Question 1

Which of the following is NOT one of the three criteria for a binomial probability experiment?

Accepted Answer

The probability of success must change from trial to trial.

Question 2

Under what conditions can the normal distribution be used to approximate the binomial distribution?

Accepted Answer

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

Question 3

Given a binomial random variable $$X$$ with $$n = 60$$ and $$p = 0.5$$, what are the mean $$\mu$$ and standard deviation $$\sigma$$ for the normal approximation?

Accepted Answer

$$\mu = 30$$, $$\sigma = 3.87$$

Question 4

Which of the following correctly applies the continuity correction for approximating $$P(X \leq 5)$$ using the normal distribution?

Accepted Answer

$$P(X \leq 5.5)$$

Question 5

A hospital administrator wants to know the probability that in a random sample of 300 people, fewer than 30 have blood type O-negative, given that 7% of the population has this blood type. Which normal approximation setup is correct?

Accepted Answer

Use $$n = 300$$, $$p = 0.07$$, $$\mu = 21$$, $$\sigma = 4.48$$, and calculate $$P(X \u003c 29.5)$$

Question 6

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

Accepted Answer

Because the binomial is discrete and the normal is continuous, so we adjust by $$0.5 to $$better approximate the probability.

Question 7

Suppose a survey finds that 63% of adults in a county favor the death penalty. In a random sample of 1000 adults, 630 favor the death penalty. What is the $$z-$$score for this sample proportion using the normal approximation?

Accepted Answer

$$z = 0$$

Question 8

If $$n = 100$$ and $$p = 0.2$$, what is the probability that $$X is $$between 15 and 25 inclusive, using the normal approximation with continuity correction?

Accepted Answer

Calculate $$P(14.5 \u003c X \u003c 25.5)$$ using $$\mu = 20$$, $$\sigma = 4$$

Question 9

Which of the following statements about the normal approximation to the binomial is TRUE?

Accepted Answer

The approximation improves as $$n$$ increases and $$p is $$not too close to 0 or 1.

Question 10

A quality control manager inspects 80 products, each with a 5% defect rate. What is the probability that at most 6 products are defective, using the normal approximation?

Accepted Answer

Use $$\mu = 4$$, $$\sigma = 1.95$$, and calculate $$P(X \leq 6.5)$$

Question 11

If $$n = 50$$ and $$p = 0.1$$, is it appropriate to use the normal approximation to the binomial? Why or why not?

Accepted Answer

Yes, because $$np = 5$$ and $$nq = 45$$, both are at least 5.

Question 12

A binomial random variable $$X$$ has $$n = 200$$ and $$p = 0.4. $$What is the standard deviation $$\sigma$$ for the normal approximation?

Accepted Answer

$$\sigma = 6.93$$

Question 13

Which of the following is a real-world scenario where the normal approximation to the binomial would be appropriate?

Accepted Answer

Estimating the probability that at least 120 out of 200 randomly selected voters support a candidate, given a 60% support rate.

Question 14

When using the normal approximation to the binomial, which formula gives the mean $$\mu$$?

Accepted Answer

$$\mu = np$$

Question 15

A researcher wants to approximate $$P(X \geq 25)$$ for a binomial variable using the normal distribution. Which is the correct continuity correction?

Accepted Answer

$$P(X \geq 24.5)$$

Question 16

If $$n = 100$$ and $$p = 0.5$$, what is the probability that $$X is $$exactly 50 using the normal approximation?

Accepted Answer

Calculate $$P(49.5 \u003c X \u003c 50.5)$$ using $$\mu = 50$$, $$\sigma = 5$$

Section 7.4: The Normal Approximation to the Binomial Probability Distribution

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