Q: A previous study found that 80% of people preferred drinking Pepsi over Coca Cola. For a random sample of 100 people, what is the mean and standard deviation of the sample proportion $$\hat{p}$$?

Mean: $$0.8$$, Standard deviation: $$\sqrt{\frac{0.8 \times 0.2}{100}}$$

Q: For a binomial random variable $$X$$ with $$n = 10$$ and $$p = 0.5$$, what is the mean and standard deviation?

Mean: $$5$$, Standard deviation: $$\sqrt{10 \times 0.5 \times 0.5}$$

Question 1

Suppose a binomial experiment has $$n = 100$$ trials and the probability of success is $$p = 0.54. $$Which formula would you use to approximate the probability that between 52 and 70 people vote for a candidate using the normal distribution?

Accepted Answer

Use $$P(52.5 \u003c X \u003c 70.5)$$ with the normal approximation and continuity correction.

Question 2

A previous study found that 80% of people preferred drinking Pepsi over Coca Cola. For a random sample of 100 people, what is the mean and standard deviation of the sample proportion $$\hat{p}$$?

Accepted Answer

Mean: $$0.8$$, Standard deviation: $$\sqrt{\frac{0.8 	imes 0.2}{100}}$$

Question 3

When using the normal approximation for a binomial distribution, which condition must be satisfied for the approximation to be valid?

Accepted Answer

$$np \geq 5$$ and $$n(1-p) \geq 5$$

Question 4

A car dealer has 76 cars produced in 2023. If 36 million out of 94 million cars were recalled in 2023, what is the probability that more than half of the dealer's cars were recalled? Which distribution should you use to approximate this probability?

Accepted Answer

Use the normal approximation to the binomial with $$n = 76$$ and $$p = \frac{36}{94}.$$

Question 5

For a binomial random variable $$X$$ with $$n = 10$$ and $$p = 0.5$$, what is the mean and standard deviation?

Accepted Answer

Mean: $$5$$, Standard deviation: $$\sqrt{10 	imes 0.5 	imes 0.5}$$

Question 6

Why is a continuity correction used when approximating binomial probabilities with the normal distribution?

Accepted Answer

To adjust for the difference between discrete and continuous distributions.

Question 7

Given a sample proportion $$\hat{p}$$, what is the formula for its standard deviation?

Accepted Answer

$$\sqrt{\frac{p(1-p)}{n}}$$

Question 8

If you want to find the probability that between 10 and 11 people in a sample of 100 prefer Coca Cola, which normal approximation formula should you use?

Accepted Answer

$$P(9.5 \u003c X \u003c 11.5)$$

Question 9

A sample of 100 voters is taken in a two-person election with $$p = 0.48. $$What is the expected number of voters for the candidate?

Accepted Answer

$$100 	imes 0.48 = 48$$

Question 10

In the context of sampling distributions, what does the sample proportion $$\hat{p}$$ represent?

Accepted Answer

The proportion of successes in a single sample.

Question 11

If you flip a coin 10 times and repeat this experiment many times, what does the distribution of sample proportions of heads look like?

Accepted Answer

It is approximately normal with mean $$p$$ and standard deviation $$\sqrt{\frac{p(1-p)}{n}}.$$

Question 12

Which of the following is NOT a valid step when using the normal approximation to estimate a binomial probability?

Accepted Answer

Use the sample mean as the population mean without checking conditions.

Question 13

A study finds that 80% of people prefer Pepsi. In a sample of 100, what is the probability that at least 85 people prefer Pepsi? Which normal approximation interval should you use?

Accepted Answer

$$P(X \geq 85)$$, so use $$P(X \u003e 84.5)$$

Question 14

Which formula would you use to convert a binomial variable $$X to a$$ $$z-$$score for the normal approximation?

Accepted Answer

$$z = \frac{X - np}{\sqrt{np(1-p)}}$$

Question 15

If $$p = 0.5$$ and $$n = 10$$, what is the probability of getting exactly 5 heads using the normal approximation?

Accepted Answer

Use $$P(4.5 \u003c X \u003c 5.5)$$ with mean $$5$$ and standard deviation $$1.58$$

Question 16

What is the main advantage of using the normal approximation for binomial probabilities in large samples?

Accepted Answer

It simplifies calculations, especially for large $$n.$$

Sampling Distribution of Sample Proportion & Normal Approximation in Statistics

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