            + Allows us to estimate probability based on distance from the mean
            + Applies to normal distribution
            + Basis for the actual decision rules
            

The 68-95-99.7 Rule

68-95-99.7 rule

From Distribution to Confidence

• We can now bridge the gap between probability distributions and our sample data.

• CLT ties our sample means to the normal distribution

• Sample stats are unbiased estimators of population parameters

• They are only point estimates, so we construct a confidence interval

• confidence interval a range of values around the point estimate that we are X% confident contains the true population parameter

Math and philosophy of the confidence interval

• Math and philosophy detail: The true population parameter is a fixed but unknown number, while our calculated interval is random but known with certainty. Our X percent confidence reflects the long-run reliability of our method. If we repeated this exact sampling process 100 times, X of the resulting intervals would successfully capture the true population parameter.

Calculating the Interval

• we need three pieces of information:

    - point estimate = sample statistics
    - standard error = the standard deviation of the sampling distribution
    - critical value derived from our desired confidence level - for normal distribution this is a z-score
    - For 95% confidence, the critical z is 1.96

• The general formula for a confidence interval is:

\[CI = \bar{x} \pm z \left( \frac{s}{\sqrt{n}} \right)\]
## CI of the variance

\[\frac{(n-1)s^2}{\chi^2_{\alpha/2}} \le \sigma^2 \le \frac{(n-1)s^2}{\chi^2_{1-\alpha/2}}\] - This use the Chi Square distribution which we will discuss after midterm

CI of the standard deviation

\[\sqrt{\frac{(n-1)s^2}{\chi^2_{\alpha/2}}} \le \sigma \le \sqrt{\frac{(n-1)s^2}{\chi^2_{1-\alpha/2}}}\] - Also uses the Chi-square distribution

Why the Chi-Square not the normal?

• When we square distances from the mean to get the variance, the distribution changes
• By eliminating all negative numbers, the distribution becomes skewed and looks like this:

Distribution Comparison

Authorship, License, Credits

• Stephen Moore code used to simulate the Law of Large Numbers

• 68-95-99 rule graphic Source:https://towardsdatascience.com/understanding-the-68-95-99-7-rule-for-a-normal-distribution-b7b7cbf760c2

• Author: Tom Hanna

• Website: tomhanna.me

• License: This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.</>