Use the normal approximation to find the indicated probability. The sample size is population proportion of successes is $p$, and $X$ is the number of successes in the sample.
$$n = 90 , p = 0.41 : P ( 32

Question

Use the normal approximation to find the indicated probability. The sample size is population proportion of successes is $p$, and $X$ is the number of successes in the sample.
$$n = 90 , p = 0.41 : P ( 32 < X < 43 )$$
A) 0.7113
B) 0.1151
C) 0.1736
D) 0.8849

Quizplus · Accepted Answer

To use the normal approximation to the binomial distribution, first calculate the mean ($\mu$) and standard deviation ($\sigma$) of the distribution:$$\mu = np = 90 \times 0.41 = 36.9$$$$\sigma = \sqrt{np(1-p)} = \sqrt{90 \times 0.41 \times (1-0.41)} \approx 4.74$$Next, convert the given range to a z-score:$$z = \frac{X - \mu}{\sigma}$$For $X = 32$:$$z = \frac{32 - 36.9}{4.74} \approx -1.03$$For $X = 43$:$$z = \frac{43 - 36.9}{4.74} \approx 1.29$$Using a standard normal distribution table or calculator, find the probabilities corresponding to these z-scores and subtract the smaller from the larger to find the probability of $X$ being between 32 and 43. This calculation yields approximately 0.7113, indicating that the probability $P(32 < X < 43)$ is about 0.7113.

Use the Normal Approximation to Find the Indicated Probability ppp , And

Use the Normal Approximation to Find the Indicated Probability $p$ , And