ECO 045 EXAM 3 Study Guide PDF

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ECO 045 EXAM 3 STUDY GUIDE5. Hypothesis Tests ● Hypothesis Test - formal procedure to answer a specificquestion about the value of population parameters ○ Possible errors in test results 𝐻𝑜 Accept𝐻𝑜 Reject𝐻𝑜True Correct Type I ErrorFalse Type II Error Correct ■ Type I Error - null hypothesis is true...

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ECO 045 EXAM 3 STUDY GUIDE 5. Hypothesis Tests ● Hypothesis Test - formal procedure to answer a specific question about the value of population parameters ○ Possible errors in test results 𝐻

Accept 𝐻

Reject 𝐻

True

Correct

Type I Error

False

Type II Error

Correct

𝑜

● ●

𝑜

■ Type I Error - null hypothesis is true, but we reject it ■ Type II Error - null hypothesis is false, but we do not reject it ■ Otherwise the test conclusion is correct Null Hypothesis (Ho) - a default assumption, typically being challenged with the analysis Alternative Hypothesis (Ha) - the logical opposite of the null hypothesis ○ Null Hypothesis and One-Sided Alternatives: ■ 𝐻 :µ ≥ µ , 𝐻 :µ < µ 0

■ ○

0

0

0

0

𝑎

0

Null Hypothesis and Two-Sided Alternatives: ■ 𝐻 :µ = µ ,𝐻 :µ ≠ µ 0

𝑎

0

Test Statistic - the statistic whose value is used to determine the outcome of the hypothesis test Level of significance ○

●

𝑎

𝐻 :µ ≤ µ ,𝐻 :µ > µ

0

● ●

𝑜

Probability of making Type I Error: 𝝰

■ Favors null hypothesis as default assumption ○ Probability of making Type II Error ■ Power of test: 𝞫 Critical Values - threshold in test statistic that is used to determine whether 𝐻 should be rejected 𝑜

●

●

Find Critical Values in the z-score: ○

One-sided alternative, where alternative is below: F(z*) = α

○

One-sided alternative, where alternative is above: 1 - F(z*) = α

○

Two-sided alternative: F(-z*) =

α 2

Find Critical Values in the t-score (t-distribution): (*use when the sample size is below 100 and σ is unknown*) ●

One-sided alternative, alternative is above null: ○ Table shows upper tail probability, P(...> t) ○

●

α

One-sided alternative, alternative is below null: ○ t-distribution is symmetric, P(...> t) = P(...< -t) ○

●

Find entry where DF=n-1 and probability =

Find entry where DF=n-1 and probability =

○ Use -t to find critical value Two-sided alternative:

α

○ ●

α

Find entry DF=n-1 and probability = 2

z-score for Hypothesis Test:

=

𝑥−µ0

○

z-score for Sample Mean: 𝑧

○

t-score for Sample Mean (use if n < 100 and σ is unknown: 𝑡 z-score for Sample Proportion:𝑧

○

σ𝑥

=

z-score for Difference in Means: 𝑧

○

=

𝑥−µ0 𝑠/ 𝑥

𝑝−𝑝0

=

σ𝑝 𝑥1 − 𝑥2 σ𝑥 − 𝑥 1

2 2

■

Standard Error of Difference in Means: σ

𝑥1− 𝑥2

=

𝑉(𝑥1 ) + 𝑉(𝑥2) =

Critical Value Approach 1. State null and alternative hypothesis 2. Specify level of significance (α = 0.01, 0.05, 0.10) 3. Designate test statistic (and data source) a. We focus on using the z-score (or t-score) as the test statistic 4. Determine critical values 5. Compute test statistic 6. Compare with critical values and conclude a. Reject 𝐻 : 0

i. z < -z* ii. z > +z* b. Do not reject 𝐻 (Accept 𝐻 ): 0

0

i. -z* < z < +z* p-Value Approach 1. State null and alternative hypothesis 2. Specify level of significance (α = 0.01, 0.05, 0.10) 3. Designate test statistic (and data source) a. We focus on using the z-score (or t-score) as the test statistic 4. Compute test statistic 5. Compute p-Value of test statistic 6. Compare with α and conclude a. Reject 𝐻 : p < α 0

b. Do not reject 𝐻 (Accept 𝐻 ): p ≥α 0

0

Goodness-of-Fit Test 1. State null and alternative hypothesis eg. 𝐻 : 𝑃 = 𝑥 , 𝑃 = 𝑦 , 𝑃 = 𝑧, etc. 0

𝐴

𝐵

𝐶

𝐻 : 𝑃 ≠ 𝑥, or 𝑃 ≠ 𝑦, or 𝑃 ≠ 𝑧,or etc. 𝑎

𝐴

𝐵

𝐶

σ1 𝑛

1

+

σ 𝑛

2 2

2

2. Specify level of significance (α = 0.01, 0.05, 0.10) (𝑓 −𝑒 )

2

𝑖

3. Designate test statistic: Chi-square distribution: χ = ∑

2

𝑖

𝑒

𝑖

𝑖

4. Determine Critical Value from Chi-square table (DF → row, α→ column) 5. Compute test statistic 6. Compare and conclude 2

2*

a. Reject 𝐻0: χ > χ

2

2*

b. Do not reject 𝐻0(Accept 𝐻0): χ ≤ χ 6. Regression Analysis ●

Sample Covariance: 𝑠

●

Correlation Coefficient:

𝑆𝑥𝑦

𝑟𝑥𝑦 =

𝑆𝑥𝑆𝑦

𝑥𝑦

=

Σ(𝑥𝑖−𝑥)(𝑦𝑖−𝑦) 𝑛−1 Σ(𝑥𝑖−𝑥)(𝑦𝑖−𝑦) 𝑛−1

= (

1 𝑛−1

2

1 𝑛−1

Σ(𝑥 −𝑥) ) 𝑖

2

2

𝑖

Regression model (population): 𝑦 = β + β 𝑥 + ϵ

●

Population regression line: 𝐸(𝑦|𝑥) = β + β 𝑥

2

2

2

[𝑛Σ𝑥 −(Σ𝑥) [𝑛Σ𝑦 −(Σ𝑦) ]

Σ(𝑦 −𝑦) )

●

0

𝑛(Σ𝑥𝑦)−(Σ𝑥)(Σ𝑦)

=

1

0

1

○ 𝐸(𝑦|𝑥) : expected value (mean) of y given x ○ β , β Parameters: 0

1

■

β : slope 1

■ β : intercept 0

● ●

Mean of 𝑦 if 𝑥 = 0

Sample regression line: 𝑦 = 𝑏 + 𝑏 𝑥 0

1

○ 𝑦 : predicted value of y given x ○ 𝑏 , 𝑏 : Parameter estimates 0

1

■ 𝑏 (slope estimate): 𝑏 1 1

=

Σ(𝑥𝑖−𝑥)(𝑦 −𝑦) 𝑖

2

Σ(𝑥 −𝑥)

𝑠𝑥𝑦

=

𝑠

2

𝑥

𝑖

■ 𝑏 (intercept estimate): 𝑏 = 𝑦 − 𝑏 𝑥 0

●

0

1

Sums of Squares: 2

○

Total Sum of Squares: TSS= Σ(𝑦𝑖 − 𝑦)

○

Sum of Squared Errors: SSE=Σ(𝑦𝑖 − 𝑦𝑖)

○

Sum of Squares due to Regression: SSR= Σ(𝑦𝑖 − 𝑦)

○

TSS = SSR + SSE

2 2

2

𝑆𝑆𝑅 𝑇𝑆𝑆

= 1−

𝑆𝑆𝐸 𝑇𝑆𝑆

2

= (𝑟𝑥𝑦)

●

Coefficient of Determination (measure of fit): 𝑟 =

●

Testing for the significance of β : Can use critical value or p-value approach 1

“statistical significance” - statistical evidence that a parameter is nonzero at significance level α ○ Hypotheses: 𝐻0:β1 = 0 𝐻𝑎:β1 ≠ 0

○

z-score for Significance Test (test statistic): 𝑧

■

Standard error of 𝑏 (𝑠 ): 1

●

𝑏1

𝑏1 𝑠

=

𝑏1

𝑠

2

ϵ

𝑠𝑏 =

2

Σ(𝑥𝑖−𝑥)

1

depends on the size of error terms vs. sample size and amount of variation in x 2

●

2

𝑠ϵ =

Σ(𝑦𝑖−𝑦𝑖)

𝑆𝑆𝐸 𝑛−2

=

𝑛−2

2

2

● Σ(𝑥𝑖 − 𝑥) = (𝑛 − 1)𝑠𝑥 ■

Sampling distribution of 𝑏1: 𝑏

1

●

●

=

𝑠𝑥𝑦 2

𝑠𝑥

2

𝑏 ~ 𝑁𝑜𝑟𝑚𝑎𝑙 (β , 𝑠 ) 1

1

𝑏1

○

Mean: β

○

Variance: 𝑠

1 2

𝑏1

Key Assumptions for Regression Analysis ○ Error terms ϵ are independent of 𝑥 ○ There is variation in 𝑥: 𝑉(𝑥) > 0 ○

2

In small samples (n...