ECON1203 Business and Economic Statistics Notes PDF

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NOTESECON1203: Business and EconomicsStatisticsUniversity of New South WalesTable of Contents 1 INTRODUCTION TO STATISTICS 1 BASIC TERMS 1 TYPES OF VARIABLES 2 MEASURES OF LOCATION, VARIATION AND ASSOCIATION 2 MEASURES OF CENTRAL TENDENCY 2 MEASURES OF VARIABILITY 2 MEASURES OF RELATIVE STANDING 2 M...

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NOTES ECON1203: Business and Economics Statistics University of New South Wales

Table of Contents 1

INTRODUCTION TO STATISTICS

1.1 1.2 2

BASIC TERMS TYPES OF VARIABLES MEASURES OF LOCATION, VARIATION AND ASSOCIATION

2.1 2.2 2.3 2.4 3

MEASURES OF CENTRAL TENDENCY MEASURES OF VARIABILITY MEASURES OF RELATIVE STANDING MEASURES OF ASSOCIATION DATA SAMPLING

3.1 3.2 4

TYPES OF DATA STEPS IN STATISTICAL ANALYSIS PROBABILITY

4.1 4.2 4.3 4.4 4.5 5

CONDITIONAL PROBABILITY JOINT PROBABILITY MUTUALLY EXCLUSIVE INDEPENDENCE MARGINAL PROBABILITY RANDOM VARIABLES

5.1 5.2 5.3 5.4 5.4.1 5.4.2 5.4.3 6

BINOMIAL DISTRIBUTION

6.1 6.2 6.3 7

ASSUMPTIONS PROBABILITY DENSITY FUNCTION UNIFORM RANDOM VARIABLES NORMAL DISTRIBUTION

7.1 7.2 7.2.1 7.3 7.4 7.4.1 8 8.1

TYPES OF RANDOM VARIABLES ASSOCIATED PROBABILITY CUMULATIVE DISTRIBUTION FUNCTIONS MEAN AND VARIANCE IN PROBABILITY RULES OF EXPECTATION COVARIANCE BETWEEN 2 DISCRETE RANDOM VARIABLES RULES OF ASSOCIATION

4 4 4 5 5 5 5 5 7 7 7 8 8 8 8 8 8 9 9 9 9 9 9 10 10 11 11 11 11 12

CHARACTERISTICS STANDARDISATION STEPS FOR STANDARDISING NORMAL DISTRIBUTION AND BINOMIAL DISTRIBUTION ESTIMATORS DESIRED PROPERTIES OF GOOD ESTIMATORS:

12 12 13 13 13 13

SAMPLE DISTRIBUTION OF THE SAMPLE MEAN

14

SAMPLING DISTRIBUTION

14

8.2 8.3 9

CENTRAL LIMIT THEOREM INTERVAL ESTIMATION

14 14

HYPOTHESIS TESTING

16

9.1 9.2 9.3

TYPE 1 ERROR STEPS IN HYPOTHESIS TESTING (REJECTION REGION APPROACH) STEPS IN HYPOTHESIS TESTING (P-VALUE APPROACH)

16 16 16

10 TYPE 2 ERROR AND T-STATISTIC

18

10.1 10.2

18 18

TYPES OF ERRORS STEPS FOR T-STATISTICS HYPOTHESIS TEST

11 SIMPLE LINEAR REGRESSION

20

11.1 11.2 11.3 11.4

20 20 20 20

RESIDUAL SUM OF SQUARES ASSUMPTIONS OF LINEAR REGRESSION SUM OF SQUARES DUMMY VARIABLES

12 MULTIPLE REGRESSION

21

12.1 12.2

21 21

INTRODUCTION AND INTERCEPTS ADJUSTED R SQUARE

13 CHI-SQUARED

22

13.1 13.2 13.3 13.4

22 22 23 23

INTRODUCTION TO CHI-SQUARE HYPOTHESIS TEST FOR CHI-SQUARED GOODNESS OF FIT HYPOTHESIS TEST INDEPENDENCE

1 Introduction to Statistics 1.1 Basic terms     

Population: Collection of everything in the universe with regards to the variable Parameter: A characteristic of the population Sample: Small group of things taken from the population Statistic: A characteristic of the sample Variable: Is a characteristic that we get data on

1.2 Types of Variables        

Discrete (cannot include every possible observation within sample space) Continuous (can include all possible observations within sample space) Qualitative (in terms of words, e.g. categories) Quantitative (in terms of numbers) Ordinal data (qualitative data with rankings) Time series data (references data with time) Descriptive statistics (summarise data in terms of numerical values) Inferential statistics (gives inferences about the populations)

When analysing histograms consider:  Symmetry (how well the right side reflects the left side),  Skewness (positive skew = tail to the right, negative skew= tail to the left)  Modal classes (most frequent class)  Number of modal classes.

2 Measures of Location, Variation and Association 2.1

Measures of Central Tendency



Mean = Average of all observations. ∑𝑁 𝑖=1 𝑥𝑖 Population mean = 𝑁 ∑𝑛𝑖=1 𝑥𝑖 Sample mean = 𝑛



Median= Middle ranking score of all observations, if there are an even number of observations, find the average of the 2 middle ranking observations. Mode= Most frequently occurring observation



2.2 Measures of Variability 1) Range= Maximum – minimum

2) Population Variance = 𝜎 3) Sample Variance = 𝑠 2

2

=

=

𝑁

∑𝑖=1 (𝑥𝑖 −μ) 2

𝑁 ∑ 𝑛𝑖=1 (𝑥𝑖 −𝑥 ) 2

𝑛 4) Population Standard deviation = 𝜎 = √𝜎 2 5) Sample standard deviation = s = √𝑠2

6) Sample co-efficient of variation (measures relative variability) = cv =

2.3 Measures of Relative Standing

𝑠 𝑥

1) Percentile = the value for which P% of observations are lower than the current one, i.e. 90th percentile means it is greater than 90% of scores. 2) Inter-quartile range = 75th percentile - 25th percentile

2.4 Measures of Association

1) Population Co-variance: 𝜎𝑥𝑦

=

∑𝑁 𝑖=1(𝑥𝑖 −𝜇𝑥 )(𝑦𝑖 −𝜇𝑦 )

𝑁 ∑ 𝑛𝑖=1(𝑥𝑖 −𝑥 )(𝑦𝑖 −𝑦)

2) Sample Covariance: 𝑠𝑥𝑦 = 𝑛 Positive covariance indicates a positive linear association, negative covariance indicates an inverse or negative linear relationship, 0 covariance indicates no linear association. 3) Correlation Co-efficient: Standardised measure of association that has a range between -1 and 1. If it is negative there is a negative relationship (inverse) if the co-efficient is positive there is a positive relationship. The closer values get to either 1 or -1 the more linear the relationship. 𝜎𝑥𝑦 Population correlation: 𝜌 = 𝜎𝑥 𝜎𝑦 𝑠𝑥𝑦 Sample correlation: 𝑟 = 𝑠𝑥 𝑠𝑦

Where: -1< 𝜌, 𝑟 A) = 𝑃(

 



  

𝐴−𝜇 𝑋−𝜇 𝐵−𝜇 ) < < 𝜎 𝜎 𝜎

𝐵−𝜇 𝐴−𝜇 < 𝑍< ) = 𝑃( 𝜎 𝜎 𝐵−𝜇 𝐴−𝜇 use the Z-table. and To find the values for 𝜎 𝜎 If A becomes negative, i.e. –A, remember the value of: −𝐴−𝜇 𝐵−𝜇 𝑃( < 𝑍< ) 𝜎 𝜎 𝐵−𝜇 −𝐴−𝜇 < 𝑍 < 0 ) + 𝑃 (0 < 𝑍 < ) = 𝑃( 𝜎 𝜎

According to the symmetry quality of a normal distribution curve, should be equal 𝐴−𝜇 𝐵−𝜇 to: 𝑃 ( < 𝑍 < 0 ) + 𝑃 (0 < 𝑍 < ) 𝜎 𝜎

When un-standardising to get back to the native distribution use the formula: 𝑍 × 𝜎+𝜇 =𝑋 What does 𝑧𝛼 mean? 𝑧𝛼 represents the Z-score that is at the 100(1-𝛼)th percentile. Therefore 𝑧0.1 would mean that only 10% of Z-scores are higher than it, and therefore it is larger than 90% of z-scores.

7.2.1 Steps for Standardising 1) Write down the notation and information that you know: 𝑋~𝑁(𝜇, 𝜎 2 ), therefore find values for the population mean 𝜇, the population variance 𝜎 2 and the population standard deviation 𝜎. These values will be used to compute Z scores. 2) Write down what we want to know: E.g. 𝑃 (𝐴 < 𝑋 < 𝐵) 3) Standardise the equation: 𝐴−𝜇 𝑃(𝐴 < 𝑋 < 𝐵) = 𝑃 ( 𝜎

<

𝑋−𝜇 𝜎

<

𝐵−𝜇 𝜎

)

4) Break down our equation if it the Z-score is negative (remembering the symmetry quality of normal distribution), and mark out the Z-scores on a normal distribution graph. Shading the area we want the probability of. 5) Use the standard normal distribution table to workout the probabilities

7.3 Normal Distribution and Binomial Distribution 

𝜇 = 𝐸(𝑋) = 𝑛𝑝 𝜎 2 = 𝑉𝑎𝑟(𝑋) = 𝑛𝑝(1 − 𝑝) Since binomial distributions are discrete random variables we use a continuity correction of -0.5 to the lower bound of ‘x’ and +0.5 to the upper bound of ‘x’. This is because 𝑃 (𝑋 = 10) in a binomial distribution would be a positive value compared 𝑃 (𝑋 = 10) in a normal distribution, which we know, being a continuous random variable, would be 0.

7.4 Estimators 

Is a Statistic that tries to estimate the parameter of the population (unknown)

 Point estimator = statistic that estimates parameter with a single number Interval estimator = statistic that estimates parameter with an interval with a degree of confidence. 7.4.1 Desired properties of Good Estimators:  Unbiased- on average the estimator should equal the parameter  Consistent – As the sample size increases the difference between the estimator and the parameter should decrease  Relative efficiency- if multiple estimators are unbiased choose the one estimator with the lowest variance.

8 Sample Distribution of the Sample Mean 8.1 Sampling Distribution 

In practice population parameters 𝜇 and 𝜎 2 are difficult to find or unknown, so



Sample mean:



 

instead we take many samples and use the sample means, 𝑥 , and sample variances, 𝑠 2 as estimators of population parameters. Therefore the key differences between the sample statistic and our population parameter is sampling error which is a natural error caused by having small sample sizes (reduced by increasing sample size), and conventional mistakes and misunderstandings. ∑𝑋 𝑋 = 𝑖; E (𝑋) = 𝜇 𝑛

𝜎2 Sample Variance: 𝑉𝑎𝑟 (𝑋) = 𝑛 ; therefore the standard error/ sample standard 𝜎 deviation of 𝑋 would be 𝑠𝑒 (𝑋) = √𝑛 𝜎 Use these values to convert: 𝑋~𝑁 (𝜇, 𝜎 2 ) into 𝑋~N (𝜇, ) 𝑛 2

𝑋 − 𝜇 𝜎 √𝑛

Therefore the standardise, into Z-scores the formula would look like this:

𝑍=

8.2 Central Limit Theorem 

When sampling from a population the sample observations will be assumed to be normally distributed if the sample size is sufficiently large, generally when 𝑛 ≥ 30.

8.3 Interval Estimation 

Provide interval estimation with a given level of confidence attached to it. Confidence intervals: confidence level = 1 – 𝛼 (𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑐𝑒 𝑙𝑒𝑣𝑒𝑙). 𝑋 −𝜇

 𝑃(−𝑧𝛼 ≤ 



0.1

2

2

because the confidence level is based on a two-tail test. So an 𝛼 = 2 would mean the lower bound would be at −𝑧0.05 and the upper bounds at

We use 𝑧0.05 .

𝛼

≤ 𝑧𝛼 ) = 1 – 𝛼

𝜎 √𝑛

 𝑃 (𝑋 − 𝑧𝛼 × 2

≤ 𝜇 ≤ 𝑋 + 𝑧𝛼 × 𝑛

𝜎

√

2

𝜎

√𝑛

)=1−𝛼

  

𝑀𝑎𝑟𝑔𝑖𝑛𝑠 𝑜𝑓 𝑒𝑟𝑟𝑜𝑟 = ±𝑧𝛼 √𝜎𝑛 × If you’re given the margin2 of error as ‘C’, the sample size required ‘n’ would just be:

𝑛=(

𝑧𝑎 × 𝜎 2 2

𝐶

)

9

Hypothesis Testing

9.1 Type 1 Error 

Type 1 error = the probability of rejecting an inherently true null hypothesis, denoted by the significance level, 𝛼.

9.2 Steps in Hypothesis Testing (Rejection Region Approach) 1. Propose Hypothesis and write down the information given: - 𝑯𝟎 : 𝑵𝒖𝒍𝒍 𝒉𝒚𝒑𝒐𝒕𝒉𝒆𝒔𝒊𝒔(𝒂𝒔𝒔𝒖𝒎𝒆𝒅 𝒔𝒕𝒂𝒕𝒆 𝒐𝒇 𝒂𝒇𝒇𝒂𝒊𝒓𝒔) - 𝑯𝟏 : 𝑨𝒍𝒕𝒆𝒓𝒏𝒂𝒕𝒆 𝒉𝒚𝒑𝒐𝒕𝒉𝒆𝒔𝒊𝒔(𝒕𝒉𝒆 𝒕𝒉𝒊𝒏𝒈 𝒚𝒐𝒖 𝒘𝒂𝒏𝒕 𝒕𝒐 𝒑𝒓𝒐𝒗𝒆) 𝜎2 - 𝑋~𝑁(𝜇, 𝜎 2 ) into 𝑋 ~N(𝜇, ) 𝑛 2. Set the Significance level (usually given) Find out what 𝛼 equals

3. Determine the associated Z-score for the rejection region, or the  𝑳 is the critical value, and we solve for 𝑿 𝑳. ‘critical value’ given 𝜶. 𝑿

For a one tail test: 𝑧𝛼

−𝑧𝛼 =

For a two tail test: Lower bounds:

=

2

𝑋 𝐿 −𝜇 𝜎

√𝑛

𝑋 𝐿 −𝜇 𝜎 √𝑛

;

Upper bounds: 𝑧𝛼

2

=

𝑋 𝐿 −𝜇 𝜎 √𝑛

4. Determine the Z-score for sample distribution mean observed, our ‘test statistic’: 𝑋 −𝜇

𝑧=

𝜎

√𝑛

5. Make a decision, reject or fail to reject the null hypothesis: General rule: if, |𝑡𝑒𝑠𝑡 𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐| > |𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑣𝑎𝑙𝑢𝑒|

= |𝑧𝑠𝑡𝑎𝑡 | > |𝑧𝑐𝑟𝑖𝑡 | Then we reject the null hypothesis in favour of the alternate, because the test statistic would be more extreme than the critical value.

6. Conclusion: the conclusion should say: - Whether you reject or fail to reject the null - Relate back to the question at hand.

9.3 Steps in Hypothesis Testing (P-Value Approach) 1) Propose Hypothesis and write down the information given: - 𝑯𝟎 : 𝑵𝒖𝒍𝒍 𝒉𝒚𝒑𝒐𝒕𝒉𝒆𝒔𝒊𝒔(𝒂𝒔𝒔𝒖𝒎𝒆𝒅 𝒔𝒕𝒂𝒕𝒆 𝒐𝒇 𝒂𝒇𝒇𝒂𝒊𝒓𝒔) - 𝑯𝟏 : 𝑨𝒍𝒕𝒆𝒓𝒏𝒂𝒕𝒆 𝒉𝒚𝒑𝒐𝒕𝒉𝒆𝒔𝒊𝒔(𝒕𝒉𝒆 𝒕𝒉𝒊𝒏𝒈 𝒚𝒐𝒖 𝒘𝒂𝒏𝒕 𝒕𝒐 𝒑𝒓𝒐𝒗𝒆)

-

𝜎2 ) 𝑋~𝑁(𝜇, 𝜎 2 ) into 𝑋 ~N(𝜇, 𝑛

2) Set the Significance level (usually given) Find out what 𝛼 equals

3) Determine the associated Z-score for the rejection region, or the ‘critical  𝑳 is the critical value, and we solve for 𝑿 𝑳. value’ given 𝜶. 𝑿

𝑧𝛼 =

𝑋 𝐿 −𝜇 𝜎 √𝑛

4) P-value of test statistic; solve for z 𝑋 −𝜇 𝑧𝑡𝑒𝑠𝑡 = 𝜎 √𝑛

Then find: 𝑃 (𝑍 > 𝑧𝑡𝑒𝑠𝑡 ) = use the table, if 𝑧𝑡𝑒𝑠𝑡 > 0 Find: 𝑃 (−𝑧𝑡𝑒𝑠𝑡 > −𝑍) = use the table, if 𝑧𝑡𝑒𝑠𝑡 < 0 5) Make a decision, reject or fail to reject the null hypothesis: General rule: Reject if: Significance level > P-value of test statistic = 𝛼 > 𝑃 (𝑍 > 𝑧_𝑡𝑒𝑠𝑡) 𝑜𝑟 𝑃 (−𝑧_𝑡𝑒𝑠𝑡 > −𝑍) 6) Conclusion: the conclusion should say: - Whether you reject or fail to reject the null - Relate back to the question at hand.

10 Type 2 Error and T-Statistic 10.1 Types of Errors  

Type 1 error= the probability of rejecting an inherently true null hypothesis, denoted by the significance level, 𝛼. Type 2 error = the probability of not rejecting an inherently false null hypothesis, denoted by 𝛽.

Changing 𝛼 for a given fixed alternative will change the value of 𝛽. Since 𝛽 = 𝑃 (𝑁𝑜𝑡 𝑟𝑒𝑗𝑒𝑐𝑡 𝐻0 𝑙 𝐻0 𝑓𝑎𝑙𝑠𝑒)

How to determine 𝜷? If the decision rule for the hypothesis test was to reject if 𝑋𝐿 > 10, we would not reject 𝐻0 when 𝑋𝐿 < 10. If then, we knew 𝐻0 was false and the true 𝜇 was 11, 𝛽 would be: 𝛽 = 𝑃(𝑋𝐿 < 10 / 𝜇 = 11)

= 𝑃 (𝑍 <

10−11 𝜎 √𝑛

)

Power of a test = 𝑃 (𝑅𝑒𝑗𝑒𝑐𝑡 𝐻0 𝑙 𝐻0 𝑓𝑎𝑙𝑠𝑒) = 1 − 𝛽 Generally when: - 𝑖𝑓 𝛼 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑠, 𝛽 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑒𝑠 - 𝑖𝑓 𝛼 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑠, 𝑝𝑜𝑤𝑒𝑟 𝑜𝑓 𝑎 𝑡𝑒𝑠𝑡 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒𝑠 In practice we usually don’t know the population variance or the population standard deviation. In large sample sizes we can invoke the CLT and replace the unknown 𝜎 with ‘s’. In large samples ‘s’ becomes a consistent estimator of 𝜎, when sample size is small we use the t-statistic. 𝑋−𝜇 𝑡 = 𝑠 ~ 𝑡𝑛−1 , we can no longer use Z-scores and normal distribution, must √𝑛

use a t-distribution with 𝑣 = n-1 degrees of freedom. The t-distribution generally has ‘fatter’ tails but as the sample size increases the distribution tends to become more normal. Confidence interval: 𝑠 < 𝜇 < 𝑋 + 𝑡𝑎 × 𝑃 (𝑋 − 𝑡𝑎 ,𝑛−1 2

√𝑛

,𝑛−1 2

×

𝑠 ) √𝑛

=1−𝛼

10.2 Steps for T-statistics Hypothesis Test

1. Propose hypotheses and write down the information given: - 𝑯𝟎 : 𝑵𝒖𝒍𝒍 𝒉𝒚𝒑𝒐𝒕𝒉𝒆𝒔𝒊𝒔(𝒂𝒔𝒔𝒖𝒎𝒆𝒅 𝒔𝒕𝒂𝒕𝒆 𝒐𝒇 𝒂𝒇𝒇𝒂𝒊𝒓𝒔) - 𝑯𝟏 : 𝑨𝒍𝒕𝒆𝒓𝒏𝒂𝒕𝒆 𝒉𝒚𝒑𝒐𝒕𝒉𝒆𝒔𝒊𝒔(𝒕𝒉𝒆 𝒕𝒉𝒊𝒏𝒈 𝒚𝒐𝒖 𝒘𝒂𝒏𝒕 𝒕𝒐 𝒑𝒓𝒐𝒗𝒆) - X~𝑡𝑛−1(𝜇, 𝑠 2 ), n =? 2. Determine the significance level: Find what 𝛼 equals to

3. Find the critical value: 𝑡𝐶𝑟𝑖𝑡 = 𝑡𝛼,𝑣 = 𝑣𝑎𝑙𝑢𝑒 𝑔𝑖𝑣𝑒𝑛 𝑖𝑛 𝑡ℎ𝑒 𝑡 − 𝑡𝑎𝑏𝑙𝑒 *Note: the column will have different values for a one-tail and two-tail test

4. Find the t-statistic: 𝑋 −𝜇 𝑡𝑠𝑡𝑎𝑡 = 𝑠 √𝑛

5. Decision rule: Reject the null hypothesis if: |𝑡𝑠𝑡𝑎𝑡 | > |𝑡𝑐𝑟𝑖𝑡 | 6. Conclusion: - Whether you reject or fail to reject the null - Relate back to the question at hand. Estimating population proportion: 𝑋 𝑝(1−𝑝) 𝑝 = 𝑡ℎ𝑒𝑛 𝑝 ~𝑁(𝑝, ) 𝑛 𝑛

𝑍=

Therefore: 𝑝−𝑝 √

𝑝(1−𝑝) 𝑛

11 Simple Linear Regression 11.1 Residual Sum of Squares 

      

In simple regression we try to draw a line of best fit for a bivariate relationship (X and Y), which aims to minimises the residual sum of squares; OLS (ordinary least squares) model. It predicts the line of best fit with the formula: 𝑖 = 𝑏0 + 𝑏1 𝑋𝑖 𝑌

 − 𝑏1 𝑋 and 𝑏1 = 𝐶𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑋 𝑎𝑛𝑑 𝑌 = Where: 𝑏0 = 𝑌 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑋

𝑆𝑥𝑦

𝑠𝑥2

=

∑𝑛𝑖=1 (𝑋𝑖 −𝑋 )(𝑌𝑖 −𝑌) ∑ 𝑛𝑖=1 (𝑋𝑖 −𝑋 )2

This is in comparison to the population regression relationship: 𝑌𝑖 = 𝛽0 + 𝛽1 𝑋𝑖 + 𝜀𝑖 OLS model will only predict with: 𝑌𝑖 = 𝑏0 + 𝑏1 𝑋𝑖 + 𝑒𝑖 𝛽1 𝑎𝑛𝑑 𝑏1 show the marginal effect of independent variable 𝑋𝑖 on 𝑌𝑖

11.2 Assumptions of Linear Regression      



Model is linear Random sampling There is some sample variation in X where values of X are not all the same The error term and the independent variable are not related Error term has a variance of 𝜎 2 = homoscedasticity Error terms are not related to each other Error terms are normally distributed with a mean of 0 and variance of 𝜎 2

11.3 Sum of Squares   

 



 )2 = SSR + SSE = total sum of square deviations of each 𝑌𝑖 from SST= ∑(𝑌𝑖 − 𝑌  the mean, 𝑌 2  SSR= ∑(𝑌𝑖 − 𝑌 ) = sum of square deviation of each predicted 𝑌𝑖 (by model) from  the mean, 𝑌

SSE= ∑ 𝑒𝑖 2 = sum of square error, parts unexplained by the model. 𝑆𝑆𝐸 SEE= estimator of 𝜎 = 𝑛−2 𝑅 2 = 𝑐𝑜 − 𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑖𝑜𝑛 = 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑠 𝑡ℎ𝑒 % 𝑜𝑓 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑌 𝑎𝑠 𝑒𝑥𝑝𝑙𝑎𝑖𝑛𝑒𝑑 𝑏𝑦 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 𝑖𝑛 𝑋.

𝑑𝑜𝑚𝑎𝑖𝑛: 0 ≤ 𝑅 2 ≤ 1

11.4 Dummy Variables 

Represent categorical variables, can only take the value of ‘1’ or ‘0’ to show the presence or absence of the variable.

12 Multiple Regression 12.1 Introduction and Intercepts      



Multiple Regression (more than 1 independent/ explanatory variable) 𝑒. 𝑔. 𝑤𝑖𝑡ℎ 2 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠 𝑌𝑖 = 𝛽0 + 𝛽1 𝑋1𝑖 + 𝛽2 𝑋2𝑖 + 𝜖𝑖 𝛽1 shows the marginal effect of a change in 𝑋1 on 𝑌𝑖 assuming 𝑋2 is held constant. 𝛽2 shows the marginal effect of a change in 𝑋2 on 𝑌𝑖 assuming 𝑋1 is held constant. The multiple regression prediction model would be: 𝑖 = 𝑏0 + 𝑏1 𝑋1𝑖 + 𝑏2 𝑋2𝑖 𝑌 On top of all the previous regression model assumptions, there are some additional one that apply to multiple regression: o Independent variables cannot have an exact linear relationship As long as 𝛽 is in the form of a linear function, there is a linear regression model, even if the independent variable is not a linear function.

12.2 Adjusted R Square 



The problem with the conventional 𝑅 2 is that adding additional variables will never cause 𝑅 2 to decrease though it may a poor explanatory variable. This is why we use ‘adjusted 𝑅 2’ for multiple regression that allow for increases of decreases in ‘adjusted 𝑅 2’ when explanatory variable are added.

Strong correlation, i.e. a high 𝑅 2 or ‘adjusted 𝑅 2’ does not always indicate causation (X is causing the change in Y). It does require common sense to consider whether or not it is feasible to state X causes variation in Y. There could be confounding factors that are overlooked.

13 Chi-Squared 13.1 Introduction to Chi-Square      

We use the chi-square to compare variability, goodness of fit and test independence of nominal variables in contingency tables Chi-square (𝜒 2 ) distribution is: Positively skewed 𝜒2 > 0 It has n-1 degrees of freedom= 𝑣 As 𝑣 increases the more symmetrical, and hence less skewed the distribution looks.

13.2 Hypothesis Test for Chi-Squared

1. Propose hypotheses: - 𝑯𝟎 : 𝑵𝒖𝒍𝒍 𝒉𝒚𝒑𝒐𝒕𝒉𝒆𝒔𝒊𝒔(𝒂𝒔𝒔𝒖𝒎𝒆𝒅 𝒔𝒕𝒂𝒕𝒆 𝒐𝒇 𝒂𝒇𝒇𝒂𝒊𝒓𝒔) E.g. 𝝈𝟐 = 𝒔𝒐𝒎𝒆𝒕𝒉𝒊𝒏𝒈 - 𝑯𝟏 : 𝑨𝒍𝒕𝒆𝒓𝒏𝒂𝒕𝒆 𝒉𝒚𝒑𝒐𝒕𝒉𝒆𝒔𝒊𝒔(𝒕𝒉𝒆 𝒕𝒉𝒊𝒏𝒈 𝒚𝒐𝒖 𝒘𝒂𝒏𝒕 𝒕𝒐 𝒑𝒓𝒐𝒗𝒆) E.g. 𝝈𝟐 ≠ 𝒔𝒐𝒎𝒆𝒕𝒉𝒊𝒏𝒈 for a two-tail test or 𝝈𝟐 𝒔𝒐𝒎𝒆𝒕𝒉𝒊𝒏𝒈 for a one-tail test 2. Find the significance level: 𝛼 Find what 𝛼 equals to, if it’s a two-tail test we use 2

3. Find the critical value/ rejection region: For two tail test: 𝑙𝑜𝑤𝑒𝑟 𝜒 2 𝐶𝑟𝑖𝑡 = 𝜒 21−𝛼 ,𝑛−1 = 𝑢𝑠𝑒 𝑡𝑎𝑏𝑙𝑒 𝑡𝑜 𝑓𝑖𝑛𝑑 𝑣𝑎𝑙𝑢𝑒 2

𝑈𝑝𝑝𝑒𝑟 𝜒 2 𝐶𝑟𝑖𝑡 = 𝜒 2 𝛼 ,𝑛−1 = 𝑢𝑠𝑒 𝑡𝑎𝑏𝑙𝑒 𝑡𝑜 𝑓𝑖𝑛𝑑 𝑣𝑎𝑙𝑢𝑒 2

4. Find the test statistic: 𝜒 2 𝑆𝑡𝑎𝑡 =

(𝑛−1)𝑠2 𝜎2

5. Decision rule: One way: 𝜒 2 𝑆𝑡𝑎𝑡 > 𝜒 2 𝐶𝑟𝑖𝑡 we reject the null hypothesis Two way: 𝜒 2 𝑆𝑡𝑎𝑡 < 𝐿𝑜𝑤𝑒𝑟 𝜒2 𝐶𝑟𝑖𝑡 or if 𝜒 2 𝑆𝑡𝑎𝑡 > 𝑈𝑝𝑝𝑒𝑟 𝜒 2 𝐶𝑟𝑖𝑡 6. Conclusion: - Whether you reject or fail to reject the null - Relate back to the question at hand. Confidence interval: 𝑃 ( (𝑛 − 1)𝑠 2 ×

𝜒2

1

𝛼 ,𝑛−1 2

1−

< 𝜎 2 < (𝑛 − 1)𝑠 2...