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## Statistics and probability

### Course: Statistics and probability>Unit 16

Lesson 1: Analysis of variance (ANOVA)

# ANOVA 3: Hypothesis test with F-statistic

Analysis of Variance 3 -Hypothesis Test with F-Statistic. Created by Sal Khan.

## Want to join the conversation?

• I have a question: is SSB similar to the Explained Sum of Squares, as well as SSW similar to the Residual Sum of Squares I see in Econometric textbooks? • Hey, can someone explain to me why t^2=F for a ANOVA analysis?
I can use it to solve problems, but I don't know why the p-values of both density curves predict the same p-values.

How can I transform the t^2 distribution to the corresponding F distribution?
Why does it work? F uses 2 different degrees (numerator and denomentator) of freedom, while T only uses 1. • At Sal roughly defines the F-stat as the ratio between the SSB and the SSW. Can someone please explain why the SSW is the denominator and not the other way around?
(1 vote) • IF I DONT HAVE F-statistic table BUT HAVE CHI-SQUARED TABLE and want to compute critical value through definition F=(Chi-sq(p)/p)/(Chi-sq(k)/k) where p, k - degrees of freedom. I have tried to do so for F(2,19) at 5% significance level and did not come to the true value of 3,52. Here are my calculations: (5,991/2)/(30.14/19)=1.888
PLEASE HELP ME TO FIND A MISTAKE AND EXPLAIN HOW TO CALCULATE F statistics WITH CHI-SQUARED DISTRIBUTION TABLE
THANK YOU)
(1 vote) • I think the mistake is with your premise: An F-distributed random variable arises from the ratio of two Chi-squares divided by their respective degrees of freedom, but it does not follow that the critical values of this random variable's distribution will be computed as the ratio of the critical values of the chi-square distributions.
• How do you get a critical value for an F-test when the actual degrees of freedom arent present in the F-table? For example if the dfs are 28 and 32. • If you really need an accurate value, you can interpolate. So if your chart only shows 25, 30, 35, ... for both df(n) and df(d), then you can first interpolate, say, between the crit values at 25 and 30 to get two values for "row" 28 in the 30 column and "row" 28 in the 35 column. Then you can interpolate between those two to get a value for 28 and 32. To interpolate, it just means you assume the values change linearly between two (nearby!) entries in the table. For example, lets say the (25,30) entry is 1.66 and the (30,30) entry is 1.61, then you can see that it changes by -.05 as the first df goes from 25 to 30, and therefore the entry at (28,30) will be very close to 1.66 + (-.05)*(28-25)/(30-25) = 1.66 + (-.05)(3/5) = 1.63. The concept is pretty simple: if it changed by an amount (-.05) when the df increased by 5, then it should change by (3/5) of (-.05) if the df increases by 3.
• How come the p-value is determined only by the right tail? Why don't we consider both right and left tails?
(1 vote) • When we're doing ANOVA, the null hypothesis is "no differences among the group means." If the null hypothesis is correct, then the F statistic will be small (if the group means are all identical, it will be 0). When the group means start to differ, the F statistic gets larger. Hence, only large values make us think the null hypothesis is wrong, and thus we only look at the right tail.
• How come you didn't use alpha/2 if the null and alternative hypotheses were equals (as opposed to < or >)? Do these problems not have two sided tests?
(1 vote) • ANOVA is inherently a 2-sided test.

Say you have two groups, A and B, and you want to run a 2-sample t-test on them, with the alternative hypothesis being: `Ha: µ.a ≠ µ.b`. You will get some test statistic, call it t, and some p-value, call it p1. If you then run an ANOVA on these two groups, you will get an test statistic, f, and a p-value p2.

If you look, then f = t² and p2 = p1. That is: the p-values are the exact same, and the test statistic for ANOVA is simply the square of the test statistic for the t-test. They are the same test, exactly.

Note: This requires using the pooled 2-sample t-test. The unpooled t-test will not be exactly equivalent to ANOVA.
• In what cases do we use ANOVA, Chi Square, t-test etce   