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Chapter 5: Problem 7

If \(Y\) is \(b\left(100, \frac{1}{2}\right)\), approximate the value of \(P(Y=50)\).

Short Answer

Expert verified
The approximation of the binomial distribution with \(n = 100\) and \(p = 1/2\) yields that the probability of obtaining exactly 50 successes, \(P(Y = 50)\), approximately corresponds to the range \(49.5 < Y < 50.5\) in our Normal approximation. This corresponds with \(P(-0.1 < Z < 0.1)\), giving us a specific result when looking up the probability in the Standard Normal Tables.

Step by step solution

01

Title: Identify the variables

The given distribution is \(Y = b(n, p)\), where \(n = 100\) (the number of trials) and \(p = 1/2\) (probability of success on each trial). In order to find \(P(Y=50)\), we need the mean and standard deviation of this distribution.
02

Title: Calculate the mean and standard deviation

The mean \(\mu\) and standard deviation \(\sigma\) of a binomial distribution can be found using the formulas \(\mu = np\) and \(\sigma = \sqrt{np(1-p)}\). From the given values, \(\mu = 100 * 1/2 = 50\) and \(\sigma = \sqrt{100*(1/2)*(1-1/2)} = 5\).
03

Title: Apply the Normal distribution approximation

The binomial distribution can be approximated by the normal distribution when the number of trials is large (n > 30). Using this approximation, we get the equivalent Z values for \(P(Y = 50)\) using the formula \(Z = (X - \mu) / \sigma\). Substituting \(X = 50\), \(\mu = 50\), and \(\sigma = 5\) gives \(Z = (50 - 50) / 5 = 0\). Now, the question becomes finding P(Z = 0).
04

Title: Interpret the Z value

The Z value relates to standard deviations. By definition, Z = 0 corresponds to the mean of the distribution. In a standard normal distribution N(0,1), P(Z = 0) is associated with the peak of the Gaussian bell, so technically it relates to the density rather than to the probability of a single discrete point. However, the context implies to approximate the probability of obtaining exactly the mean value in our original problem, so we assume a bin width of 1 around our target value (49.5 < Y < 50.5).
05

Title: Find the probability

To find the approximate probability that Y is exactly 50, \(P(49.5 < Y < 50.5)\) using z-scores, we instead find \(P(-0.1 < Z < 0.1)\). Looking this up in the standard normal tables (or using a calculator or software package), we find a probability.

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