The sample size \( n \), probability of success \( p \), and probability of failure \( q \) are given for a binomial experiment. Decide whether you can use the normal distribution to approximate the random variable \( x \). Can the normal distribution be used to approximate the random variable \( x \) ? A. Yes, because \( n p \geq 5 \) and \( n q \geq 5 \). B. No, because \( n p<5 \). C. No, because \( n q<5 \). D. No, because \( n p<5 \) and \( n q<5 \).

To determine if we can use the normal distribution to approximate a binomial distribution for the random variable \( x \), we look at the conditions involving the sample size \( n \), probability of success \( p \), and probability of failure \( q \). The rule of thumb states that both \( n p \) and \( n q \) should be at least 5. Thus, the correct answer is: **A. Yes, because \( n p \geq 5 \) and \( n q \geq 5 \).** The normal approximation is a handy tool! It simplifies calculations for large sample sizes, allowing us to use the elegant properties of the bell curve instead of the more cumbersome binomial formula. So, in scenarios where \( n \) is large and both probabilities are satisfied, feel free to unleash your inner statistician and enjoy the smoothness of normal distribution! Also, if you're venturing into the world of statistical experiments, remember that using the normal approximation can greatly streamline your calculations. Keep an eye on those \( n p \) and \( n q \) thresholds, and you’ll be sailing smoothly through your statistical seas!