Q: Which of the following conditions must be satisfied to use the normal distribution as an approximation to the binomial distribution?

$$n \times p \geq 5$$ and $$n \times q \geq 5$$

Q: Given a binomial experiment with $$n = 80$$ and $$p = 0.1$$, can you use the normal approximation? Why or why not?

Yes, because $$n \times p = 8$$ and $$n \times q = 72$$, both are greater than or equal to 5.

Q: What is the mean $$\mu$$ and standard deviation $$\sigma$$ for a binomial distribution with $$n = 50$$ and $$p = 0.6$$?

$$\mu = 30$$, $$\sigma = \sqrt{50 \times 0.6 \times 0.4}$$

Q: A survey finds that 40% of adults exercise regularly. If you randomly select 150 adults, what is the standard deviation of the number who exercise regularly?

$$\sqrt{150 \times 0.4 \times 0.6}$$

Q: Given $$n = 200$$, $$p = 0.52$$, and $$q = 0.48$$, what is the $$z-$$score for $$x = 90.5 ($$after continuity correction) when approximating the probability of at most 90 successes?

$$z = \frac{90.5 - 104}{\sqrt{200 \times 0.52 \times 0.48}}$$

Q: A poll finds that 70% of voters support a policy. If 90 voters are sampled, can the normal approximation be used? Show your reasoning.

Yes, because $$n \times p = 63$$ and $$n \times q = 27$$, both are at least 5.

Q: A company finds that 15% of its shipments are delayed. If 200 shipments are tracked, what is the standard deviation for the number of delayed shipments?

$$\sqrt{200 \times 0.15 \times 0.85}$$

Q: If $$n = 50$$ and $$p = 0.02$$, can the normal approximation be used for the binomial distribution? Why or why not?

No, because $$n \times p = 1$$ which is less than 5.

Question 1

Which of the following conditions must be satisfied to use the normal distribution as an approximation to the binomial distribution?

Accepted Answer

$$n 	imes p \geq 5$$ and $$n 	imes q \geq 5$$

Question 2

Given a binomial experiment with $$n = 80$$ and $$p = 0.1$$, can you use the normal approximation? Why or why not?

Accepted Answer

Yes, because $$n 	imes p = 8$$ and $$n 	imes q = 72$$, both are greater than or equal to 5.

Question 3

What is the mean $$\mu$$ and standard deviation $$\sigma$$ for a binomial distribution with $$n = 50$$ and $$p = 0.6$$?

Accepted Answer

$$\mu = 30$$, $$\sigma = \sqrt{50 	imes 0.6 	imes 0.4}$$

Question 4

A survey finds that 40% of adults exercise regularly. If you randomly select 150 adults, what is the standard deviation of the number who exercise regularly?

Accepted Answer

$$\sqrt{150 	imes 0.4 	imes 0.6}$$

Question 5

When using the normal approximation for the probability of getting between 20 and 35 successes inclusive in a binomial experiment, what interval should you use after applying the continuity correction?

Accepted Answer

$$19.5 to$$ $$35.5$$

Question 6

Suppose you want to approximate the probability of getting at least 25 successes in a binomial experiment. Which interval should you use for the normal approximation after applying the continuity correction?

Accepted Answer

$$x \geq 24.5$$

Question 7

A researcher wants to find the probability that at most 60 out of 120 randomly selected people prefer a new product. What value should be used for the upper bound in the normal approximation after continuity correction?

Accepted Answer

$$60.5$$

Question 8

Given $$n = 200$$, $$p = 0.52$$, and $$q = 0.48$$, what is the $$z-$$score for $$x = 90.5 ($$after continuity correction) when approximating the probability of at most 90 successes?

Accepted Answer

$$z = \frac{90.5 - 104}{\sqrt{200 	imes 0.52 	imes 0.48}}$$

Question 9

In a clinical trial, 25% of patients are expected to respond to a new drug. If 120 patients are treated, what is the mean number of expected responders?

Accepted Answer

$$120 	imes 0.25$$

Question 10

A quality control manager wants to know the probability that between 15 and 25 (inclusive) out of 100 products are defective. Which interval should be used for the normal approximation?

Accepted Answer

$$14.5 to$$ $$25.5$$

Question 11

Which of the following best describes the purpose of the continuity correction when using the normal approximation to the binomial?

Accepted Answer

To adjust for the difference between discrete and continuous distributions by adding or subtracting $$0.5 to $$the endpoints

Question 12

A poll finds that 70% of voters support a policy. If 90 voters are sampled, can the normal approximation be used? Show your reasoning.

Accepted Answer

Yes, because $$n 	imes p = 63$$ and $$n 	imes q = 27$$, both are at least 5.

Question 13

If you want to approximate $$P(x = 10)$$ for a binomial variable using the normal distribution, what interval should you use after applying the continuity correction?

Accepted Answer

$$9.5 \u003c x \u003c 10.5$$

Question 14

A company finds that 15% of its shipments are delayed. If 200 shipments are tracked, what is the standard deviation for the number of delayed shipments?

Accepted Answer

$$\sqrt{200 	imes 0.15 	imes 0.85}$$

Question 15

Which of the following is the correct formula for the $$z-$$score when using the normal approximation to the binomial?

Accepted Answer

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

Question 16

A study finds that 35% of teens text while driving. In a sample of 100 teens, what is the probability that at most 30 admit to texting while driving? (Set up the normal approximation, but do not compute the final probability.)

Accepted Answer

Find $$z$$ for $$x = 30.5$$ using $$\mu = 35$$ and $$\sigma = \sqrt{100 	imes 0.35 	imes 0.65}$$

Question 17

If $$n = 50$$ and $$p = 0.02$$, can the normal approximation be used for the binomial distribution? Why or why not?

Accepted Answer

No, because $$n 	imes p = 1$$ which is less than 5.

Normal Approximation to the Binomial Distribution

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