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 7% 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). 0.0154 (round to four decimal places as needed.) sketch the graph of the normal distribution with the indicated probability shaded. (c) find the probability that the number of people who say auto racing is their favorite sport is between 40 and 46, inclusive. (round to four decimal places as needed.)

Explanation:

Step1: Check normal - approximation conditions

For a binomial distribution $X\sim B(n,p)$, we can approximate it with a normal distribution $N(np,np(1 - p))$ if $np\geq5$ and $n(1 - p)\geq5$. Here, $n = 600$ and $p=0.07$. So, $np=600\times0.07 = 42\geq5$ and $n(1 - p)=600\times(1 - 0.07)=600\times0.93 = 558\geq5$. The normal - approximation is valid, and the normal distribution is $N(np = 42,np(1 - p)=42\times0.93 = 39.06)$.

Step2: Use continuity correction

To find $P(40\leq X\leq46)$ for the binomial distribution approximated by a normal distribution $N(42,\sqrt{39.06})$, for $x = 40$, the corrected value for the normal distribution is $x_1=39.5$, and for $x = 46$, the corrected value is $x_2 = 46.5$.

Step3: Standardize the values

The z - score is given by $z=\frac{x-\mu}{\sigma}$, where $\mu = 42$ and $\sigma=\sqrt{39.06}\approx6.25$. For $x_1 = 39.5$, $z_1=\frac{39.5 - 42}{6.25}=\frac{- 2.5}{6.25}=-0.4$. For $x_2 = 46.5$, $z_2=\frac{46.5 - 42}{6.25}=\frac{4.5}{6.25}=0.72$.

Step4: Find the probabilities

We want $P(-0.4\leq Z\leq0.72)$. Using the standard normal table, $P(Z\leq0.72)=\Phi(0.72)=0.7642$ and $P(Z\leq - 0.4)=\Phi(-0.4)=0.3446$. So, $P(-0.4\leq Z\leq0.72)=P(Z\leq0.72)-P(Z\leq - 0.4)=0.7642-0.3446 = 0.4196$.

Answer:

$0.4196$