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Chapter 7: Problem 93

According to studies done in the 1940 s, \(29 \%\) of people dream in color. Assuming this is still true, find the probability that in a random sample of 200 independent people, \(50 \%\) or more will report dreaming in color. Start by checking the conditions to see whether the Central Limit Theorem applies.

Short Answer

Expert verified
The probability that in a random sample of 200 independent people, 50% or more will report dreaming in color is virtually 0.

Step by step solution

01

Identify Constants

In this problem, we have that the proportion of people who dream in color, \(p\), is 0.29. We also have a sample size of 200, denoted \(n\). The number of successes we want to find the probability for is 50% of 200, which equals 100, denoted \(k\).
02

Check Conditions for Central Limit Theorem

We need to check two conditions for the Central Limit Theorem: \nThe sample size is large enough, \(np>10\) and \(n(1-p)>10\). \nFor this problem: \n\(np= 200 * 0.29 = 58 >10\) and \n\(n(1-p) = 200 * (1-0.29) =142 >10\). \nBoth conditions are met, so we can proceed.
03

Apply the Normal Approximation

We can use normal approximation to calculate the probability of getting 100 or more people who dream in color. However, we need to work with continuous data, and the normal distribution is a continuous probability distribution. We use a continuity correction and look for 99.5 or more. The mean \(\mu = np = 200*0.29 = 58\) and the standard deviation \(\sigma = \sqrt{np(1 鈥� p)} = \sqrt{200*0.29*0.71} = 7.43\). The z-score can be calculated as \(z = (X - \mu)/ \sigma = (99.5 -58) / 7.43 = 5.58\). Looking this z-score up in the z-table gives a probability of 1. So the probability of getting a z-score of 5.58 or less is 1. However, we want the probability of getting a z-score of 5.58 or more, which is 1 - 1 = 0.

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Most popular questions from this chapter

A poll on a proposition showed that we are \(95 \%\) confident that the population proportion of voters supporting it is between \(40 \%\) and \(48 \%\). Find the margin of error.

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