HUM 101 Cognitive Bias EssayDirections:Write an essay that addresses the following issues and questions:Requirements:You may wish to review the APA Template Paper (Links to an external site.) for help formatting your essay according to the requirements. If you need assistance with your writing style or you need writing tips or tutorials, visit the CSU Global Writing Center (Links to an external site.). Review the grading rubric to see how you will be graded for this assignment.
1.Let the function f: R 2 → R be given by f (x, y) = x 3 – 3x – y 3 + 3y.(a) Find the partially derived of 1st and 2nd order.(b) In which direction does the function grow most in point (2,2)?(c) Find and classify the stationary points of the function.
Assume that a stochastic variable X is normally distributed with unknown mean µ andunknown variance σ 2 . You make 10 measurements on X and get the following result:58,73,60,64,68,60,63,67,64,77(a) Find a 95% confidence interval for µ.(b) Assume that you are told that X has known variance σ 2 = 36 = 6 2 .Find a 95% confidence interval for µ. Explain in words why it is naFortunately, this confidence interval is less than the interval you found above.(c) How many tests must be performed to ensure that the length of thethe interval should be less than 1?(d) Is it possible to determine with certainty how many attempts must beperform to determine a confidence interval of length 1 if X has unknownvariance? Base the answer.
In the development of a vaccine against Covid-19, researchers at a pharmaceutical companytry out a test vaccine on a group of people. All the people in the grouptested negative for Covid-19 when the experiment started.They divide the group of people involved in the experiment into 2 parts. One half(control group) receive placebo. After one month, 1% of the control group testedpositive for Covid-19. The other half of the group receives the test vaccine. LetX be the stochastic variable that describes how many of those who receivedvaccine that tests positive for Covid-19 after one month. The researchers assume thatThe number of people who test positive for Covid-19 is binomially distributed.To continue the development of the vaccine, the researchers require that the test vaccine beincluded 95% security should have a positive effect against the virus.In the experiment, 10,000 people received a placebo, and 10,000 received the test vaccine. In theanalysis of such an attempt should be based on the assumption that the vaccine does not work, andon that hypothesis does not hold is the basis for continuing development.(a) Find values for p and n such that X ∼ bin (n, p) if the vaccine does not work.(b) Explain why one can approximate the binomially distributed variable X witha normal distribution with expectation µ = 100 and variance σ 2 = 99.(c) Set up a hypothesis test that is relevant to the experiment.(d) Of the 10,000 people who received the test vaccine, 78 tested positivefor Covid-19 after one month. Assess whether there is a basis for continuingthe development of the vaccine.
The definition range of the function f is the convergence range of the power series:(a) Calculate the open interval where the series converges using ratiostest.(b) Find an expression for the function f.(c) Write down the sequence of numbers that has the function f as its generating function.
In a money game, three cards are drawn from a deck of cards. With bet 1 youwin x if there is a spade among the three cards, 2x if there are two spades and 5xwhether there are three spades.(a) Explain why X = number of spades is a hypergeometrically distributed variable, andwhat values the parameters have.(b) If the game is to be a zero-sum game, the expected payout must be equal to the bet.Calculate what x must be for this game to be a zero-sum game.(c) An unlucky player must play five times. Let Y be the number of losses (nonespar). What distribution does Y have? What is the probability that there will be a losson exactly four of the five games?
A language in the alphabet {0,1} has regular expression(a) Write down all the strings in the language that have five or fewer symbols.(b) Find a final state machine that accepts the language.(c) Find a regular grammar that generates the language.
In a set of outcomes from an experiment, we have the three subsets A, B and C.We know about these quantitiesi) P (A) = 0.3ii) P (A ∩ B) = 0.06iii) A and B are independentiv) P (C) = 0.4v) P (A ∩ C) = 0.12vi) P (B ∩ C) = 0.08vii) P (A ∪ B ∪ C) = 0.35(a) Are A and C independent?(b) Are B and C independent?(c) Calculate P (A ∪ B).(d) Calculate P (A ∩ B ∩ C).(e) Are A and B ∪ C independent?
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