Different types of statistical inference - Lecture 16, 17

Statistical Inference

Using sample distribution to draw conclusion about population at large.

Type of inference: Confidence Interval

An interval around the proportion that we believe contains the possibility, with a level of confidence that we can specify

When to use Confidence interval?

Estimate a population parameter or range of plausible values; Questions with keywords: estimate, confidence interval, range of values

Another type of inference: Test of significance/ hypothesis testing

A decision about whether p equals some hypothesized value or whether we should reject the hypothesized value because the observed proportion would be too improbable if that were the case- the main objective is to test a claim about parameter.

When to use hypothesis testing?

Purpose- test a claim or make a decision about a hypothesis; Questions with keywords: test the claim, is there evidence, significance level.

Critical Value (z*)

For an X% confidence interval; The number of standard deviations (or standard errors) to the left and right of the mean to bracket X% of the area under a normal curve.

Margin of Error

The actual distance to the left and right of the mean to bracket x% of the area under the curve

How to solve for X% confidence interval for p

1. find the proportion first - the subset we’re looking at over the entire population

2. Find the critical value - look at the z-distribution table with your z-score

3. Solve for your Standard Error formula

4. Multiple the critical value and the standard error for MOE

5. Solve for CI → [proportion - MOE, proportion + MOE]

90% Confidence level & its critical value

z* = 1.645

95% Confidence Level and its critical value

z*= 1.96

99% Confidence Level & its critical value

z*= 2.576

Interpretation of X% Confidence Interval

“We are X% confident that between the [confidence interval] of all the population doing the observation.”

Steps to solve for Hypothesis testing

1. Define the null & alternative hypothesis

2. Chose the significance level - this determines the threshold for rejecting the null hypothesis

3. Select the appropriate statistical test based on data type & sample size - For means, use the z-test (population standard deviation is known & n is large) or t-test (standard deviation is unknown or n is small); For proportion, use the z-test; For two samples use two-sample t-test or z-test

4. Compute the test statistic based on your test type

5. Determine the P-value or the critical value

6. Make a decision

   1. For P Value→ if the p-value is less than the significance level, reject H_o. If the p-value is greater than significance level, fail to reject H_o.

   2. For the critical value, if the test statistic falls beyond the critical value, reject H_o. Otherwise, fail to reject H_o.