How to solve a cyclic quintic in radicals?
Galois theory tells us that
can be solved in radicals because its group is solvable. Actually performing the calculation is beyond me, though - here what I have got so far:
Let the roots be , following Gauss we can split the problem into solving quintics and quadratics by looking at subgroups of the roots. Since 2 is a generator of the group [2,4,8,5,10,9,7,3,6,1] we can partition into the five subgroups of conjugate pairs [2,9],[4,7],[8,3],[5,6],[10,1].
Once one has one easily gets . It's easy to find . The point is that takes to and so takes to . Thus can be written down in terms of rationals (if that's your starting field) and powers of . Alas, here is where the algebra becomes difficult. The coefficients of powers of in are complicated. They can be expressed in terms of a root of a "resolvent polynomial" which will have a rational root as the equation is cyclic. Once one has done this, you have as a fifth root of a certain explicit complex number. Then one can express the other in terms of . The details are not very pleasant, but Dummit skilfully navigates through the complexities, and produces formulas which are not as complicated as they might be. Alas, I don't have the time nor the energy to provide more details.
Find the length of the confidence interval given the following data
S=3 n=275 confidence level 95 %
Simple Linear Regression - Difference between predicting and estimating?
Here is what my notes say about estimation and prediction:
Estimating the conditional mean
"We need to estimate the conditional mean at a value , so we use as a natural estimator." here we get
with a confidence interval for is
where Where these results are found by looking at the shape of the distribution and at and
"We want to predict the observation at a value "
Hence a prediction interval is of the form
Sally has caught covid but doesn’t know it yet. She is testing herself with rapid antigen kits which have an 80% probability of returning a positive result for an infected person. For the purpose of this question you can assume that the results of repeated tests are independent.
a) If sally uses 3 test kits what is the probability that at least one will return a positive result?
b) In 3 tests, what is the expected number of positive results?
c) Sally has gotten her hands on more effective tests, these ones have a 90% probability of returning a positive result for an infected person. If she tested herself
twice with the new tests, how many positive results would she expect to see?