# Hypothesis Testing & Confidence Intervals

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Hypothesis Testing & Confidence Intervals

## 1. p Value

### 1.1. What?

1.1.1. p value = P(X-bar >= Sample mean value) under H0

### 1.2. So What?

1.2.1. Probability of making the Type 1 error

1.2.1.1. Rejecting the H0 when it is TRUE

1.2.2. Used to reject the H 0

1.2.3. Gives indication of statistical significance

1.2.3.1. if statistically strong (5%), reject H0 and Accept H(a)

1.2.3.2. Levels

1.2.3.2.1. <5% (0.05)

1.2.3.2.2. <1% (0.01)

1.2.3.2.3. <10%

1.2.3.3. How to set the level of signicance

1.2.3.3.1. Seriousness of the Type 1 Errors

1.2.3.3.2. Size of remediation/recovery efforts

## 2. Hypothesis

### 2.1. Types

2.1.1. Null Hypothesis

2.1.1.1. What?

2.1.1.1.1. claim that begs evidence AGAINST

2.1.1.1.2. test for "="

2.1.1.2. So What?

2.1.2. Alternative Hypothesis

2.1.2.1. What?

2.1.2.1.1. what we what to prove

2.1.2.1.2. type

### 2.2. HT Errors

2.2.1. Type 1

2.2.1.1. Reject H0 when it is TRUE

2.2.2. Type 2

2.2.2.1. NOT reject when it FALSE

### 2.3. Assumptions

2.3.1. Sample is normally distributed (spl, n<30)

2.3.1.1. with μ

2.3.1.2. and σ

## 3. t-distribution

### 3.1. What?

3.1.1. another form of normal dist => bell

3.1.2. based on CLT => inference of population based on the sample

3.1.3. T=(x bar - μ) / (s√n)

3.1.3.1. s => sample sd not pop sd

3.1.3.2. s vs σ is the difference

3.1.3.2.1. higher standard deviation then Std.ND => higher spread => higher uncertainty

3.1.4. Excel

3.1.4.1. t > 0

3.1.4.2. Left tailed prob

3.1.4.2.1. T.DIST

3.1.4.3. 2 Tailed prob

3.1.4.3.1. T.DIST.2T

3.1.4.3.2. Always TWICE the value of one sided

3.1.4.4. Right tailed prob

3.1.4.4.1. T.DIST.RT

3.1.4.5. When?

3.1.4.5.1. when done manually, StatTools does it auto.

### 3.2. Why?

3.2.1. to perform HT for the pop mean μ when pop sd σ is unknown & estimated by sample sd S

3.2.1.1. NO σ NO Z score

3.2.1.1.1. I.e. in practice is Z score is not that useful

### 3.4. So What (t dist)

3.4.1. when the n<30. t dist. replaces normal dist.

## 5. Unit

### 5.1. So What

5.1.1. HT => Let the data speak

5.1.1.1. Establish a tentative claim

5.1.1.2. Test the validity of the clain

5.1.1.3. Accept/Reject/Adjust the claim

5.1.2. CI => So what's plausible and how confident are you?

### 5.2. Outcomes

5.2.1. hypothesis testing jargon in a single sample of data

5.2.2. 2 decision making errors w/ hypotheses

5.2.3. hypothesis test & confidence intervals with μ when σ=given

5.2.4. hypothesis test & confidence intervals with μ when σ=?

5.2.5. interpret(key) p-value, the