Q: Which of the following best describes the expected value $$E(X)$$ for a binomial distribution with $$n=10$$ and $$p=0.45$$?

$$E(X) = 10 \times 0.45 = 4.5$$

Q: During a clinical drug trial, the probability that a patient experiences a side effect is 0.05. If 450 patients participate, what is the expected number of patients to experience the side effect?

$$E(X) = 450 \times 0.05 = 22.5$$

Q: A local university finds that 15% of students live in dormitories. In a random sample of 50 students, what is the standard error of the sample proportion?

$$SE = \sqrt{\frac{0.15 \times 0.85}{50}}$$

Q: Your business wants to study the proportion of office workers who use standing desks. You want to be 95% confident that your estimate is within 0.06 of the true proportion. If you estimate $$p = 0.44$$, what is the minimum sample size required?

$$n = \left( \frac{1.96^2$ \times 0.44 \times 0.56}{0.06^2$} \right) \approx 264$$

Q: At a local hockey arena, the food and beverage department wants to predict the revenue for the season. A random sample of 25 fans has an average spend of $$35 with a sample standard deviation of 1.15. $$What is the 90% confidence interval for the population mean?

$$35 \pm 1.645 \times \frac{1.15}{\sqrt{25}}$$

Q: Which of the following is the correct formula for the margin of error for a confidence interval for a population mean when the population standard deviation is unknown?

$$ME = t^* \times \frac{s}{\sqrt{n}}$$

Question 1

A recent study revealed that 60% of business travelers make their own travel arrangements. If a random sample of 12 frequent business travelers is selected, what is the probability that exactly 9 of them make their own travel arrangements? Use the binomial formula.

Accepted Answer

$$P(X=9) = \binom{12}{9} (0.60)^9$ (0.40)^3$

Question 2

Which of the following best describes the expected value $$E(X)$$ for a binomial distribution with $$n=10$$ and $$p=0.45$$?

Accepted Answer

$$E(X) = 10 	imes 0.45 = 4.5$$

Question 3

Suppose 45% of California residents have adequate earthquake supplies. In a random sample of 10 residents, what is the probability that exactly 4 have adequate supplies?

Accepted Answer

$$P(X=4) = \binom{10}{4} (0.45)^4$ (0.55)^6$

Question 4

Twenty-eight percent of a business’s new clients will not return after one year. If you randomly interview 3 clients, what is the probability that at least 2 will not return?

Accepted Answer

$$P(X \geq 2) = P(X=2) + P(X=3) = \binom{3}{2} (0.28)^2$ (0.72)^1$ + \binom{3}{3} (0.28)^3$

Question 5

During a clinical drug trial, the probability that a patient experiences a side effect is 0.05. If 450 patients participate, what is the expected number of patients to experience the side effect?

Accepted Answer

$$E(X) = 450 	imes 0.05 = 22.5$$

Question 6

A local university finds that 15% of students live in dormitories. In a random sample of 50 students, what is the standard error of the sample proportion?

Accepted Answer

$$SE = \sqrt{\frac{0.15 	imes 0.85}{50}}$$

Question 7

If the sample proportion of students living in dormitories is 0.20, and the standard error is $$0.0505$$, what is the $$z-$$score for a sample proportion of $$0.172$$?

Accepted Answer

$$z = \frac{0.172 - 0.20}{0.0505} = -0.554$$

Question 8

Your business wants to study the proportion of office workers who use standing desks. You want to be 95% confident that your estimate is within 0.06 of the true proportion. If you estimate $$p = 0.44$$, what is the minimum sample size required?

Accepted Answer

$$n = \left( \frac{1.96^2$ 	imes 0.44 	imes 0.56}{0.06^2$} ight) \approx 264$$

Question 9

At a local hockey arena, the food and beverage department wants to predict the revenue for the season. A random sample of 25 fans has an average spend of $$35 with a sample standard deviation of 1.15. $$What is the 90% confidence interval for the population mean?

Accepted Answer

$$35 \pm 1.645 	imes \frac{1.15}{\sqrt{25}}$$

Question 10

Which of the following is the correct formula for the margin of error for a confidence interval for a population mean when the population standard deviation is unknown?

Accepted Answer

$$ME = t^* 	imes \frac{s}{\sqrt{n}}$$

Question 11

If a 95% confidence interval for a population proportion is $$(0.35, 0.55)$$, what does this mean in context?

Accepted Answer

We are 95% confident that the true population proportion is between 0.35 and 0.55.

Step-by-Step Guidance for Binomial Distributions, Sampling, and Confidence Intervals

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