Let ?1,?2,…,?? be a collection of independent discrete random variables that all take the value 1 with probability p and take the value 0 with probability (1-p). a) Compute the mean and the variance of ?1 (which is the same for ?2, ?3, etc.) b) Use your answer to (a) to compute the mean and variance of ?̂ = 1/n (?1 + ?2 + ⋯+ ??), which is the proportion of “ones” observed in the n instances of ??. c) Suppose n = 10,000. Use Chebyshev’s inequality to provide an upper bound for the probability that the difference between ?̂ and ? exceeds 0.05. d) Use calculus to show that if ? is a number between 0 and 1, then ?(1 − ?) ≤ 1/4 . e) Use your answers to (c) and (d) to provide an upper bound, that does not depend on ?, for the probability that the difference between ?̂ and ? exceeds 0.05. f) Interpret this problem in the context of randomly sampling 10,000 people from a large population, asking them a yes-no question, and using the result to make an inference about the whole population.
Contingency Table
A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.
Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
PLEASE ANSWER f)
Let ?1,?2,…,?? be a collection of independent discrete random variables that all take the value 1 with
a) Compute the mean and the variance of ?1 (which is the same for ?2, ?3, etc.)
b) Use your answer to (a) to compute the mean and variance of ?̂ = 1/n (?1 + ?2 + ⋯+ ??), which is the proportion of “ones” observed in the n instances of ??.
c) Suppose n = 10,000. Use Chebyshev’s inequality to provide an upper bound for the probability that the difference between ?̂ and ? exceeds 0.05.
d) Use calculus to show that if ? is a number between 0 and 1, then ?(1 − ?) ≤ 1/4 .
e) Use your answers to (c) and (d) to provide an upper bound, that does not depend on ?, for the probability that the difference between ?̂ and ? exceeds 0.05.
f) Interpret this problem in the context of randomly sampling 10,000 people from a large population, asking them a yes-no question, and using the result to make an inference about the whole population.
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