Statistics

Inferential Statistics

• used to learn about the population from a sample

2 main applications of Inferential Statistics

1. Estimation

2. Hypothesis Testing

A survey was conducted on 500 students from a population of 20,000 students at a university. It found that 74% of students were employed during the semester. Identify the following:

What is the Population, Sample, Statistic, Parameter

Population: 20,000

Sample: 500

Statistics: 74%

Parameter: Unknown statistics of Parameter

Representative Sample:

• It’s the sample that acts as a smaller version of the whole group

• For example, if the population is 60% men and 40% women, the sample should also be 60% men and 40% women.

EPSEM

• Random Selection

• AKA Probability Sample:

• - Every person has an equal chance of being selected.

• EQUAL PROBABILITY SELECTION METHODS

SRS

• Simple Sampling Sample

• The most basic form of random sampling

• Has 2 versions: Stratified and cluster

2 Versions of SRS

1. Stratified: Divide people into different groups and then pick a person from each group

2. Cluster: Divde people into groups and choose one groupV

Convincence Sample:

• Based on Convenience and not an equal chance of being selected

• Ex: choosing all the kids who show up early to class.

• It’s used when you don’t want to infer.

Sampling Distribution

Involves repeatedly taking a sample from a population and calculating a statistic for each individual sample, and then combining that information to create a distribution.

Inferential Statistics has 3 different distributions

1. Sample Distribution

2. Sampling Distribution

3. Population Distribution

Central Limit Theorem

• If the sample size (n) is large enough, then the sampling distribution of the sample mean will be approximately normal.

• Your sample size has to be 100 or more to reach a bell curve.

First theorem:

• The true population mean has to be the same as the sample mean (sampling distribution)

• Both should be a bell curve.

• The less data you have, the fatter the curve is

• The more data you have the skinner the curve is

Standard Error

• measures how far the estimated mean is from the true mean

Checklist to see if your sample is big enough

Formula nPu and n(1-Pu)

• If your answer is equal to or more than 15, then it is big enough.

  • Appendix A

Pu =

Proportion mean (population percentage)

Mp

Population mean

𝜎𝑝

Proportion Standard Deviation

Logic of Estimation

• Information from samples is used to estimate information about the population

Point of Estimate:

• Your best single guess for a whole group based on a small sample you surveyed.

• A number in the middle of the Confidence interval

• Confidence Interval: 27% - 33%

• Point of Estimate: 30%

• The margin of Erroe is 3%

Confidence Interval

• How accurate our estimate is likely to be
  

• you peek at a small bit, make a smart guess for a range, and the confidence number tells you how often that method will get a range that includes the true answer.

It’s usually a range: 27% - 33%

I know the truth varies from 27% - 33%

Lower Bound: 27%

Upper Bound: 33%

Steps to building the confidence intervals.

1. Set the alpha (choose the level of risk that you’re willing to take that your range will miss the true answer)

2. Find the Z Score asccoiated witht the alpha

3. Contruct the confidence interval.

Most commoly used alpha level

0.05 = 5% chance risk of a potential error that your interval doesn’t contain the true population value.

P value:

Type I error

• Falso Positive

• Rejecting the null hypothesis (H0) when it is actually true.

• You claim there is a pattern when there isn’t

• Solution: make the alpha smaller.

Type II Error

• False Negative

Failing to reject the null hypothesis (H0) when it is actually false.

As the alpha level decreases the less likely it’ll fall in the critical region.