Decide whether you can use the normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use the binomial distibution to find the indicated probabilities. A survey of adults found that \( 8 \% \) say their favorite sport is auto racing. You randomly select 300 adults and ask them to name their favorite spert. Complete parts (a) through (d) Determine whether a normal distribution can be used to approximate the binomial distribution. Choose the correct answer below. A. No, because np \( <5 \). B. No, because nq \( <5 \). C. Yes, because both \( n p \geq 5 \) and nq \( \geq 5 \).

To determine whether you can use the normal distribution to approximate the binomial distribution, you need to check the conditions involving \( np \) and \( nq \). Here, \( n = 300 \), \( p = 0.08 \), and \( q = 1 - p = 0.92 \). Calculating \( np = 300 \times 0.08 = 24 \) and \( nq = 300 \times 0.92 = 276 \): - Since both \( np \) and \( nq \) are greater than 5, we conclude that the normal distribution can indeed be used to approximate the binomial distribution. Therefore, the correct choice is C. Now, as for real-world applications, consider how this statistical approach aids businesses. For instance, if a company promotes a new auto racing product to a group of 300 adults, knowing the percentage who favor this sport can guide marketing strategies effectively. Collecting data and applying this approximation can result in tailored campaigns that resonate with their target audience, maximizing engagement and sales. For a fun twist, dive into the historical application of probability! The binomial distribution has roots in the work of mathematicians like Jacob Bernoulli and Pierre-Simon Laplace, who laid the groundwork for modern statistics. Imagine these pioneers crunching numbers long before computers were a twinkle in anyone's eye, pondering how many times a coin would land heads or tails when flipped—a classic original problem that led to the vast realm of probability we use today!