An experiment is conducted to select a suitable catalyst for the commercial production of ethylenediamine (EDA), a product used in soaps. Suppose a chemical engineer randomly selects 5 catalysts without replacement for testing among a group of 116 catalysts,58 of which have low acidity and 58 of which have high acidity. Let denote the number of high acidic catalysts. a)Plot the probability mass function (pmf) of considering its range. What is the distribution of X? Is this a valid probability mass function? b)What is the probability that exactly 0 highly acidic catalysts are selected? c)What is the probability that at least 4 highly acidic catalysts are selected?
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.
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An experiment is conducted to select a suitable catalyst for the commercial production of ethylenediamine (EDA), a product used in soaps. Suppose a chemical engineer randomly selects 5 catalysts without replacement for testing among a group of 116 catalysts,58 of which have low acidity and 58 of which have high acidity.
Let denote the number of high acidic catalysts.
a)Plot the
b)What is the probability that exactly 0 highly acidic catalysts are selected?
c)What is the probability that at least 4 highly acidic catalysts are selected?
d)Calculate the mean, variance and standard deviation of .
Now suppose that sampling is considered to be done with replacement. Answer the following questions considering the new situation with approximation to both a discrete distribution and a continuous distribution.
e)Plot the pmf of for the discrete approximation considering its range. What are the suitable approximated distributions of ? Are they valid probability distributions?
f)Using both approximations, what is the probability that exactly 0 highly acidic catalysts are selected? Plot the probability found by continuous approximation on the
g)Using both approximations, what is the probability that at least 4 highly acidic catalysts are selected? Plot the probability found by continuous approximation on the pdf plot.
h)Compare the results obtained without using approximation and the results obtained by using both approximations. Is this problem suitable for applying approximation? Interpret the results briefly to decide which approximation is suitable.
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