Question

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 distribution to find the indicated probabilities. a survey of adults found that 9% say their favorite sport is auto racing. you randomly select 600 adults and ask them to name their favorite sport. complete parts (a) through (d). (a) find the probability that the number of people who say auto racing is their favorite sport is at most 63. (round to four decimal places as needed.) sketch the graph of the normal distribution with the indicated probability shaded.

Explanation:

Step1: Check normal approximation conditions

For a binomial distribution \(X\sim B(n,p)\) (here \(n = 600\), \(p=0.09\)), we check \(np\) and \(n(1 - p)\).
\(np=600\times0.09 = 54\) and \(n(1 - p)=600\times(1 - 0.09)=600\times0.91 = 546\). Since \(np\geq5\) and \(n(1 - p)\geq5\), we can use the normal approximation. The mean of the normal distribution is \(\mu=np = 54\), and the standard deviation is \(\sigma=\sqrt{np(1 - p)}=\sqrt{600\times0.09\times0.91}=\sqrt{49.14}\approx7.01\).

Step2: Continuity correction

For \(P(X\leq63)\) (binomial), using the normal approximation \(Y\sim N(\mu = 54,\sigma\approx7.01)\), we apply the continuity correction. We find \(P(Y\leq63 + 0.5)=P(Y\leq63.5)\).

Step3: Standardize the value

We use the formula \(z=\frac{x-\mu}{\sigma}\). Substitute \(x = 63.5\), \(\mu = 54\), and \(\sigma\approx7.01\) into the formula: \(z=\frac{63.5 - 54}{7.01}=\frac{9.5}{7.01}\approx1.36\).

Step4: Find the probability

Using the standard - normal table \(P(Z\leq1.36)\). Looking up the value in the standard - normal table (or using a calculator with a normal - distribution function, e.g., for a TI - 84: normalcdf(-100,1.36)), we get \(P(Z\leq1.36)=0.9131\).

Answer:

\(0.9131\)