Assume that x is a binomial random variable with n = 400 and p = 0.30. Use a normal approximation to find P(x 100).
A)0.5854
B)0.9834
C)0.9875
D)0.5910

Question

Assume that x is a binomial random variable with n = 400 and p = 0.30. Use a normal approximation to find P(x > 100).
A)0.5854
B)0.9834
C)0.9875
D)0.5910

Quizplus · Accepted Answer

To approximate the probability $P(x > 100)$ for a binomial random variable using a normal distribution, we first calculate the mean ($\mu$) and standard deviation ($\sigma$) of the binomial distribution. The formulas for these are $\mu = np$ and $\sigma = \sqrt{np(1-p)}$, respectively. Given $n = 400$ and $p = 0.30$, we find:$$\mu = 400 \times 0.30 = 120$$$$\sigma = \sqrt{400 \times 0.30 \times (1 - 0.30)} = \sqrt{400 \times 0.30 \times 0.70} = \sqrt{84} \approx 9.165$$To find $P(x > 100)$, we convert the binomial problem into a normal distribution problem and use the Z-score formula $Z = \frac{X - \mu}{\sigma}$. For $X = 100$:$$Z = \frac{100 - 120}{9.165} \approx -2.18$$Using a Z-table or calculator, we find the probability of $Z > -2.18$, which corresponds to the area to the right of $Z = -2.18$ on the standard normal distribution curve. This area represents $P(x > 100)$. The Z-score of -2.18 corresponds to a left-tail probability, but since we're interested in the right-tail (greater than 100), we subtract this value from 1. The exact value might slightly vary depending on the Z-table or calculator used, but the correct approximation given the options is closest to 0.9834.

Assume That X Is a Binomial Random Variable with N