Considering imaginary solutions is definitely important! However, I don't think that here it's a concern since we don't have x^(n/2). This can be problematic because if n is odd, we are taking the square root of x to some odd power, which would necessitate that x >= 0. However, here, since we have x^(2n), whose domain in this context is the set of all real numbers, it wouldn't be a problem. Also, even if -sqrt(3) gave an imaginary number, I don't think that sqrt(3) / 2 would solve that problem. But thanks for sharing your thought on this!

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You're correct! I had the same thought process initially, which is why I couldn't seem to figure out the correct answer at first. The key lies in the fact that we are conducting a hypothesis test. When we find the expected individuals for each phenotype, we use expected values for p^2, 2pq, and q^2 as you can see in the table I've drawn rather than the experimental values. That's the whole point of doing the chi-squared test: to see whether the observed numbers are really "different enough" from the expected numbers. Perhaps I could have explained that better in the video. I hope that answers your question!

Glad to hear that!

So glad to hear that! May we ask what class your exam is in?

It's our pleasure! We're glad you were able to benefit from our video!

It's actually not that formula. For wind turbines, the power output is 1/2 of the product of the wind density, the area swept by the blades of the turbine, and the wind speed cubed. So P = (1/2)(rho)(A)(v^3) This can be derived by considering P = (Change in Kinetic Energy)/time. Consider a volume of the wind that is pushed by the turbine. KE = (1/2)mv^2. But mass is just the density (rho) times Volume. But volume is just the product of the area swept by the turbine and the distance of the wind moved. Then, m = (rho)(A)(d) where d is the distance of the wind moved. Then, P = (1/2)(rho)(A)(d)(v^2)/t. By grouping (d/t) together, we get another v (because the wind speed is the distance travelled by the wind in time t). Therefore, P = (1/2)(rho)(A)(v^3)

We should have specified- the formula you stated that P = Fv is a correct formula. It's just that in this case, force is not constant, and thus power is not proportional to wind speed. When the proper substituons are made, the dependence found is that power is proportional to wind speed cubed, as stated in the video.

Answered in your other comment :)

p is the frequency of the dominant ALLELE. So Aa contains one copy of the dominant allele and AA contains two copies of the dominant allele. That is why we must consider Aa in addition to AA for p.

Do you have a question about that part that we can help you with?

Yes- the null hypothesis (which would follow the expected frequencies from the Hardy Weinberg model) is rejected due to the high chi-squared value greater than the critical value for that particular confidence level and degrees of freedom. Since the observed frequencies are significantly different than the expected frequencies (as shown by the GOF chi-squared analysis), the population does not fit Hardy Weinberg equilibrium

@@revisnow thank you so much!!!

Hi there! Generally, such a low chi-squared value means that there is very little difference between the expected and observed values. At this point, we would search for the critical value on a chi-squared table for a particular significance level (usually for p=0.05) and degrees of freedom (df) in accordance with the situation. If your calculated chi-squared value is greater than the critical value found on the table, we reject the null hypothesis and accept the alternative hypothesis. Otherwise (which is likely in your case where the chi-squared value is very very small and the differences between the observed and accepted values are thus very very small), we cannot reject the null hypothesis. In this case, I like to think of it as "the differences between observed and expected are not big enough to be statistically significant."

@@revisnow Hi! Thank you so much for your reply. So if a question like "Which stimulation do you think evolution is occurring" do we look for lowest chi-square value if your comparing it to others?

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Yessir