A public health researcher is investigating the effectiveness of fluoride in preventing cavities. Two communitiesare studied, one with a fluoridiated water supply and the second with non-fluoridated water. A randomsample of n1 =found that Y1400 and n2 == 348 residents in the first community and y2900 residents are obtained from these two communities, respectively. It was819 residents in the second communityhad cavities. Consider modeling these counts by two binomial distributions with the probability of success(presence of cavities) as 71 and 72 for communities with fluoridated and non-fluoridated water supplies,respectively. (i.e. consider the random variables YBin(n1, T1) and Y2 ~ Bin(n2, T2).)a. Let P = Y1/n1 and P2with cavities in each population. Approximate the sampling distributions of P1 and P2.b. Based on your answer from (a), approximate the sampling distribution for the difference in sampleproportions, D = P1 – P2. (Hint: think of this as a sum of normal random variables.)c. Suppose we use the statistic D as an estimator for the true difference 71 – 72. Calculate the bias andstandard error of this estimator. (Note, these may be a function of 11, 72, n1, and n2.)d. Assume that 71 = 12 = 0.9. Then, using your approximation in (b), calculate P(|D| > d), whered = y1/n1 – Y2/n2 is the observed difference found by the researcher.e. Based on your answer in part (d), do you think the assumption that 71 = T2 = 0.9 is contradicted bythe data? Briefly explain why/why not. (Note: there is no right/wrong answer here, but you need togive some probabilistic reasoning. This reasoning may be similar to the logic behind hypothesis testing,even though you're not performing a formal statistical test in this problem.)Y2/n2 be random variables denoting the proportion of sampled residents

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A public health researcher is investigating the effectiveness of fluoride in preventing cavities. Two communities are studied, one with a fluoridiated water supply and the second with non-fluoridated water. A random sample of n1 = 400 and n2 = 900 residents are obtained from these two communities, respectively. It was found that y1 = 348 residents in the first community and y2 = 819 residents in the second community had cavities. Consider modeling these counts by two binomial distributions with the probability of success (presence of cavities) as ī1 and 72 for communities with fluoridated and non-fluoridated water supplies respectively. (i.e. consider the random variables Y1 ~ Bin(n1,T1) and Y2 ~ Bin(n2, 12).)

A public health researcher is investigating the effectiveness of fluoride in preventing cavities. Two communities are studied, one with a fluoridiated water supply and the second with non-fluoridated water. A random sample of n1 = 400 and n2 = 900 residents are obtained from these two communities, respectively. It was found that y1 = 348 residents in the first community and y2 = 819 residents in the second community had cavities. Consider modeling these counts by two binomial distributions with the probability of success (presence of cavities) as ī1 and 72 for communities with fluoridated and non-fluoridated water supplies respectively. (i.e. consider the random variables Y1 ~ Bin(n1,T1) and Y2 ~ Bin(n2, 12).)

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Compound Probability

Compound probability can be defined as the probability of the two events which are independent. It can be defined as the multiplication of the probability of two events that are not dependent.

Tree diagram

Probability theory is a branch of mathematics that deals with the subject of probability. Although there are many different concepts of probability, probability theory expresses the definition mathematically through a series of axioms. Usually, these axioms express probability in terms of a probability space, which assigns a measure with values ranging from 0 to 1 to a set of outcomes known as the sample space. An event is a subset of these outcomes that is described.

Conditional Probability

By definition, the term probability is expressed as a part of mathematics where the chance of an event that may either occur or not is evaluated and expressed in numerical terms. The range of the value within which probability can be expressed is between 0 and 1. The higher the chance of an event occurring, the closer is its value to be 1. If the probability of an event is 1, it means that the event will happen under all considered circumstances. Similarly, if the probability is exactly 0, then no matter the situation, the event will never occur.

Question

100%

A public health researcher is investigating the effectiveness of fluoride in preventing cavities. Two communities
are studied, one with a fluoridiated water supply and the second with non-fluoridated water. A random
sample of n1 =
found that Y1
400 and n2 =
= 348 residents in the first community and y2
900 residents are obtained from these two communities, respectively. It was
819 residents in the second community
had cavities. Consider modeling these counts by two binomial distributions with the probability of success
(presence of cavities) as 71 and 72 for communities with fluoridated and non-fluoridated water supplies,
respectively. (i.e. consider the random variables Y
Bin(n1, T1) and Y2 ~ Bin(n2, T2).)
a. Let P = Y1/n1 and P2
with cavities in each population. Approximate the sampling distributions of P1 and P2.
b. Based on your answer from (a), approximate the sampling distribution for the difference in sample
proportions, D = P1 – P2. (Hint: think of this as a sum of normal random variables.)
c. Suppose we use the statistic D as an estimator for the true difference 71 – 72. Calculate the bias and
standard error of this estimator. (Note, these may be a function of 11, 72, n1, and n2.)
d. Assume that 71 = 12 = 0.9. Then, using your approximation in (b), calculate P(|D| > d), where
d = y1/n1 – Y2/n2 is the observed difference found by the researcher.
e. Based on your answer in part (d), do you think the assumption that 71 = T2 = 0.9 is contradicted by
the data? Briefly explain why/why not. (Note: there is no right/wrong answer here, but you need to
give some probabilistic reasoning. This reasoning may be similar to the logic behind hypothesis testing,
even though you're not performing a formal statistical test in this problem.)
Y2/n2 be random variables denoting the proportion of sampled residents

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