Question Solved1 Answer A balanced coin is tossed three times. Let Y equal the number of heads observed. (a) Use the formula for the binomial probability distribution to calculate the probabilities associated with Y = 0, 1, 2, and 3. PCY - 0) = P(Y = 1) = P(Y = 2) = P(Y = 3) - (b) Construct the probability distribution below. y ply) 0 1 2 3 (c) Find the expected value and standard deviation of Y, using the formulas E(Y) = np and ViY) = npg. (Round your answers to three decimal places.) u = (d) Using the probability distribution from part (b), find the fraction of the population measurements lying within 1 standard deviation of the mean. How do your results compare with the results of Tchebysheff's theorem and the empirical rule? This is consistent with the empirical rule. This is consistent with Tchebysheff's theorem. This is consistent with the empirical rule and Tchebysheff's theorem. This is not consistent with either the empirical rule or Tchebysheff's theorem. Using the probability distribution from part (b), find the fraction of the population measurements lying within 2 standard deviations of the mean. How do your results compare with the results of Tchebysheff's theorem and the empirical rule? This is consistent with the empirical rule. This is consistent with Tchebysheff's theorem. This is consistent with the empirical rule and Tchebysheff's theorem. This is not consistent with either the empirical rule or Tchebysheff's theorem.

GS6IAA The Asker · Probability and Statistics

Transcribed Image Text: A balanced coin is tossed three times. Let Y equal the number of heads observed. (a) Use the formula for the binomial probability distribution to calculate the probabilities associated with Y = 0, 1, 2, and 3. PCY - 0) = P(Y = 1) = P(Y = 2) = P(Y = 3) - (b) Construct the probability distribution below. y ply) 0 1 2 3 (c) Find the expected value and standard deviation of Y, using the formulas E(Y) = np and ViY) = npg. (Round your answers to three decimal places.) u = (d) Using the probability distribution from part (b), find the fraction of the population measurements lying within 1 standard deviation of the mean. How do your results compare with the results of Tchebysheff's theorem and the empirical rule? This is consistent with the empirical rule. This is consistent with Tchebysheff's theorem. This is consistent with the empirical rule and Tchebysheff's theorem. This is not consistent with either the empirical rule or Tchebysheff's theorem. Using the probability distribution from part (b), find the fraction of the population measurements lying within 2 standard deviations of the mean. How do your results compare with the results of Tchebysheff's theorem and the empirical rule? This is consistent with the empirical rule. This is consistent with Tchebysheff's theorem. This is consistent with the empirical rule and Tchebysheff's theorem. This is not consistent with either the empirical rule or Tchebysheff's theorem.
More
Transcribed Image Text: A balanced coin is tossed three times. Let Y equal the number of heads observed. (a) Use the formula for the binomial probability distribution to calculate the probabilities associated with Y = 0, 1, 2, and 3. PCY - 0) = P(Y = 1) = P(Y = 2) = P(Y = 3) - (b) Construct the probability distribution below. y ply) 0 1 2 3 (c) Find the expected value and standard deviation of Y, using the formulas E(Y) = np and ViY) = npg. (Round your answers to three decimal places.) u = (d) Using the probability distribution from part (b), find the fraction of the population measurements lying within 1 standard deviation of the mean. How do your results compare with the results of Tchebysheff's theorem and the empirical rule? This is consistent with the empirical rule. This is consistent with Tchebysheff's theorem. This is consistent with the empirical rule and Tchebysheff's theorem. This is not consistent with either the empirical rule or Tchebysheff's theorem. Using the probability distribution from part (b), find the fraction of the population measurements lying within 2 standard deviations of the mean. How do your results compare with the results of Tchebysheff's theorem and the empirical rule? This is consistent with the empirical rule. This is consistent with Tchebysheff's theorem. This is consistent with the empirical rule and Tchebysheff's theorem. This is not consistent with either the empirical rule or Tchebysheff's theorem.
See Answer
Add Answer +20 Points
Community Answer
PCL2C2 The First Answerer
See all the answers with 1 Unlock
Get 4 Free Unlocks by registration

A balanced coin is tossed 3 times. Denote,  H: Occurrence of Head T: Occurrence of Tail Then, the probability of head P(H)=1//2=P(T) and Y = number of heads observed Clearly,  Y∼Binomial(n=3,p=0.5) and the pdf of Y is given by, f_(Y)(y)=([3],[y])(0.5)^(y)(1-0.5)^(3-y)=([3],[y])(0.5)^(3) a)  P[Y=0]=([3],[0])(0.5)^(3)=0.125P[Y=1]=([3],[1])(0.5)^(3)=0.375P[Y=2]=([3],[2])(0.5)^(3)=0.375P[Y=3]=([3],[3])(0.5)^(3)=0.125 b) t tt ttt ttty  ttt tttp(y) tt tt ttt0 ttt0.125 tt tt ttt ttt1 ttt ttt0.375 tt tt ttt2 ttt0.375 tt tt ttt3 ttt ttt0.125 ttt tt t   c) mu=E(Y) E(Y)=np=3xx0.5=1.5 Variance of Y is given by V(Y)=npq (where q=1-p=1-0.5=0.5 here) So, V(Y)=3xx0.5 xx0.5=0.75 Hence, the standard deviation of Y is given by,  So, sqrt(V(Y))=sigma=sqrt0.75=0.866 d) 1 sd of mean  implies the interval  (mu-sigma,mu+sigma)=(1.5-0.866,1.5+0.866)=(0.634,2.366) Then  fraction of population measurements lying within the interval ( 0.634, 2.366) is  P(Y in(0.634,2.366))=P(Y=1)+P(Y=2)=0.375+0.375=0.75 Tchebysheff's theorem says that at least 1-1//k^(2)& ... See the full answer