Q: A medical researcher tests a new drug on 200 patients. The probability of success is $$p = 0.2. $$What is the standard deviation for the normal approximation?

$$\sigma = \sqrt{200 \times 0.2 \times 0.8} = 5.66$$

Q: In a survey, 47% of 45 students say they get fair or poor grades. What is the mean number of students expected to respond 'yes'?

$$\mu = 45 \times 0.47 = 21.15$$

Question 1

Which of the following conditions must be met to use the normal approximation to a binomial distribution?

Accepted Answer

$$np \geq 5$$ and $$nq \geq 5$$

Question 2

Given a binomial experiment with $$n = 100$$ and $$p = 0.04$$, can you use the normal approximation? Why or why not?

Accepted Answer

No, because $$np = 4$$ and $$nq = 96$$, so $$np \u003c 5$$

Question 3

For a binomial distribution with $$n = 50$$ and $$p = 0.6$$, what is the mean $$\mu$$ and standard deviation $$\sigma$$ for the normal approximation?

Accepted Answer

$$\mu = 30$$, $$\sigma = 3.46$$

Question 4

When approximating the probability of 'at least $$c$$' successes in a binomial distribution using the normal distribution, which continuity correction should be applied?

Accepted Answer

Use $$P(x \u003e c - 0.5)$$

Question 5

A medical researcher tests a new drug on 200 patients. The probability of success is $$p = 0.2. $$What is the standard deviation for the normal approximation?

Accepted Answer

$$\sigma = \sqrt{200 	imes 0.2 	imes 0.8} = 5.66$$

Question 6

In a survey, 47% of 45 students say they get fair or poor grades. What is the mean number of students expected to respond 'yes'?

Accepted Answer

$$\mu = 45 	imes 0.47 = 21.15$$

Question 7

If you want to approximate the probability of 'fewer than 20' successes in a binomial distribution using the normal distribution, which value should you use for the continuity correction?

Accepted Answer

Use $$x = 19.5$$

Question 8

A factory produces light bulbs with a 95% success rate. If 100 bulbs are tested, what is the probability that at most 93 bulbs are successful? Which normal interval should you use?

Accepted Answer

Use $$P(x \u003c 93.5)$$

Question 9

Which formula is used to calculate the $$z-$$score when using the normal approximation to the binomial?

Accepted Answer

$$z = \frac{x - \mu}{\sigma}$$

Question 10

A school administrator wants to know the probability that more than 30 out of 50 students pass an exam, with $$p = 0.6. $$What is the correct normal interval to use?

Accepted Answer

Use $$P(x \u003e 30.5)$$

Question 11

Why is a continuity correction necessary when using the normal approximation for binomial probabilities?

Accepted Answer

Because the binomial distribution is discrete and the normal distribution is continuous

Question 12

A researcher wants to find the probability of exactly 25 successes in a binomial experiment. What normal interval should be used after applying the continuity correction?

Accepted Answer

Use $$P(24.5 \u003c x \u003c 25.5)$$

Question 13

If $$n = 80$$, $$p = 0.3$$, and you want the probability of 'at most 20' successes, what is the mean and standard deviation for the normal approximation?

Accepted Answer

$$\mu = 24$$, $$\sigma = 4.10$$

Question 14

A binomial experiment has $$n = 60$$ and $$p = 0.5. $$What is the probability that at least 40 are successful? Which normal interval should you use?

Accepted Answer

Use $$P(x \u003e 39.5)$$

Question 15

In a binomial experiment, $$n = 30$$ and $$p = 0.2. Is $$the normal approximation appropriate?

Accepted Answer

Yes, because both $$np = 6$$ and $$nq = 24$$ are greater than 5

Question 16

A company finds that 80% of its products pass quality control. If 200 products are tested, what is the mean and standard deviation for the normal approximation?

Accepted Answer

$$\mu = 160$$, $$\sigma = 5.66$$

Question 17

Which of the following best describes the process for using the normal approximation to a binomial probability?

Accepted Answer

Verify $$np \geq 5$$ and $$nq \geq 5$$, calculate $$\mu$$ and $$\sigma$$, apply continuity correction, find $$z-$$score, and use the standard normal table

Question 18

A scientist wants to know the probability that more than 15 out of 40 samples show a positive reaction, with $$p = 0.4. $$What is the correct normal interval to use?

Accepted Answer

Use $$P(x \u003e 15.5)$$

Question 19

If you want to approximate the probability of 'at most $$c$$' successes in a binomial distribution, which continuity correction should you apply?

Accepted Answer

Use $$P(x \u003c c + 0.5)$$

Normal Approximation to the Binomial Distribution

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