Slides for different statistics tests., Lecture notes of Statistics

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Tests for Mean & Variance

Tests One Sample One sample z test One sample t test Two Samples Two sample z test Two sample t test Paired t test Two sample standard deviation More than 2 samples ANOVA

Errors of Statistical Tests

True State of Nature H 0 Is true H a Is true Conclusion Support H 0 / Reject H a Correct Conclusion Type II Error Support H a / Reject H 0 Type I Error Correct Conclusion (Power)

Significance Level

Level of Confidence / Confidence Interval: C = 0.90, 0.95, 0.99 (90%, 95%, 99%) Level of Significance: α = 1 – C (0.10, 0.05, 0.01)

Power

❖ Power = 1 – β (or 1 - type II error) ❖ Type II Error: Failing to reject null hypothesis when null hypothesis is false. ❖ Power: Likelihood of rejecting null hypothesis when null hypothesis is false. ❖ Or: Power is the ability of a test to correctly reject the null hypothesis.

p Value

❖ p value is the lowest value of alpha for which the null hypothesis can be rejected. (Probability that the null hypothesis is correct) ❖ If p = 0.01 you can reject the null hypothesis at α = 0. ❖ p is low the null must go / p is high the null fly.

Hypothesis Testing

1. State the Alternate Hypothesis.
2. State the Null Hypothesis.
3. Select a probability of error level (alpha level). Generally 0.
4. Select and compute the test statistic (e.g t or z score)
5. Critical test statistic
6. Interpret the results.

Hypothesis Testing

❖ Two Tail Tests ❖ H 0 : μ = 150cc ❖ H a : μ ≠ 150cc

Calculate Test Statistic

❖ Single sample ❖ z = (x- μ ) / σ ❖ Mean of Multiple samples ❖ z = (x̄ - μ) / ( σ / √n)

One Sample t Test

t critical = 3.

One Sample t Test

❖ Calculated value ❖ t = [x̄ - μ ] / [s / sqrt( n ) ] ❖ Example: Perfume bottle producing 150 cc, 4 bottles are randomly picked and the average volume was found to be be 151 cc and sd of sample was 2 cc. Has mean volume changed? ( 95 % confidence) ❖ t cal = ( 151 - 150 )/[ 2 / sqrt( 4 ) ] = 1 / 1 = 1 ❖ t critical = 3.182 > Fail to reject Ho

One Sample Chi Square

❖ For testing the population variance against a specified value σ

One Sample Chi Square

❖ Example: A sample of 25 bottles was selected. The variance of these 25 bottles as 5 cc. Has it increased from established 4 cc? 95% confidence level. ❖ X 2 = 24x5 / 4 = 30 ❖ What is critical value of Chi Square for 24 degrees of freedom?

One Sample Chi Square

❖ Example: A sample of 25 bottles was selected. The variance of these 25 bottles as 5 cc. Has it increased from established 4 cc? 95% confidence level. ❖ X 2 = 24x5 / 4 = 30 ❖ Critical value of Chi Square for 24 degrees of freedom = 36. ❖ Fail to reject H 0

Tests for M ean & Varianc e

Tests One Sample One sample z test One sample t test Two Samples Two sample z test Two sample t test Paired t test Two sample standard deviation More than 2 samples ANOVA