Questions: Determine 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 in a region found that 32% name professional football as their favorite sport. You randomly select 12 adults in the region and ask them to name their favorite sport. Complete parts (a) through (d) below Determine whether a normal distribution can be used to approximate the binomial distribution. Choose the correct answer below. A. No, np<5 B. No, nq<5 C. Yes, both np ≥ 5 and nq ≥ 5

Determine 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 in a region found that 32% name professional football as their favorite sport. You randomly select 12 adults in the region and ask them to name their favorite sport. Complete parts (a) through (d) below

Determine whether a normal distribution can be used to approximate the binomial distribution. Choose the correct answer below.
A. No, np<5
B. No, nq<5
C. Yes, both np ≥ 5 and nq ≥ 5

Transcript text: Determine 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 in a region found that $32 \%$ name professional football as their favorite sport. You randomly select 12 adults in the region and ask them to name their favorite sport. Complete parts (a) through (d) below Determine whether a normal distribution can be used to approximate the binomial distribution. Choose the correct answer below. A. No, $\mathrm{np}<5$ B. No, $\mathrm{nq}<5$ C. Yes, both $n p \geq 5$ and $n q \geq 5$

Solution

Solution Steps

Step 1: Check Normal Approximation Conditions

To determine whether we can use the normal distribution to approximate the binomial distribution, we calculate $ np $ and $ nq $:

\[ np = n \cdot p = 12 \cdot 0.32 = 3.84 \] \[ nq = n \cdot q = 12 \cdot (1 - 0.32) = 12 \cdot 0.68 = 8.16 \]

Since $ np < 5 $, we cannot use the normal distribution for approximation.

Step 2: Use Binomial Distribution

Since the normal approximation is not valid, we will use the binomial distribution to find the probability of exactly 3 successes. The probability mass function for a binomial distribution is given by:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

For $ n = 12 $, $ x = 3 $, $ p = 0.32 $, and $ q = 0.68 $:

\[ P(X = 3) = \binom{12}{3} \cdot (0.32)^3 \cdot (0.68)^{12-3} \]

Calculating this gives:

\[ P(X = 3) \approx 0.2241 \]