Q: A survey finds that 69% of students graduate within six years. If you randomly select 45 students, what is the mean $$\mu$$ and standard deviation $$\sigma$$ for the normal approximation to the binomial distribution?

$$\mu = 45 \times 0.69 = 31.05$$, $$\sigma = \sqrt{45 \times 0.69 \times 0.31} \approx 3.36$$

Q: A study finds that 6% of college graduates value prestige most. If you randomly select 40 graduates, can you use the normal approximation to the binomial distribution? Why or why not?

No, because $$np = 40 \times 0.06 = 2.4 \u003c 5$$

Q: In a binomial experiment with $$n = 100$$ and $$p = 0.34$$, what is the standard deviation $$\sigma$$ for the normal approximation?

$$\sigma = \sqrt{100 \times 0.34 \times 0.66} \approx 4.74$$

Question 1

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

Accepted Answer

$$np \u003e 5$$ and $$nq \u003e 5$$

Question 2

A survey finds that 69% of students graduate within six years. If you randomly select 45 students, what is the mean $$\mu$$ and standard deviation $$\sigma$$ for the normal approximation to the binomial distribution?

Accepted Answer

$$\mu = 45 	imes 0.69 = 31.05$$, $$\sigma = \sqrt{45 	imes 0.69 	imes 0.31} \approx 3.36$$

Question 3

Why is a continuity correction necessary when using the normal distribution to approximate a binomial probability?

Accepted Answer

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

Question 4

When approximating the probability of getting between 270 and 310 successes (inclusive) in a binomial experiment, what is the correct interval to use for the normal approximation with continuity correction?

Accepted Answer

$$269.5 \u003c x \u003c 310.5$$

Question 5

A study finds that 6% of college graduates value prestige most. If you randomly select 40 graduates, can you use the normal approximation to the binomial distribution? Why or why not?

Accepted Answer

No, because $$np = 40 	imes 0.06 = 2.4 \u003c 5$$

Question 6

In a binomial experiment with $$n = 100$$ and $$p = 0.34$$, what is the standard deviation $$\sigma$$ for the normal approximation?

Accepted Answer

$$\sigma = \sqrt{100 	imes 0.34 	imes 0.66} \approx 4.74$$

Question 7

A study finds that 65% of U.S. adults drive at least once a week. If you randomly select 200 adults, what is the probability that at least 125 will say yes? (Use the normal approximation and continuity correction.)

Accepted Answer

Convert to $$P(x \u003e 124.5)$$, then find the z-score and use the standard normal table.

Question 8

Which of the following best describes the shape of the binomial distribution as $$n$$ increases?

Accepted Answer

It becomes more bell-shaped and approaches the normal curve.

Question 9

If you want to approximate the probability of getting fewer than 63 successes in a binomial experiment, what is the correct interval for the normal approximation with continuity correction?

Accepted Answer

$$x \u003c 62.5$$

Question 10

A study of former NFL players finds that 34.4% believe they have CTE. If you randomly select 100 players, what is the probability that exactly 40 will say yes? (Describe the steps for the normal approximation.)

Accepted Answer

Use the interval $$39.5 \u003c x \u003c 40.5$$, find the corresponding z-scores, and use the standard normal table to find the probability.

Question 11

Which formula gives the mean $$\mu of a $$binomial distribution with $$n$$ trials and probability $$p of $$success?

Accepted Answer

$$\mu = n p$$

Question 12

Which formula gives the standard deviation $$\sigma of a $$binomial distribution with $$n$$ trials, probability $$p of $$success, and $$q = 1 - p$$?

Accepted Answer

$$\sigma = \sqrt{n p q}$$

Question 13

In a binomial experiment, what is the effect of increasing $$n$$ while keeping $$p$$ constant on the normal approximation?

Accepted Answer

The approximation becomes more accurate as the distribution becomes more normal.

Question 14

If you want to approximate the probability of getting at least 158 successes in a binomial experiment, what is the correct interval for the normal approximation with continuity correction?

Accepted Answer

$$x \u003e 157.5$$

Question 15

A binomial distribution is represented by a probability histogram. What does the area under the normal curve represent when using the normal approximation?

Accepted Answer

The approximate probability of the event occurring.

Normal Approximations to Binomial Distributions: Study Notes

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