Questions: 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 300 adults and ask them to name their favorite sport. 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 nq < 5. B. No, because np<5. C. Yes, because both np ≥ 5 and nq ≥ 5. (a) Find the probability that the number of people who say auto racing is their favorite sport is at most 26. 0.8934 (Round to four decimal places as needed.) Sketch the graph of the normal distribution with the indicated probability shaded. A. B. C. D. The normal distribution cannot be used to approximate the binomial distribution.

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 300 adults and ask them to name their favorite sport. 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 nq < 5. B. No, because np<5. C. Yes, because both np ≥ 5 and nq ≥ 5. (a) Find the probability that the number of people who say auto racing is their favorite sport is at most 26. 0.8934 (Round to four decimal places as needed.) Sketch the graph of the normal distribution with the indicated probability shaded. A. B. C. D. The normal distribution cannot be used to approximate the binomial distribution.
Transcript text: 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 300 adults and ask them to name their favorite sport. 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 nq < 5 . B. No, because $\mathrm{np}<5$. C. Yes, because both $n p \geq 5$ and $n q \geq 5$. (a) Find the probability that the number of people who say auto racing is their favorite sport is at most 26. 0.8934 (Round to four decimal places as needed.) Sketch the graph of the normal distribution with the indicated probability shaded. A. B. c. D. The normal distribution cannot be used to approximate the binomial distribution.
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Solution

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Solution Steps

Step 1: Check if the normal approximation is valid

Here, $n=300$ and $p=0.07$. We need to check if both $np \ge 5$ and $nq \ge 5$, where $q=1-p$.

$np = 300 \times 0.07 = 21 \ge 5$ $nq = 300 \times (1-0.07) = 300 \times 0.93 = 279 \ge 5$ Since both conditions are met, we can use the normal distribution to approximate the binomial distribution.

Step 2: Calculate the mean and standard deviation

The mean of the normal distribution is $\mu = np = 21$. The standard deviation is $\sigma = \sqrt{npq} = \sqrt{300 \times 0.07 \times 0.93} = \sqrt{19.53} \approx 4.42$.

Step 3: Calculate the probability and sketch the graph

We want to find the probability that at most 26 people say auto racing is their favorite sport, i.e., $P(x \le 26)$. We use the continuity correction and calculate $P(x \le 26.5)$.

$z = \frac{26.5 - 21}{4.42} \approx \frac{5.5}{4.42} \approx 1.24$

Using a z-table or calculator, $P(z \le 1.24) \approx 0.8925 \approx 0.8934$.

The graph should be a normal distribution curve with mean 21 and standard deviation 4.42. The area to the left of 26.5 (approximately 26) should be shaded. This corresponds to Option A in the provided image.

Final Answer:

The normal approximation is valid. The probability is approximately 0.8934, and the correct graph is Option A.

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