Mathematics, 23.02.2021 04:10 ceceshelby1130

A university considers giving only pass/fail grades to freshmen to reduce competition and stress. The student newspaper interviews faculty members and reports their opinions of the proposed policy. Suppose that 70% of the faculty favor the pass/fail proposal. Assuming 50 faculty members are interviewed. The goal is to find the probability that a majority (26 or more) will favor the proposal.

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Answer from: angsoccer02

0.9983 = 99.83% probability that a majority will favor the proposal.

Step-by-step explanation:

We use the normal approximation to the binomial to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

E(X) = np

The standard deviation of the binomial distribution is:

$\sqrt{V(X)} = \sqrt{np(1-p)}$

Normal probability distribution

Problems of normally distributed distributions can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

When we are approximating a binomial distribution to a normal one, we have that $\mu = E(X)$ , $\sigma = \sqrt{V(X)}$ .

Suppose that 70% of the faculty favor the pass/fail proposal. Assuming 50 faculty members are interviewed.

This means, respectively, that p = 0.7, n = 50

Mean and standard deviation:

$\mu = E(X) = np = 50*0.7 = 35$

$\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{50*0.7*0.3} = 3.24$

Probability that a majority (26 or more) will favor the proposal.

Using continuity correction, this is $P(X \geq 26 - 0.5) = P(X \geq 25.5)$ , which is 1 subtracted by the pvalue of Z when X = 25.5. So

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{25.5 - 35}{3.24}$

Z = -2.93

Z = -2.93 has a pvalue of 0.0017

1 - 0.0017 = 0.9983

0.9983 = 99.83% probability that a majority will favor the proposal.

Answer from: Quest

Bis your answer. it only holds the decimal numerals
Which of the following is not a possible value for a probability?

Answer from: Quest

step-by-step explanation: the declaration of the rights of man and of the citizen (french: déclaration des droits de l'homme et du citoyen de 1789), set by france's national constituent assembly in 1789, is a human civil rights document from the french revolution.[1]

the declaration was drafted by the abbé sieyès and the marquis de lafayette, in consultation with thomas jefferson.[2] influenced by the doctrine of "natural right", the rights of man are held to be universal: valid at all times and in every place, pertaining to human nature itself. it became the basis for a nation of free individuals protected equally by the law. it is included in the beginning of the constitutions of both the fourth french republic (1946) and fifth republic (1958) and is still current. inspired by the enlightenment philosophers, the declaration was a core statement of the values of the french revolution and had a major impact on the development of freedom and democracy in europe and worldwide.[3]

the 1789 declaration, together with the 1215 magna carta, the 1689 english bill of rights (1689), the 1776 united states declaration of independence, and the 1789 united states bill of rights, inspired in large part the 1948 united nations universal declaration of human rights