Question

According to the 2011 Gallup daily tracking polls (www.gallup, February 3, 2012), Mississippi is the most conservative U.S. state, with 53.4 percent of its residents identifying themselves as conservative. What is the probability that fewer than 45 respondents of a random sample of 100 Mississippi residents do not identify themselves as conservative

Answer 1

Answer:

33.72% probability that fewer than 45 respondents of a random sample of 100 Mississippi residents do not identify themselves as conservative

Step-by-step explanation:

I am going to 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 samples 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 $\mu = E(X)$, $\sigma = \sqrt{V(X)}$.

In this problem, we have that:

$n = 100, p = 1 - 0.534 = 0.466$

So

$\mu = E(X) = np = 100*0.466 = 46.6$

$\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{100*0.534*0.466} = 4.99$

What is the probability that fewer than 45 respondents of a random sample of 100 Mississippi residents do not identify themselves as conservative

Using continuity correction, this is $P(X < 45 - 0.5) = P(X < 44.5)$, which is the pvalue of Z when X = 44.5. So

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

$Z = \frac{44.5 - 46.6}{4.99}$

$Z = -0.42$

$Z = -0.42$ has a pvalue of 0.3372

33.72% probability that fewer than 45 respondents of a random sample of 100 Mississippi residents do not identify themselves as conservative