Test $H_0$ hypothesis that the population variances are equal Here is the problem: Suppose you analyze potato prices in Brno and Prague. For Brno you analyze 18 shops and find sample variance
45. For Prague you analyze 27 shops and find sample variance 75. Test the equality of price variances using the
5% significance level.

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*First, as this is two tailed test, the regions of rejection on either side should be equal to $\alpha / 2 = 0.025$. Then for null hypothesis to be rejected, the test statistic $F = \frac{s_1^2}{s^2_2} = \frac{45}{75} = 0.6$ should be less than $F_{1-\alpha/2, \nu_1, \nu_2}$ or greater than $F_{\alpha/2, \nu_1, \nu_2}$. $F_{1-\alpha/2, \nu_1, \nu_2} = 0.396$ and $F_{\alpha/2, \nu_1, \nu_2} = 2.313. $ So, with confidence interval $\alpha = 5% $ we cannot reject the hypothesis, that the price variances are not equal.

The p-value given for the ratio and given $\nu_1 , \nu_2$ is equal to $0.863 < 0.95~~\Longrightarrow~~H_0~$ cannot be rejected. But is there a way to find p-value from tables? Also, how to know how to graph the given F distribution ?
 A: Your calculations are nearly correct (The accurate critical values should be $0.394$ and $2.335$), and your inference is also correct. I'm not sure if I understand your question properly of not, but anyway, I presume that you're asking how can you plot the density of $F_{\nu_1 , \nu_2}$ distribution, and how to find p-value from an F-table.

For given values of $\nu_1$ and $\nu_2$ , and $x \in [0 , \infty)$ , the random variable $X \sim F_{\nu_1 , \nu_2}$ distribution has a density function : $$f_X(x) = \frac{1}{B(\nu_1,\nu_2)}\cdot \dfrac{\left(\frac{\nu_1}{\nu_2}\right)^{\frac{\nu_1}{2}} x^{\frac{\nu_1}{2} - 1}}{\left[\left(\frac{\nu_1}{\nu_2}\right)x + 1 \right]^\frac{\nu_1+\nu_2}{2} }$$ As you can see, the density of $X$ looks quite horrendous, and it's nearly impossible to figure out manually how the density actually looks like. The best way to plot the density is to use either some graphing calculator (e.g. Desmos, Geogebra) or programming languages (e.g. R, Python, MATLAB).

Now, let us talk about your question regarding p-value. The way you asked it is telling me that your understanding of p-value may be a little bit problematic. Remember that on the basis of an observation $c$ (Here, $c=0.6$) , the p-value is the probability that the random variable takes value "at least as extreme as $c$" . So, here it's nothing but : $$P(X \geqslant 0.6) = 1 - P(X \leqslant 0.6) = 1 - 0.138 = 0.862$$ So basically, from the table you'll have to check the value of $P(X \leqslant 0.6)$ which should be readily available in the F-table. However, for more precision, you may use R or Python. For example, in R, you may write :

install.packages("VaRES")
library(VaRES)
pf(0.6 , df1 = 17 , df2=26 , lower.tail = FALSE)

This'll give you the p-value, and hence you'll be able to infer whether to accept or reject $H_0$ .
A: Finding the p-value for a two-tailed test can be confusing.  Some Intro to Statistics textbooks say to double the p-value and compare it to alpha, while other texts say to us the p-value as-is and compare it to half-alpha.  Either method will provide the same results, but any discussion of the findings that included the p-value should indicate which method was used.
