Q: In a singing competition with 16 finalists, how many ways can the top four singers be selected if the order matters?

$$16 \times 15 \times 14 \times 13$$

Q: A sample of 100 adults is surveyed, and 18% say they smoke. Find a 95% confidence interval for the proportion of smokers in the population.

$$0.18 \pm 1.96 \sqrt{\frac{0.18 \times 0.82}{100}}$$

Q: What is the minimum sample size required to estimate a population mean within $$\pm 2$$ units with 95% confidence, given $$\sigma = 10$$?

$$n = \left( \frac{1.96 \times 10}{2} \$$right)^2$$

Q: A random sample of 1327 U.S. adults finds that 18% have two jobs. What is the standard error of the sample proportion?

$$\sqrt{\frac{0.18 \times 0.82}{1327}}$$

Question 1

In a singing competition with 16 finalists, how many ways can the top four singers be selected if the order matters?

Accepted Answer

$$16 	imes 15 	imes 14 	imes 13$$

Question 2

A jury selection requires a group of 12 people to be selected from 18 candidates. How many different ways can this jury be chosen?

Accepted Answer

$$\binom{18}{12}$$

Question 3

If a random variable $$X$$ follows a binomial distribution with $$n=10$$ and $$p=0.3$$, what is the probability that $$X=4$$?

Accepted Answer

$$\binom{10}{4} (0.3)^4$ (0.7)^6$

Question 4

A factory produces light bulbs with a mean life of 800 hours and a standard deviation of 50 hours. What is the probability that a randomly selected bulb lasts more than 850 hours? Assume normal distribution.

Accepted Answer

$$P(Z \u003e \frac{850-800}{50})$$

Question 5

A Poisson distribution models the number of emails received per hour. If the average rate is 5 emails per hour, what is the probability of receiving exactly 3 emails in one hour?

Accepted Answer

$$\frac{5^3$ e$$^{-5}$$}{3!}$$

Question 6

A sample of 100 adults is surveyed, and 18% say they smoke. Find a 95% confidence interval for the proportion of smokers in the population.

Accepted Answer

$$0.18 \pm 1.96 \sqrt{\frac{0.18 	imes 0.82}{100}}$$

Question 7

What is the minimum sample size required to estimate a population mean within $$\pm 2$$ units with 95% confidence, given $$\sigma = 10$$?

Accepted Answer

$$n = \left( \frac{1.96 	imes 10}{2} \$$right)^2$$

Question 8

A researcher wants to test if the mean test score of a sample of 50 students is different from the national mean of 75. The sample mean is 78, and the population standard deviation is 10. What is the value of the test statistic?

Accepted Answer

$$z = \frac{78 - 75}{10 / \sqrt{50}}$$

Question 9

A random sample of 100 medical school applicants has a mean MCAT score of 521. The population mean is 520 with a standard deviation of 8. What is the $$z-$$score for the sample mean?

Accepted Answer

$$z = \frac{521 - 520}{8 / \sqrt{100}}$$

Question 10

A hypothesis test is performed at the 0.05 significance level. The $$p-$$value is 0.03. What is the correct conclusion?

Accepted Answer

Reject the null hypothesis

Question 11

A study finds that the mean price of a three-year-old used car is $$24,625 with a standard deviation of 1$$,910. What is the probability that a randomly selected car costs more than $28,000?

Accepted Answer

$$P\left(Z \u003e \frac{28000 - 24625}{1910}\right)$$

Question 12

Which of the following is NOT an assumption for using the Central Limit Theorem?

Accepted Answer

The population must be normally distributed for any sample size

Question 13

A scientist claims that the mean incubation period for swan eggs is less than 40 days. What is the null hypothesis $$H_0$$?

Accepted Answer

$$H_0$$: \mu = 40$$

Question 14

A random sample of 1327 U.S. adults finds that 18% have two jobs. What is the standard error of the sample proportion?

Accepted Answer

$$\sqrt{\frac{0.18 	imes 0.82}{1327}}$$

Question 15

Which of the following best describes a Type I error in hypothesis testing?

Accepted Answer

Rejecting the null hypothesis when it is actually true

Question 16

A researcher wants to estimate the mean cholesterol level in a population. The sample mean is 200 mg/dL, the population standard deviation is 25 mg/dL, and the sample size is 100. What is the 99% confidence interval for the mean?

Accepted Answer

$$200 \pm 2.576 	imes \frac{25}{\sqrt{100}}$$

Question 17

Which formula is used to calculate the margin of error for a population proportion at 95% confidence?

Accepted Answer

$$E = 1.96 \sqrt{\frac{p(1-p)}{n}}$$

Question 18

A hypothesis test for the mean is performed and the test statistic is $$z = 2.5. $$The critical value for a two-tailed test at $$\alpha = 0.05 is$$ $$\pm 1.96. $$What is the correct decision?

Accepted Answer

Reject the null hypothesis

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