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Summary statistics full notes - sta1000 complete notes for distinction

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  • Course
  • STA1000 (STA1000S)
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  • University Of Cape Town (UCT)

these are notes that cover the entire course with solutions to questions . in depth explanations that cover all topics and would guarantee an a*. Contains both information from lectures , videos , textbook and from tutors.

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  • January 9, 2023
  • 73
  • 2023/2024
  • Summary

Subjects

  • stats
  • statistics
  • sta1000
  • sta1000s
  • sta1006
  • sta1006s

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  • University of Cape Town (UCT)
  • STA1000 (STA1000S)
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STA1000S
Notes

,Table of Contents
Week 1 .......................................................................................................................................... 6
Probability Versus Odds ......................................................................................................................... 6
Statistical Distributions .......................................................................................................................... 6
Excel Formulas ................................................................................................................................... 6
Fair Game ............................................................................................................................................... 6
Win Percentage ...................................................................................................................................... 6
House Advantage ................................................................................................................................... 7
Expected Gain/Loss ............................................................................................................................ 7
Week 2 .......................................................................................................................................... 8
Counting Rules ....................................................................................................................................... 8
Permutations .......................................................................................................................................... 8
Combinations ......................................................................................................................................... 8
Counting Rules ....................................................................................................................................... 8
Counting Rule 1: ................................................................................................................................. 8
Counting Rule 2: ................................................................................................................................. 8
Counting Rule 3: ................................................................................................................................. 8
Counting Rule 4: ................................................................................................................................. 8
Conditional Probability ........................................................................................................................... 9
Bayes’ Theorem .................................................................................................................................. 9
If A and B are Unrelated/Independent: .............................................................................................. 9
The Table Method for Bayes Theorem ................................................................................................. 10
Week 3 ........................................................................................................................................ 12
Qualitative data .................................................................................................................................... 12
Quantitative data: ................................................................................................................................ 12
Ordinal data: ........................................................................................................................................ 12
Excel ................................................................................................................................................. 13
Exploratory Data Analysis..................................................................................................................... 13
Visually Displaying Data ....................................................................................................................... 13
Skewness .............................................................................................................................................. 14
Five-Number Data Summaries ............................................................................................................. 14
o Median .................................................................................................................................... 14
o Lower-quartile: ........................................................................................................................ 15
o Upper quartile ......................................................................................................................... 15
Constructing Box and Whisker Plots: ............................................................................................... 15
......................................................................................................................................................... 15
Five-Number Summary in Excel ........................................................................................................... 15
Summary Statistics ............................................................................................................................... 16
Standard Deviation in Excel ............................................................................................................. 16
Formulas for Mean and Variance ......................................................................................................... 16
Measures of Location and Spread ........................................................................................................ 16
Location ........................................................................................................................................... 16
Spread .............................................................................................................................................. 17
For more notes, videos and explanations: ........................................................................................... 17
Week 4 ........................................................................................................................................ 18
Random Variable .................................................................................................................................. 18
Probability Mass Functions/Discrete Random Variables ..................................................................... 18
Probability Density Functions/Continuous Random Variables............................................................. 18
Expected Values of PDFs and PMFs ...................................................................................................... 19

,Variance of Random Variable X ............................................................................................................ 19
Probability Mass Function (Discrete): .............................................................................................. 19
Probability Density Function (Continuous): ...................................................................................... 19
Coefficient of Variation ........................................................................................................................ 20
Expected Winnings/Loss ...................................................................................................................... 20
Combining Random Variables .............................................................................................................. 20
Week 5 ........................................................................................................................................ 21
Probability Distribution ........................................................................................................................ 21
Uniform Distribution ........................................................................................................................ 21
• Expected Value ........................................................................................................................ 21
• Variance .................................................................................................................................. 21
• Graph of Uniform Distribution................................................................................................. 22
Binomial Distribution ....................................................................................................................... 23
• Expected Value ........................................................................................................................ 23
• Variance .................................................................................................................................. 23
Week 6 ........................................................................................................................................ 24
Probability Distributions....................................................................................................................... 24
Poisson Distribution ............................................................................................................................. 24
• Expected Value ........................................................................................................................ 24
• Variance .................................................................................................................................. 24
• Graph....................................................................................................................................... 24
......................................................................................................................................................... 24
Exponential Distribution....................................................................................................................... 25
• Expected Value ........................................................................................................................ 25
• Variance .................................................................................................................................. 25
......................................................................................................................................................... 25
Central Limit Theorem...................................................................................................................... 26
The Normal Distribution ................................................................................................................... 26
Calculating Probability in Normal Distributions ................................................................................... 27
Things to Remember ........................................................................................................................... 28
Subtracting / Adding / Multiplying Normal Distributions ................................................................ 28
Lower/Upper Quartiles with Normal Distributions .......................................................................... 28
Example of Normal Distribution Question ........................................................................................... 29
Week 7 ........................................................................................................................................ 30
Sample v Population ........................................................................................................................ 30
Percentage Point Notation ................................................................................................................... 33
Confidence Intervals............................................................................................................................. 33
Point Estimate .................................................................................................................................. 33
Interval ............................................................................................................................................. 33
Confidence Interval Formula: ............................................................................................................... 33
Width of Confidence Interval: .......................................................................................................... 33
Determining Sample Size...................................................................................................................... 34
General Sample Size Formula ........................................................................................................... 34
Sample Size Formula When Trying to Achieve ‘L’ Accuracy ............................................................. 34
Some Common Z Values: ..................................................................................................................... 34
Some Things to Remember: ................................................................................................................. 34
Week 8 ........................................................................................................................................ 35
The Hypothesis Test ............................................................................................................................. 35
2-Sided Test .......................................................................................................................................... 37

, Rejection Region in a 2-Sided Test ................................................................................................... 37
Which Level of Significance (a) to Use? ............................................................................................... 37
Some Things to Remember: ................................................................................................................. 38
Comparing 2 Sample Means ................................................................................................................ 39
Subtracting Distributions ................................................................................................................. 39
Finding the Z Value for Calculating the Test Statistic ....................................................................... 39
The Modified Approach........................................................................................................................ 39
The P Value Explained ...................................................................................................................... 41
• Example of P-Value Question .................................................................................................. 41
Week 9 ........................................................................................................................................ 42
Unknown Population Variances ........................................................................................................... 42
Finding the t-value: .......................................................................................................................... 42
Comparing to Z-Table ........................................................................................................................... 43
Confidence Interval Without Knowing Population Variance ................................................................ 43
Testing the Mean: ................................................................................................................................ 43
Two-Sided Test with Same Example ................................................................................................. 46
The Modified Approach........................................................................................................................ 46
P-Value with One-Sided.................................................................................................................... 46
P-Value with Two-Sided ................................................................................................................... 47
• P-Value Example ...................................................................................................................... 47
What are Degrees of Freedom? ........................................................................................................... 47
The Degree of Freedom “Rule”......................................................................................................... 47
Some Things to Remember .................................................................................................................. 48
Comparing Two Means with the T-Distribution (6 Step Approach) ..................................................... 48
Finding a T-Value when Dealing with Two Means ........................................................................... 48
Comparing Two Means with the T-Distribution (Modified Approach)................................................. 49
Finding the P-Value .......................................................................................................................... 49
Week 10 ...................................................................................................................................... 50
Comparing two Means in Paired Data Sets (6 Step Approach) ............................................................ 50
Comparing two Means in Paired Data Sets (Modified Approach)........................................................ 50
Calculate P-Value ............................................................................................................................. 50
P-Value and Rejecting or Accepting H0 ............................................................................................ 50
Excel and the T-Distribution ................................................................................................................. 51
- Right-Tailed Test ( > )............................................................................................................... 51
- Left-Tailed Test ( < ) ................................................................................................................. 51
- Two-Tailed Test ( < > ) ............................................................................................................. 51
Confidence Intervals Under the T-Test ................................................................................................ 51
Some Things to Remember .................................................................................................................. 52
Example of a Question ......................................................................................................................... 53
Goodness-of-Fit-Test: Whether Data Fits Various Distributions .......................................................... 54
Goodness-of-Fit-Test Under the 6-Step Approach ........................................................................... 54
Chi-squared Distribution ...................................................................................................................... 54
Getting Correct Degrees of Freedom for Chi-Squared ...................................................................... 55
What Does the Critical Value Mean? ............................................................................................... 55
Test Statistic Formula for Chi-Squared ............................................................................................. 56
Goodness-of-Fit Test Under the Modified Approach ........................................................................ 57
- Finding the P-Value ................................................................................................................. 57
A Note on Both the Modified and 6-Step Approach ............................................................................ 58
................................................................................................................. Error! Bookmark not defined.
Some Things to Remember .................................................................................................................. 59

,Week 11 ...................................................................................................................................... 60
Testing for an Association Between Two Categorical Variables........................................................... 60
Testing for Association Using the 6-Step Approach ............................................................................. 60
Getting Correct Degrees of Freedom Under Two Variable Association Test .................................... 61
Degrees of Freedom for Tests of Association....................................................................................... 62
Testing for Association Using the Modified Approach ......................................................................... 64
Finding the P-Value .......................................................................................................................... 64
Excel and the Chi-Squared Test of Association ................................................................................. 64
- P-Value .................................................................................................................................... 64
- Critical Value ........................................................................................................................... 65
- Test Statistic ............................................................................................................................ 65
Testing for a Linear Relationship Between Two Variables ................................................................... 65
Is There a Linear Relationship Between X and Y? ............................................................................. 65
The Coefficient of Determination: R2 ................................................................................................... 66
Using X to Predict Y .............................................................................................................................. 66
Straight Line Formula with True Paramters ..................................................................................... 66
Straight Line Formula with Estimated Paramters ............................................................................ 66
Hypothesis Test About 𝛽 Using the 6-Step Approach .......................................................................... 68

, Week 1
𝑛𝑜. 𝑜𝑓 𝑒𝑞𝑢𝑎𝑙𝑙𝑦 𝑙𝑖𝑘𝑒𝑙𝑦 𝑤𝑎𝑦𝑠 𝑜𝑓 𝑔𝑒𝑡𝑡𝑖𝑛𝑔 𝑜𝑢𝑡𝑐𝑜𝑚𝑒
Pr 𝑂𝑢𝑡𝑐𝑜𝑚𝑒 =
𝑛𝑜. 𝑜𝑓 𝑒𝑞𝑢𝑎𝑙𝑙𝑦 𝑙𝑖𝑘𝑒𝑙𝑦 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠

Probability Versus Odds

Probability reflects the number of ways of getting specific outcome relative to the
total number of ways of conducting the experiment.

Odds reflect the number of ways that give you the event of interest relative to the
number of ways that don’t give you the event of interest.

Statistical Distributions

Statistical distributions are long-run patterns of various outcomes – used to predict
etc.

Excel Formulas

= rand()
® Generates a random number between 0 and 1. By pressing F9, random
numbers are re-generated.
= countif (A1:A20;1)
® “A1:A20” refers to range of data.
® “1” refers to what you’re looking for.
By pressing F4, you lock the data set to specific numbers.

Fair Game

Nobody is expected to win and nobody is expected to lose in the long run.

(Example):

I have a stall where people come and bet on the numbers from die throws. Each
bet is R1. If Amy bets R1 on all 6 numbers, in order for it to be a fair game, she
should win R6 if she’s correct – what she pays = what she wins.
Win Percentage

𝑡𝑜𝑡𝑎𝑙 𝑝𝑎𝑦𝑜𝑢𝑡 𝑓𝑜𝑟 𝑤𝑖𝑛𝑛𝑖𝑛𝑔
𝑊𝑖𝑛 𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 = 𝑋 100
𝑎𝑚𝑜𝑢𝑛𝑡 𝑏𝑒𝑡 𝑎𝑐𝑟𝑜𝑠𝑠 𝑎𝑙𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠

, Indicates the percentage of what we have received in the form of bets that we pay
back in winnings.

The amount we pay back of what we take in.

House Advantage

This = the amount we [the stall owners] retain.

𝐻𝑜𝑢𝑠𝑒 𝐴𝑑𝑣 = 100 − 𝑊𝑖𝑛 𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒

Expected Gain/Loss

𝐸 𝐺 𝑜𝑟 𝐿 = 𝑇𝑜𝑡𝑎𝑙 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝐵𝑒𝑡 − (Pr 𝑊𝑖𝑛𝑛𝑖𝑛𝑔 × 𝑇𝑜𝑡𝑎𝑙 𝑃𝑎𝑦𝑜𝑢𝑡)




𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝐵𝑒𝑡 = 𝑅2
1
Pr 𝑊𝑖𝑛𝑛𝑖𝑛𝑔 = ( )×2
12
𝑇𝑜𝑡𝑎𝑙 𝑃𝑎𝑦𝑜𝑢𝑡 = 𝑅11 × 𝑅2

Therefore:
1
𝐸 𝐺 𝑜𝑟 𝐿 = 2 − (( )×22)
12
𝐸 𝐺 𝑜𝑟 𝐿 = 0.166

*If E(G or L) <1, then it’s a loss!

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