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**Statistics Are Fun! Analysis of Variance**

Chapter 12 Analysis of Variance Statistics Is Fun!

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**Goals List the characteristics of the F distribution**

Statistics Are Fun! Goals List the characteristics of the F distribution Conduct a test of hypothesis to determine whether the variances of two populations are equal Discuss the general idea of analysis of variance Organize data into a ANOVA table Conduct a test of hypothesis among three or more treatment means Statistics Is Fun!

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Statistics Are Fun! F Distribution Used to test whether two samples are from populations having equal variances Applied when we want to compare several population means simultaneously to determine if they came from equal population ANOVA Analysis of variance In both situations: Populations must be Normally distributed Data must be Interval-scale or higher Statistics Is Fun!

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**Characteristics Of The F Distribution**

Statistics Are Fun! Characteristics Of The F Distribution Family of F Distributions Each is determined by: df in numerator comes from pop. 1 which has larger sample variation df in denominator comes from pop. 2 which has smaller sample variation F distribution is continuous F Value can assume an infinite number of values from 0 to ∞ Value for F Distribution cannot be negative Smallest value = 0 Positively skewed Long tail is always to right As # of df increases in both the numerator and the denominator, the distribution approaches normal Asymptotic As X increases the F curve approaches the X-axis Statistics Is Fun!

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Statistics Are Fun! Why Do We Want To Compare To See If Two Population Have Equal Variances? What if two machines are making the same part for an airplane? Do we want the parts to be identical or nearly identical? Yes! We would test to see if the means are the same: Chapter 10 & 11 We would test to see if the variation is the same for the two machines: Chapter 12 What if two stocks have similar mean returns? Would we like to test and see if one stock has more variation than the other? Statistics Is Fun!

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**Remember Chapter 11: Assumptions for small sample tests of means:**

Statistics Are Fun! Why Do We Want To Compare To See If Two Population Have Equal Variances? Remember Chapter 11: Assumptions for small sample tests of means: Sample populations must follow the normal distribution Two samples must be from independent (unrelated) populations The variances & standard deviations of the two populations are equal Statistics Is Fun!

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**larger sample variance**

Statistics Are Fun! Conduct A Test Of Hypothesis To Determine Whether The Variances Of Two Populations Are Equal To conduct a test: Conduct two random samples List population 1 as the sample with the largest variance: n1 = # of observations s1^2 = sample variance n1 – 1 = df1 = degree of freedom (numerator for critical value lookup) List population 2 as the sample with the smaller variance: n2 = # of observations s2^2 = sample variance n2 – 1 = df2 = degree of freedom (denominator for critical value lookup) Always list the sample with the larger sample variance as population 1 (allows us to use fewer tables) Statistics Is Fun!

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**Step 1: State null and alternate hypotheses**

Statistics Are Fun! Step 1: State null and alternate hypotheses List the population with the suspected largest variance as population 1 Because we want to limit the number of F tables we need to use to look up values, we always put the larger variance in the numerator and the smaller variance in the denominator This will force the F value to be at least 1 We will only use the right tail of the F distribution Examples of Step 1: Statistics Is Fun!

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**Step 2: Select a level of significance:**

Statistics Are Fun! Step 2: Select a level of significance: Appendix G only lists significance levels: .05 and .01 Significance level = .10 .10/2 = .05 Use .05 table in Appendix G Significance level = .05 Use .05 table in Appendix G Statistics Is Fun!

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Statistics Are Fun! Step 3: Identify the test statistic (F), find critical value and draw picture Look up Critical value in Appendix G and draw your picture If you have a df that is not listed in the border, calculate your F by estimating a value between two values. HW #5: df = 11, use value Between 10 & 12 Book says: ( )/2 = 3.105 3.10 Statistics Is Fun!

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**Step 4: Formulate a decision rule:**

Statistics Are Fun! Step 4 Step 4: Formulate a decision rule: Example: If our calculated test statistic is greater than 3.87, reject Ho and accept H1, otherwise fail to reject Ho Statistics Is Fun!

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**Larger variance in numerator,**

Statistics Are Fun! Step 5 Step 5: Take a random sample, compute the test statistic, compare it to critical value, and make decision to reject or not reject null and hypotheses Test Statistic F: Example Conclusion for a two tail test: Fail to reject null “The evidence suggests that there is not a difference in variation” Reject null and accept alternate “The evidence suggests that there is a difference in variation” Larger variance in numerator, always!! Let’s Look at Handout Statistics Is Fun!

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Statistics Are Fun! Colin, a stockbroker at Critical Securities, reported that the mean rate of return on a sample of 10 software stocks was 12.6 percent with a standard deviation of 3.9 percent. The mean rate of return on a sample of 8 utility stocks was 10.9 percent with a standard deviation of 3.5 percent. At the .05 significance level, can Colin conclude that there is more variation in the software stocks? Example 1 Statistics Is Fun!

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**Step 3: The test statistic is the F distribution.**

Statistics Are Fun! Step 1: The hypotheses are Step 2: The significance level is .05. Step 3: The test statistic is the F distribution. Example 1 continued Statistics Is Fun!

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**Step 5: The value of F is computed as follows.**

Statistics Are Fun! Step 4: H0 is rejected if F>3.68 or if p < The degrees of freedom are n1-1 or 9 in the numerator and n1-1 or 7 in the denominator. Step 5: The value of F is computed as follows. H0 is not rejected. There is insufficient evidence to show more variation in the software stocks. Example 1 continued Statistics Is Fun!

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**ANOVA Analysis Of Variance**

Statistics Are Fun! ANOVA Analysis Of Variance Technique in which we compare three or more population means to determine whether they could be equal Assumptions necessary: Populations follow the normal distribution Populations have equal standard deviations () Populations are independent Why ANOVA? Using t-distribution leads to build up of type 1 error “Treatment” = different populations being examined Statistics Is Fun!

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**Case Where Treatment Means Are Different**

Statistics Are Fun! Case Where Treatment Means Are Different Statistics Is Fun!

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**Case Where Treatment Means Are The Same**

Statistics Are Fun! Case Where Treatment Means Are The Same Statistics Is Fun!

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**Example Of ANOVA Test To See If Four Treatment Means Are The Equal**

Statistics Are Fun! Example Of ANOVA Test To See If Four Treatment Means Are The Equal 22 students earned the following grades in Professor Rad’s class. The grades are listed under the classification the student gave to the instructor Is there a difference in the mean score of the students in each of the four categories? Use significance level α = .01 Statistics Is Fun!

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**Conduct A Test Of Hypothesis Among Four Treatment Means**

Statistics Are Fun! Conduct A Test Of Hypothesis Among Four Treatment Means Step 1: State H0 and H1 H0 : µ1 = µ2 = µ3 = µ4 H1 : The Mean scores are not all equal (at least one treatment mean is different) Step 2: Significance Level? α = .01 Statistics Is Fun!

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**Step 3: Determine Test Statistic And Select Critical Value**

Statistics Are Fun! Step 3: Determine Test Statistic And Select Critical Value Statistics Is Fun!

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**Step 4: State Decision Rule**

Statistics Are Fun! Step 4: State Decision Rule If our calculated test statistic is greater than we reject H0 and accept H1, otherwise we fail to reject H0 Now we move on to Step 5: Select the sample, perform calculations, and make a decision… Are you ready for a lot of procedures?!! Statistics Is Fun!

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**The F critical value will determined whether we are close to 1 or not**

Statistics Are Fun! The idea is: If we estimate variation in two ways and use one estimate in the numerator and the other estimate in the denominator: If we divide and get 1 or close to 1, the sample means are assumed to be the same If we get a number far from 1, we say that the means are assumed to be different The F critical value will determined whether we are close to 1 or not Statistics Is Fun!

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**Let’s go calculate this!**

Statistics Are Fun! ANOVA Table So Far Let’s go calculate this! Statistics Is Fun!

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**Calculation 1: Treatment Means and Overall Mean**

Statistics Are Fun! Calculation 1: Treatment Means and Overall Mean Statistics Is Fun!

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**Calculation 2: Total Variation**

Statistics Are Fun! Calculation 2: Total Variation Statistics Is Fun!

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**Calculation 2: Total Variation**

Statistics Are Fun! Calculation 2: Total Variation Statistics Is Fun!

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**Let’s go calculate this!**

Statistics Are Fun! ANOVA Table So Far Let’s go calculate this! Statistics Is Fun!

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**Calculation 3: Random Variation**

Statistics Are Fun! Calculation 3: Random Variation Statistics Is Fun!

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**Calculation 3: Random Variation**

Statistics Are Fun! Calculation 3: Random Variation Statistics Is Fun!

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**Let’s go calculate this!**

Statistics Are Fun! ANOVA Table So Far Let’s go calculate this! Statistics Is Fun!

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**Calculation 4: Treatment Variation**

Statistics Are Fun! Calculation 4: Treatment Variation Simple Subtraction! Statistics Is Fun!

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**Calculation 5: Mean Square (Estimate of Variation)**

Statistics Are Fun! Calculation 5: Mean Square (Estimate of Variation) Statistics Is Fun!

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Statistics Are Fun! Calculation 6: F Statistics Is Fun!

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Statistics Are Fun! Step 5: Make A Decision Because is less than 5.09, we fail to reject H0 The evidence suggests that the mean score of the students in each of the four categories are equal (no difference) Statistics Is Fun!

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**Summarize Chapter 12 List the characteristics of the F distribution**

Statistics Are Fun! Summarize Chapter 12 List the characteristics of the F distribution Conduct a test of hypothesis to determine whether the variances of two populations are equal Discuss the general idea of analysis of variance Organize data into a ANOVA table Conduct a test of hypothesis among three or more treatment means Statistics Is Fun!

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