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). (c) find the probability that the number of people who say auto racing is their favorite sport is between 51 and 58, inclusive. (round to four decimal places as needed.)

Explanation:

Step1: Check if normal approximation is valid

For a binomial distribution \(X\sim B(n,p)\), we can use normal approximation \(X\sim N(np,np(1 - p))\) if \(np\geq5\) and \(n(1 - p)\geq5\).
Here, \(n = 600\), \(p=0.09\).
\(np=600\times0.09 = 54\geq5\)
\(n(1 - p)=600\times(1 - 0.09)=600\times0.91 = 546\geq5\)
So, we can use normal approximation \(X\sim N(\mu = np=54,\sigma=\sqrt{np(1 - p)})\)
\(\sigma=\sqrt{600\times0.09\times(1 - 0.09)}=\sqrt{54\times0.91}=\sqrt{49.14}\approx7.01\)

Step2: Apply continuity correction

For \(P(51\leq X\leq58)\) (since \(X\) is binomial and we are using normal approximation for a discrete - to - continuous conversion), we use \(P(50.5\lt Y\lt58.5)\) where \(Y\sim N(54,7.01^{2})\)

Step3: Standardize the values

The \(z\) - score formula is \(z=\frac{x-\mu}{\sigma}\)
For \(x = 50.5\), \(z_1=\frac{50.5 - 54}{7.01}=\frac{- 3.5}{7.01}\approx - 0.50\)
For \(x = 58.5\), \(z_2=\frac{58.5 - 54}{7.01}=\frac{4.5}{7.01}\approx0.64\)

Step4: Find the probabilities

We know that \(P(a\lt Z\lt b)=\Phi(b)-\Phi(a)\) where \(\Phi(z)\) is the cumulative distribution function of the standard normal distribution.
\(P(-0.50\lt Z\lt0.64)=\Phi(0.64)-\Phi(-0.50)\)
From the standard normal table, \(\Phi(0.64) = 0.7389\), \(\Phi(-0.50)=0.3085\)
\(P(-0.50\lt Z\lt0.64)=0.7389 - 0.3085=0.4304\)

Answer:

\(0.4304\)