Question

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

Short answer

The approximate value of \(P(Y=50)\) is 0.0797.

Step-by-step solution

Calculate the mean and standard deviation

Before proceeding with the approximation, it's necessary to calculate the mean \(\mu\) and standard deviation \(\sigma\) from the binomial formula. The mean \(\mu\) for a binomial distribution is given by the formula \(\mu = np\), and the standard deviation \(\sigma\) is given by \(\sigma = \sqrt{np(1-p)}\). For the given values, \(\mu = 100*0.5 = 50\), and the standard deviation \(\sigma = \sqrt{100*0.5*0.5} = 5\).

Approximate the binomial distribution using a normal distribution

The significant principle behind this step is the central limit theorem, which states that if sample size is large enough, the sampling distribution of the mean will be approximately normally distributed. So, \(Y\) can be approximated by \(N(\mu,\sigma^2)\), a normal distribution with a mean of 50 and variance of \(5^2\).

Calculate the required probability

The required probability \(P(Y=50)\) in the standard normal distribution would be \(P(49.5<Y<50.5)\) because of the Continuity Correction Factor, due to the fact we are approximating a discrete distribution using a continuous distribution. Converting to z-scores: \(P(\frac{49.5-\mu}{\sigma}<Z<\frac{50.5-\mu}{\sigma})\), substituting the known values we get \(P(-0.1<Z<0.1)\). Using Standard Normal tables, or as calculated by software or calculators, one finds that this corresponds to a probability of approximately 0.0797.