Although the binomial distribution is discrete, in many cases it can be modeled using a normal distribution.If X is binomial with mean = μx and standard deviation = σx, then we can define a new random variable Y that is normal with the same mean and same standard deviation.In this case under certain assumptions,P(X ≤ x) ≈ P(Y ≤ x).This relationship is most useful when X has a large number of trials.If X is binomial with a mean of 25 and a standard deviation of 4.33 (n = 100, p = 0.25), what is P(X ≤ 23) using the normal approximation discussed here?
A) 0.3228
B) 0.4300
C) 0.4500
D) 0.6772

Question

Quizplus · Accepted Answer

To use the normal approximation to the binomial distribution, we first convert the binomial random variable X to the standard normal variable Z using the formula $Z = \frac{X - \mu}{\sigma}$. Given $\mu = 25$ and $\sigma = 4.33$, for $X = 23$, $Z = \frac{23 - 25}{4.33} \approx -0.462$. Looking up this Z-value in the standard normal distribution table or using a calculator, we find the probability $P(Z \leq -0.462) \approx 0.3228$, which corresponds to $P(X \leq 23)$ in the normal approximation.

Although the Binomial Distribution Is Discrete, in Many Cases It