Summary

STA1000S/F Course Notes Summary

0 purchase

Course
STA1000 (STA1000S)

Institution
University Of Cape Town (UCT)

These Notes provide a summarized version with step-by-step instructions on how to complete various calculations such as Hypothesis tests, T-Tests etc. Please note the notes teach an approach on how to tackle calculations using Excel (Includes what excel formulas to use and not manual calculation.) ...

[Show more]

Preview 5 out of 16 pages

View example

Uploaded on April 13, 2024
Number of pages 16
Written in 2023/2024
Type Summary

sta1000

Institution
University of Cape Town (UCT)
Course
STA1000 (STA1000S)

elizabeth6666

Member since 3 year 6 documents sold

$5.59

Add to cart

Add to wishlist

100% satisfaction guarantee
Immediately available after payment
Both online and in PDF
No strings attached

STA100S Notes
Module 1: Introducing Probability
Random Experiment: When the result from a trial cannot be predetermined.

𝑁𝑜. 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒
𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 =
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑖𝑎𝑙𝑠

We can use an observed pa7ern to conclude what we expect to happen

𝑂𝑑𝑑𝑠 = 𝑁𝑜 𝑤𝑎𝑦𝑠 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 ∶ 1 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑟𝑖𝑎𝑙𝑠

Fair Game:

• P(W) : Probability of winning
• Bet amount: Price to play
• Actual payout: What they pay you if you win
• The E(X) = 0
• Fair payout: What they should pay you if you win (if the game is fair)
• AP<FP

𝟏
𝒇𝒑 = × 𝑩𝒆𝒕 𝒂𝒎𝒐𝒖𝒏𝒕
𝑷(𝑾)

• House Advantage:

𝒇𝒑 − 𝒂𝒑
𝑯𝑨% = × 𝟏𝟎𝟎
𝒇𝒑

𝑯𝑨% = 𝟏𝟎𝟎 − 𝑾𝒊𝒏 %

𝑨𝒄𝒕𝒖𝒂𝒍 𝒑𝒂𝒚𝒐𝒖𝒕
𝑾% = × 𝟏𝟎𝟎
𝑭𝒂𝒊𝒓 𝒑𝒂𝒚𝒐𝒖𝒕

Risk averse: Doesn’t like risk

Sets:
Subset: A set whose element are all contained within another set
Eg. CÌA
Empty set: A set which has no elements
Intersec?on: elements that are common to both A and B
Union: combining elements that are in either or both A and B
Mutually exclusive sets: When intersecKon is empty set
Pairwise mutually exclusive and exhaus?ve: Two sets make up the sample space

1

,Exhaus?ve: nothing in sample space
Complement: All elements that ae not in the set

NOTE!!!: Use a Venn diagram for these types of ques?ons.
Combinations and Permutations:

• PermutaKon: Arrangement of objects when Order maKers

• CombinaKon: Order doesn’t ma7er

• Arrangement: n! = number of arrangements with no repe??on
o If is repeKKon remove the repeKKon by dividing by number repe??ons

o If want things to be together you treat it as a single thing(remember to take
into consideraKon of swapping posiKons)

o When they are not supposed to be together you take:

𝐻𝑜𝑤 𝑚𝑎𝑛𝑦 𝑤𝑎𝑦𝑠 𝑐𝑎𝑛 𝑎𝑟𝑟𝑎𝑛𝑔𝑒(𝑛𝑜 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠) − 𝑊ℎ𝑒𝑛 𝑡ℎ𝑒𝑦 𝑎𝑟𝑒 𝑡𝑜𝑔𝑒𝑡ℎ𝑒𝑟

o If something needs to be in a speciﬁc posiKon say 1 x (n-1)!

• Permuta?on: nPr = number ways ordering r objects from n object with no repe??on
o Order ma7ers
o No Arrangements as there is only 1 correct answer
o Select in a speciﬁc order

• Combina?on: SelecKon + Arrangement

• Selec?on: nCr = number ways selecKng/choosing set r objects from n with no
repleKon

• nr= number of ways of arranging n objects taken r at a Kme with repe??on

Conditional Probability:

Use Tree Diagrams!! Don’t need Bayes Theorem

𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵)

!(#∩%) !(#∩%)
Condi?onal Probability: 𝑃(𝐴|𝐵) = !(%)
OR 𝑃(𝐵|𝐴) = !(#)

If have same condiKonal informaKon: 𝑃(𝐴|𝐵) = 1 − 𝑃(𝐴̅|𝐵)

Know that:
𝑃(𝐷) = 𝑃(𝐷|𝑆)𝑃(𝑆) + 𝑃(𝐷|𝑆̅)𝑃(𝑆̅)

2

,Independent Events:

Use Arrangements with Independent events!!

• 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) × 𝑃(𝐵)

• 𝑃(𝐴|𝐵) = 𝑃(𝐴)

Module 2: Graphical Data
NOTE!!: Mostly theory ques?ons with mul?ple choice
Graphs:
Pie Chart: Showing composiKon
Histogram: Showing quanKtaKve

Qualita?ve data: Categorical data(put into categories and can we counted)

Quan?ta?ve data: Count or a measure of data(interval or raKo)

Ordinal: ranked or ordered data

('()*+'(,-)
Interval Width: √-

Trend Line: linear regression of y=ax+b

Numerical summaries:

5 number summary:
• Min/ QuarKle 0
• Lower QuarKle 1
• Median/ QuarKle 2
• Upper QuarKle 3
• Max/ QuarKle 4

Some sta?s?cs that can be drawn:
• Median: Number in the middle (ascending or descending order) – 50% MARK

• Mean: Average of the data

• Mean=Median: The data is symmetrical
• Mean>Median: The data is skewed to the right/posiKvely skewed

• Mean<Median: The data is skewed to the lea/negaKvely skewed

3

,• Variance: How variable your data is

o high = data not close to mean, more outliers
o low = data close to mean, few or no outliers
o Range also indicated variance

• Lower Quar?le: The 25% mark for the data

• Upper Quar?le: The 75% mark for the data

• Regression: Tells us about the relaKonship between data

o R/Mul?ple R = -1<R<1
§ R = 1 : perfect posiKve correlaKon
§ 0.5<R<1 : strong posiKve
§ 0<R<0.5 : weak posiKve correlaKon
§ -0.5<R<0 : weak negaKve correlaKon
§ -1<R<-0.5 : strong negaKve correlaKon
§ R = -1 : perfect negaKve correlaKon
§ Determine if its +/- by looking at slope (y=ax+b)
2
o R = Represents the proporKon of the dependent variable that can be
explained by the independent variable (relaKonship between 2 variables)

• Regression Line: y = a + bx
o a = y intercept
o b = slope – if posiKve slants up ; if negaKve slopes down
§ Posi?ve or nega?ve rela?onship

• Standard devia?on: How far out data deviates from mean

• Strays: Median ± 3(Median – Lower/Upper QuarKle)

• Outliers: Median ± 6(Median – Lower/Upper QuarKle)

o Greater the diﬀerence between the mean and median the more outliers we
have

• Interquar?le Range: Upper - Lower

• Lower fence: Q1−1.5 × IQR

• Upper fence: Q1+1.5 × IQR

4

, Module 3: Discrete and Continuous Random Variables

Probability Mass Functions (Discrete):

If given a PMF use a TABLE!!

It will be given as either Formula or a Table

The general format for the table
X X1 X2 X3 X4
P(X=x) P1 P2 P3 P4

• P1+P2+P3+P4 = 1

• The Expected Value E(X) is the sum of all the probabiliKes Kmes x à
X1p1 +x2p2 +x3p3 … (This is also known as the mean or average)

• The Variance is the sum of all the probabiliKes Kmes 𝑥 / minus the E(X) squared à
𝒙𝟐 1p1 + 𝒙𝟐 2p2 … - (E(X))2 (If they ask for standard devia?on just square root √ )

Linear Combination of Random Discrete Variables:

• E(ax +by) = aE(X) + bE(Y)

• E(ax -by) = aE(X) - bE(Y)

• E(b) = b

• Var(ax±by) = 𝒂𝟐 𝑽𝒂𝒓(𝑿) + 𝒃𝟐 𝑽𝒂𝒓(𝒀)

o Workout in variance and then convert to standard deviaKon.

o Either decide to square or not

o Sums: (NOT Square)
§ Independent variables tell you not to square
§ If you have to ﬁgure out formula you may not square variance
§ eg. 20R = 20 rusks

o Mul?ples: (DO Square)
§ When you need to mulKply(constant mulKplied by variable) you do
square
§ If you are given the formula it is mulKple so you square
§ eg. Kg = 0.45(W))

• Var(b) = 0

5

The benefits of buying summaries with Stuvia:

Guaranteed quality through customer reviews

Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.

Quick and easy check-out

You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.

Focus on what matters

Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!

Frequently asked questions

What do I get when I buy this document?

You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.

Satisfaction guarantee: how does it work?

Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.

Who am I buying these notes from?

Stuvia is a marketplace, so you are not buying this document from us, but from seller elizabeth6666. Stuvia facilitates payment to the seller.

Will I be stuck with a subscription?

No, you only buy these notes for $5.59. You're not tied to anything after your purchase.

Can Stuvia be trusted?

4.6 stars on Google & Trustpilot (+1000 reviews)

69411 documents were sold in the last 30 days

Founded in 2010, the go-to place to buy study notes for 15 years now

Start selling

Summary

STA1000S/F Course Notes Summary

Document information

Subjects

Written for

Seller

Reviews received

Content preview