If \( P(A) = 0.4 \), \( P(B) = 0.5 \), and \(A\) and \(B\) are independent events, what is the value of \( P(A \cup B) \)?

• (A) \(0.60\)
• (B) \(0.65\)
• (C) \(0.70\)
• (D) \(0.75\)

For a Poisson distribution where the mean is \(4\), what is the value of the third central moment?

• (A) \(4\)
• (B) \(8\)
• (C) \(16\)
• (D) \(64\)

Which property of an estimator is satisfied if its expected value equals the population parameter?

• (A) Consistency
• (B) Efficiency
• (C) Unbiasedness
• (D) Sufficiency

If a matrix \(A\) has eigenvalues \(2\) and \(3\), what are the eigenvalues of the matrix \(A^2\)?

• (A) \(2, 3\)
• (B) \(4, 9\)
• (C) \(5, 6\)
• (D) \(8, 27\)

In a Normal distribution, what percentage of data falls within two standard deviations of the mean?

• (A) \(68%\)
• (B) \(95%\)
• (C) \(99.7%\)
• (D) \(90%\)

What is the rank of a \(3 \times 3\) identity matrix added to a \(3 \times 3\) null matrix?

• (A) \(0\)
• (B) \(1\)
• (C) \(2\)
• (D) \(3\)

In Simple Random Sampling Without Replacement (SRSWOR), what is the probability of selecting a specific unit at the second draw?

• (A) \( \dfrac{1}{N} \)
• (B) \( \dfrac{1}{N-1} \)
• (C) \( \dfrac{2}{N} \)
• (D) \( \dfrac{1}{N^2} \)

If the correlation coefficient between \(X\) and \(Y\) is \(0.8\), what is the coefficient of determination?

• (A) \(0.64\)
• (B) \(0.80\)
• (C) \(0.16\)
• (D) \(1.60\)

A bag contains \(5\) red and \(7\) blue balls. If two balls are drawn at random, what is the probability that both are red?

• (A) \( \dfrac{5}{66} \)
• (B) \( \dfrac{10}{66} \)
• (C) \( \dfrac{5}{33} \)
• (D) \( \dfrac{25}{66} \)

Which test is most appropriate for testing the significance of the difference between two small sample means?

• (A) Z-test
• (B) t-test
• (C) Chi-square test
• (D) F-test

If the null hypothesis \(H_0\) is rejected when it is actually true, what type of error has been committed?

• (A) Type I Error
• (B) Type II Error
• (C) Sampling Error
• (D) Standard Error

What is the value of the integral \( \displaystyle \int_{-\infty}^{\infty} e^{-x^2}\,dx \)?

• (A) \( \sqrt{\pi} \)
• (B) \( \pi \)
• (C) \( 1 \)
• (D) \( \dfrac{\sqrt{\pi}}{2} \)

In a negatively skewed distribution, what is the correct relationship between the Mean, Median, and Mode?

• (A) Mean \(>\) Median \(>\) Mode
• (B) Mean \(<\) Median \(<\) Mode
• (C) Mean \(=\) Median \(=\) Mode
• (D) Mode \(<\) Median \(<\) Mean

Find the limit of \( \left(1 + \frac{1}{n}\right)^n \) as \(n \to \infty\).

• (A) \(1\)
• (B) \(e\)
• (C) \(0\)
• (D) \(2\)

What is the degree of freedom for a Chi-square test used in a \(3 \times 4\) contingency table?

• (A) \(6\)
• (B) \(8\)
• (C) \(9\)
• (D) \(12\)