Hypothesis Testing (Statistical Significance)1Hypo.docx
- 1. Hypothesis Testing (Statistical Significance) 1 Hypothesis Testing Goal: Make statement(s) regarding unknown population parameter values based on sample data Elements of a hypothesis test: Null hypothesis - Statement regarding the value(s) of unknown parameter(s). Typically will imply no association between explanatory and response variables in our applications (will always contain an equality) Alternative hypothesis - Statement contradictory to the null hypothesis (will always contain an inequality) The level of significant (Alpha) is the maximum probability of committing a type I error. P(type I error)= alpha Definitions Rejection (alpha, α) Region: Represents area under the curve that is used to reject the null hypothesis Level of Confidence, 1 - alpha (a): Also known as fail to reject (FTR) region Represents area under the curve that is used to fail to reject the null hypothesis
- 2. FTR H0 α/2 α/2 3 1 vs. 2 Sided Tests Two-sided test No a priori reason 1 group should have stronger effect Used for most tests Example H0: μ1 = μ2 HA: μ1 ≠ μ2 One-sided test Specific interest in only one direction Not scientifically relevant/interesting if reverse situation true Example H0: μ1 ≤ μ2 HA: μ1 > μ2 4 Example: It is believed that the mean age of smokers in San Bernardino is 47. Researchers from LLU believe that the average age is different than 47. Hypothesis H0:μ = 47 HA: μ ≠ 47
- 3. μ = 47 α /2 = 0.025 Fail to Reject (FTR) α /2 = 0.025 5 Three Approaches to Reject or Fail to Reject A Null Hypothesis: 1a. Confidence interval Calculate the confidence interval Decision Rule: a. If the confidence interval (CI) includes the null, then the decision must be to fail to reject the H0. b. If the confidence interval (CI) does not include the null, then the decision must be to reject the H0. 6 1b. Confidence interval to compare groups Calculate the confidence interval for each group Decision Rule:
- 4. a. If the confidence interval (CI) overlap, then the decision must be to fail to reject the H0. b. If the confidence interval (CI) do not include the null, then the decision must be to reject the H0. 7 2.Test Statistic Calculate the test statistic (TS) Obtain the critical value (CV) from the reference table Decision Rule: a. If the test statistic is in the FTR region, then the decision must be to fail to reject the H0. b. If the test statistic is in the rejection region, then the decision must be to reject the H0. FTR CV TS Since the test statistic is in the rejection region, reject the H0 FTR CV
- 5. Since the test statistic is in the fail to reject region, fail to reject the H0 TS CV CV 8 3. P-Value Choose α Calculate value of test statistic from your data Calculate P- value from test statistic Decision Rule: a. If the p-value is less than the level of significance, α, then the decision must be to reject H0. b. If the p-value is greater than or equal to the level of significance ,α, then the decision must be to fail to reject H0. FTR CV TS FTR
- 6. CV TS P-value P-value 9 Types of Errors! Types of Errors TruthHypothesis TestingDecision Based on a Random Sample1-α (Correct Decision)Type II error (β) Type I error (α) 1-β ( Power) (Correct Decision) Fail to Reject H0 Reject H0 The Null Hypothesis (H0) is True
- 7. The Null Hypothesis (H0) is False The level of significant (Alpha) is the maximum probability of committing a type I error. P(type I error)= alpha 11 FTR CV H0 is True ts Since the H0 is true and we decide to accept it, we have thus made a correct decision Correct Decision 12 FTR ts CV H0 is True
- 8. Since the H0 is true and we decide to reject it, we have thus made an incorrect decision leading to Type I error Alpha (α) Error 13 ts FTR CV H0 is False Since the H0 is False and we decide to reject it, we have thus made a correct decision Power 14 FTR ts
- 9. CV H0 is False Since the H0 is False and we decide to accept it, we have thus made an incorrect decision leading to type II error. Beta, β, Error 15 Null Hypothesis True Fail to reject Reject False Reject Correct Decision
- 10. Type I Error Fail to Reject Type II Error Correct Decision
- 11. How to Reduce Errors Alpha error is reduced by increasing the confidence interval or reducing bias Beta error is reduced by increasing the sample size Alpha and beta are inversely related Example What type of error was possibly committed in the above example? How would you reduce the error? ANOVA GROUPS 763.000 2 381.500 31.918 .000 251.000 21 11.952 1014.000 23 Between Groups Within Groups Total Sum of Squares df
- 12. Mean Square F Sig. Understanding Inferential Statistics—Estimation Types of StatisticsThe choice of a type of analysis is based on: Research questions. The type of data collected. Audience who will receive the results. Descriptive & Inferential Statistics Statistical Methods * Inference Process Population Sample Sample statistic (`X, Ps )
- 13. Estimation & Hypothesis testing * Point Estimating &Population Parameters Population Parameters µ = Population mean σ = Population standard deviation σ2 = Population variance π = Population proportion N = The size of the population you can generalize to Sample Statistics (Point Estimates) = Mean point Estimate S = Standard deviation point estimate S2 = Variance point estimate P = Proportion point estimate n = The size of a sample taken from a population Population Parameter is Unknown Sample Statistics Population Parameters are usually represented by Greek lettersPoint estimates are usually represented by Roman letters *
- 14. Point Estimating &Population ParametersCharacteristic measuresPoint estimates (Sample)Parameters (Population)MeanµStandard deviationSσVarianceS2σ2ProportionPπ Population Parameters are usually represented by Greek lettersPoint estimates are usually represented by Roman letters * Point estimation involves the use of sampledata to calculate a single value (known as a statistic) which is to serve as a "best guess" for an unknown population parameter. Interval estimation is the use of sampledata to calculate an interval of possible (or probable) values of an unknown population parameter. * Example 1: The College Board reports that the scores on the 2010 SAT mathematics test were normally distributed. A sample of 25
- 15. scores had a mean of 510. Assume the population standard deviation is 100. Construct a 95% confidence interval for the population mean score on the 2010 SAT math test. Interval Estimation of Population Mean *For α = 0.05 (95% CI), we get Zα/2 = Z0.025 = 1.96. Interval Estimation of Population Mean (µ) with Known Variance (σ Known) Interpretation: We are 95% confidant that the population mean SAT score on the 2010 mathematics SAT test lies between 470.8 and 549.2 Solution : n = 25, = 510, σ = 100
- 16. Example 2: Estimate with 95% confidence interval the mean cholesterol level for freshman nursing students using a sample of 30 students who have an average cholesterol of 180mg/dl and a standard deviation of 34mg/dl. Interval Estimation of Population Mean (µ) with Unknown Variance (σ Unknown) Recall, Note: Since σ ( Standard deviation of the population) is unknown, we will use s (standard deviation of the sample) in place of σ. When s is used instead of σ, an error is introduced because s is only an estimate of σ. We will substitute the Z value with a another value called the student’s t or just t to account for this additional error. If σ is known:
- 17. If σ is unknown: Thus: d.f * = n-1 = 30-1=29 * Degrees of freedom (d.f) is the number of values that are free to vary when computing a statistic Interpretation: we are 95% confidant that the freshman nursing students population mean cholesterol level is between 167.31 and 192.69