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Chapter 5: Problem 17

Find the mean and standard deviation for a binomial distribution with \(n=100\) and these values of \(p:\) a. \(p=.01\) b. \(p=.9\) c. \(p=.3\) d. \(p=.7\) e. \(p=.5\)

Short Answer

Expert verified
Question: Calculate the mean and standard deviation for a binomial distribution with n=100 and the following values of p: a) 0.01, b) 0.9, c) 0.3, d) 0.7, e) 0.5. Answer: a. For p = 0.01, Mean = 1, Standard Deviation = 0.995 b. For p = 0.9, Mean = 90, Standard Deviation = 3 c. For p = 0.3, Mean = 30, Standard Deviation = 4.58 d. For p = 0.7, Mean = 70, Standard Deviation = 4.58 e. For p = 0.5, Mean = 50, Standard Deviation = 5.

Step by step solution

01

a. Mean and standard deviation for \(p = 0.01\)

Calculate the mean using \(\mu = np\) and standard deviation using \(\sigma = \sqrt{np(1-p)}\) for \(p = 0.01\). Mean: $$\mu = (100)(0.01) = 1$$ Standard Deviation: $$\sigma =\sqrt{(100)(0.01)(1-0.01)} = \sqrt{0.99} \approx 0.995$$
02

b. Mean and standard deviation for \(p = 0.9\)

Calculate the mean and standard deviation for \(p = 0.9\). Mean: $$\mu = (100)(0.9) = 90$$ Standard Deviation: $$\sigma =\sqrt{(100)(0.9)(1-0.9)} =\sqrt{9} = 3$$
03

c. Mean and standard deviation for \(p = 0.3\)

Calculate the mean and standard deviation for \(p = 0.3\). Mean: $$\mu = (100)(0.3) = 30$$ Standard Deviation: $$\sigma =\sqrt{(100)(0.3)(1-0.3)} =\sqrt{21} \approx 4.58$$
04

d. Mean and standard deviation for \(p = 0.7\)

Calculate the mean and standard deviation for \(p = 0.7\). Mean: $$\mu = (100)(0.7) = 70$$ Standard Deviation: $$\sigma =\sqrt{(100)(0.7)(1-0.7)} =\sqrt{21} \approx 4.58$$
05

e. Mean and standard deviation for \(p = 0.5\)

Calculate the mean and standard deviation for \(p = 0.5\). Mean: $$\mu = (100)(0.5) = 50$$ Standard Deviation: $$\sigma =\sqrt{(100)(0.5)(1-0.5)} =\sqrt{25} = 5$$ So, we have calculated the mean and standard deviation for each value of \(p\) as follows: a. \(p=.01\) | Mean = 1 | Standard Deviation = 0.995 b. \(p=.9\) | Mean = 90 | Standard Deviation = 3 c. \(p=.3\) | Mean = 30 | Standard Deviation = 4.58 d. \(p=.7\) | Mean = 70 | Standard Deviation = 4.58 e. \(p=.5\) | Mean = 50 | Standard Deviation = 5

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Most popular questions from this chapter

A candy dish contains five blue and three red candies. A child reaches up and selects three candies without looking. a. What is the probability that there are two blue and one red candies in the selection? b. What is the probability that the candies are all red? c. What is the probability that the candies are all blue?

Use the formula for the binomial probability distribution to calculate the values of \(p(x)\), and construct the probability histogram for \(x\) when \(n=6\) and \(p=.2\). [HINT: Calculate \(P(x=k)\) for seven different values of \(k\).]

Gender Bias? A company has five applicants for two positions: two women and three men. Suppose that the five applicants are equally qualified and that no preference is given for choosing either gender. Let \(x\) equal the number of women chosen to fill the two positions. a. Write the formula for \(p(x)\), the probability distribution of \(x\) b. What are the mean and variance of this distribution? c. Construct a probability histogram for \(x\).

Evaluate these binomial probabilities: a. \(C_{0}^{8}(.2)^{0}(.8)^{8}\) b. \(C_{1}^{8}(.2)^{1}(.8)^{7}\) c. \(C_{2}^{8}(.2)^{2}(.8)^{6}\) d. \(P(x \leq 1)\) when \(n=8, p=.2\) e. \(P(\) two or fewer successes)

Under what conditions can the Poisson random variable be used to approximate the probabilities associated with the binomial random variable? What application does the Poisson distribution have other than to estimate certain binomial probabilities?
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