Q: A researcher wants to estimate the proportion of college students who own a laptop. She surveys 200 students and finds that 150 own a laptop. Construct a 95% confidence interval for the population proportion.

$$\hat{p} = 0.75$$, $$SE = \sqrt{\frac{0.75 \times 0.25}{200}}$$, CI: $$0.75 \pm 1.96 \times SE$$

Q: A sample of 36 students has a mean test score of 82 with a population standard deviation of 12. What is the 95% confidence interval for the population mean?

CI: $$82 \pm 1.96 \times \frac{12}{\sqrt{36}}$$

Q: Which of the following is the correct formula for the probability of at least one success in $$n$$ independent Bernoulli trials with probability $$p of $$success?

$$P(\text{at least one success}) = 1 - (1-p)^n$$

Question 1

A pharmaceutical company claims that their new drug is effective in curing a disease in 80% of cases. In a clinical trial, 15 patients are treated with the drug. What is the probability that exactly 12 patients are cured? Use the binomial probability formula.

Accepted Answer

$$P(X = 12) = \binom{15}{12} (0.8$$)^{12}$$ (0.2)^3$

Question 2

Which of the following statements about the Central Limit Theorem is correct?

Accepted Answer

As the sample size increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of the population's distribution.

Question 3

A researcher wants to estimate the proportion of college students who own a laptop. She surveys 200 students and finds that 150 own a laptop. Construct a 95% confidence interval for the population proportion.

Accepted Answer

$$\hat{p} = 0.75$$, $$SE = \sqrt{\frac{0.75 	imes 0.25}{200}}$$, CI: $$0.75 \pm 1.96 	imes SE$$

Question 4

Which of the following is the correct formula for the standard error of the mean when sampling from a population with standard deviation $$\sigma$$ and sample size $$n$$?

Accepted Answer

$$SE = \frac{\sigma}{\sqrt{n}}$$

Question 5

A hospital administrator wants to compare the average length of stay for patients in two different wards. Which statistical test should be used to determine if there is a significant difference between the two means?

Accepted Answer

Two-sample t-test

Question 6

A fair six-sided die is rolled 10 times. What is the probability of getting exactly 3 sixes? Use the binomial probability formula.

Accepted Answer

$$P(X = 3) = \binom{10}{3} (\frac{1}{6})^3$ (\frac{5}{6})^7$

Question 7

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 8

A sample of 36 students has a mean test score of 82 with a population standard deviation of 12. What is the 95% confidence interval for the population mean?

Accepted Answer

CI: $$82 \pm 1.96 	imes \frac{12}{\sqrt{36}}$$

Question 9

Which of the following is NOT an assumption for using the normal approximation to the binomial distribution?

Accepted Answer

The probability of success changes from trial to trial.

Question 10

A researcher is testing $$H_0$$: \mu = 100$$ vs. H_a$$: \mu 
eq 100$$ using a sample mean of 104, sample standard deviation of 10, and sample size of 25. What is the value of the test statistic?

Accepted Answer

$$z = \frac{104 - 100}{10/\sqrt{25}} = 2$$

Question 11

Which of the following is the correct formula for the probability of at least one success in $$n$$ independent Bernoulli trials with probability $$p of $$success?

Accepted Answer

$$P(	ext{at least one success}) = 1 - (1-p)^n$$

Question 12

A 95% confidence interval for a population mean is calculated as (50, 60). Which of the following statements is correct?

Accepted Answer

We are 95% confident that the population mean lies between 50 and 60.

Question 13

Which of the following is the correct formula for the variance of the sum of two independent random variables $$X$$ and $$Y$$?

Accepted Answer

$$Var(X + Y) = Var(X) + Var(Y)$$

Question 14

A company claims that at least 60% of its customers are satisfied. A random sample of 100 customers shows 54 are satisfied. Test the hypothesis at the 0.05 significance level. What is the value of the test statistic?

Accepted Answer

$$z = \frac{0.54 - 0.60}{\sqrt{0.6 	imes 0.4 / 100}}$$

Question 15

Which of the following is the correct formula for the expected value of a discrete random variable $$X$$ with possible values $$x_i$$ and probabilities $$p_i$$?

Accepted Answer

$$E(X) = \sum x_i p_i$$

Question 16

A researcher wants to test if the mean cholesterol level in a population is different from 200 mg/dL. Which of the following is the correct set of hypotheses?

Accepted Answer

$$H_0$$: \mu = 200$$, H_a$$: \mu 
eq 200$$

Question 17

Which of the following is the correct formula for the binomial coefficient?

Accepted Answer

$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$

Question 18

A normal distribution has mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What is the $$z-$$score for a value of 130?

Accepted Answer

$$z = \frac{130 - 100}{15} = 2$$

Question 19

Which of the following is the correct formula for the probability of $$k$$ successes in $$n$$ independent Bernoulli trials with probability $$p of $$success?

Accepted Answer

$$P(X = k) = \binom{n}{k} p^k (1-p$$)^{n-k}$$

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