Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Student's t-test wikipedia, lookup

Taylor's law wikipedia, lookup

Bootstrapping (statistics) wikipedia, lookup

Resampling (statistics) wikipedia, lookup

Misuse of statistics wikipedia, lookup

Degrees of freedom (statistics) wikipedia, lookup

Psychometrics wikipedia, lookup

Foundations of statistics wikipedia, lookup

Analysis of variance wikipedia, lookup

Transcript

One-Way ANOVA Page 1 Testing Claims about Differences between Three or More Means Step 0: Verify Assumptions The hypothesis test of the difference of three or more population means has four assumptions. 1. The k samples are each obtained using simple random sampling. 2. The k samples data independent of each other within and among the samples. 3. The k populations are normally distributed. 4. The k populations have equal variances. Step 1: State the Hypothesis A claim is made regarding the three or more population means. This claim is used to determine the following null and alternative hypotheses. H0: µ1 = µ2 = µ3 = … = µk H1: At least one of the population means is different from the others. Step 2: Select a Level of Significance The selection of the level of significance α is done based on the seriousness of making a Type I error. (The typical value of α is 0.05.) Step 3: Calculate the Test Statistic The test statistic follows the F-distribution with k – 1 degrees of freedom in the numerator and n – k degrees of freedom in the denominator. The F-statistic is calculated as follow: a. Find the sample mean of the combined data: b. Find the sample mean for each individual sample: , , , …, c. Find the sample standard deviations of each individual sample: , , , …, d. Compute the mean square due to treatment: ∑ ( − ) = −1 e. Compute the mean square due to error: ∑( − 1) = − f. Compute the test statistic: = Robert A. Powers University of Northern Colorado One-Way ANOVA Page 2 Step 4: Determine the Decision Criterion The Classical Approach: Find the Critical Value The level of significance is used to determine the critical value, represented by the F-value in the figure below. All one-way ANOVA tests use a right-tailed test, so the critical value is ,, with k – 1 degrees of freedom in the numerator and n – k degrees of freedom in the denominator. Fα , k −1, n − k The Modern Approach: Find the p-Value Based on the test statistic , determine the probability that the ratio of the mean square due to treatment and the mean square due to error is more extreme than is found. This is represented by the shaded region under the F-distribution with k – 1 degrees of freedom in the numerator and n – k degrees of freedom in the denominator in the figure below. F0 Step 5: Make a Decision Reject the null hypothesis if the test statistics lies in the critical region or the probability associated with the test statistic is less than the level of significance. Do not reject the null hypothesis if the test statistic does not lie in the critical region or the probability associated with the test statistic is greater than or equal to the level of significance. Step 6: State the Conclusion State the conclusion of the hypothesis test based on the decision made and with respect to the original claim. Reject H0 Do Not Reject H0 Original Claim is H0 There is sufficient evidence (at the α level) to reject the claim that … . There is not sufficient evidence (at the α level) to reject the claim that … . Robert A. Powers Original Claim is H1 There is sufficient evidence (at the α level) to support the claim that … . There is not sufficient evidence (at the α level) to support the claim that … . University of Northern Colorado