1. Let $X$ be a continuous random variable and $F(x) = F(X \leq x)$, where $x$ is a real number. Which one of the following statements is true?
(A) \(F(x)\) is non-decreasing in \(x\)
(B) \(\sum_{x} F(x) = 1\)
(C) \(x\geq 0\)
(D) \(\sum_{x} x = 1\)
(E) \(F(0) = 0.5\)

2. 100 students participated in a statistics test. Test scores are shown a normal distribution with mean 50 and standard deviation 10. Approximately how many students whose scores are between 40 and 60? (A)34 (B) 50 (C) 68 (D) 75 (E) 95

3. In an instant lottery, your chances of winning are 0.2. If you play the lottery five times and outcomes are independent, the probability that you win at least once is (A) 0.0819 (B) 0.6723 (C) 0.4096 (D) 0.7373 (E) 0.8125

4. The time to complete a standardized exam is approximately normal with a mean of 70 minutes and a standard deviation of 10 minutes. Using the 68-95-99.7 rule, what percent of students will complete the exam in under an hour? (A) \(68\%\) (B) \(32\%\) (C) \(16\%\) (D) \(5\%\) (E) \(1\%\)

5. Assume that you have a binomial experiment with $p = 0.5$ and a sample size of 100. The expected value of this distribution is (A) 0.50 (B) 0.30 (C) 100 (D) 50 (E) 25

6. A standard normal distribution is a normal distribution (A) With a mean of 1 and a standard deviation of 0 (B) With a mean of 0 and a standard deviation of 1 (C) With any mean and a standard deviation of 1 (D) With any mean and any standard deviation (E) With a mean of 0 and a standard deviation of 0

7. The probability distribution of random variable, Xᵢ is defined as follows: | X | 0 | 1 | 2 | 3 | 4 | | --- | --- | --- | --- | --- | --- | | Probability | ? | 0.3 | 0.1 | 0.3 | 0.3 | What is the missing value? (A) 0 (B) 0.1 (C) 0.2 (D) 0.3 (E) 0.4

8. A box contains 10 bulbs, 2 of which are defective. If 3 bulbs are picked randomly, what is the probability that at least one defective is picked? (A) \(\frac{1}{6}\) (B) \(\frac{8}{15}\) (C) \(\frac{2}{3}\) (D) \(\frac{7}{15}\) (E) \(\frac{5}{6}\)

9. Independently toss a fair dice 4 times. Suppose that Xᵢ is the outcome of the iᵘ toss, and X̄ = (X₁ + X₂ + X₃ + X₄) / 4. Then Var(X̄) is: (A) \(\frac{35}{12}\) (B) \(\frac{35}{24}\) (C) \(\frac{35}{36}\) (D) \(\frac{35}{48}\) (E) \(\frac{35}{60}\)

10. As the sample size increases, the (A) Standard deviation of the population decreases (B) Population mean increases (C) Sample mean increases (D) Standard error of the sample mean increases (E) Standard error of the sample mean decreases

11. A random sample of size 25 is to be taken from a population that is normally distributed with mean 60 and standard deviation 10. The average $\overline{x}$ of the observations in our sample is to be computed. The sampling distribution of $\overline{x}$ is (A) Normal with mean 60 and standard deviation 10 (B) Normal with mean 60 and standard deviation 2 (C) Normal with mean 60 and standard deviation 0.4 (D) Normal with mean 60 and standard deviation 25 (E) Normal with mean 60 and standard deviation 2.5

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