Q: A fair coin is flipped 10 times. What is the probability of getting exactly 6 heads?

$$P = \frac{10!}{6!4!} \times (0.5)^6$ \times (0.5)^4$ = 210 \times (0.5$$)^{10}$$ = 0.205$$

Q: If $$P(A) = 0.4$$ and $$P(B) = 0.5$$, and $$A$$ and $$B$$ are independent, what is $$P(A \text{ and } B)$$?

$$P(A \text{ and } B) = P(A) \times P(B) = 0.4 \times 0.5 = 0.2$$

Q: Given the regression equation $$\hat{Y} = 2 + 3X$$, what is the predicted value of $$Y$$ when $$X = 4$$?

$$\hat{Y} = 2 + 3 \times 4 = 14$$

Q: A random variable $$X$$ has the following probability distribution: $$P(X=1)=0.2$$, $$P(X=2)=0.5$$, $$P(X=3)=0.3. $$What is the expected value $$E(X)$$?

$$E(X) = 1 \times 0.2 + 2 \times 0.5 + 3 \times 0.3 = 2.1$$

Q: If $$P(A) = 0.7$$ and $$P(B|A) = 0.5$$, what is $$P(A \text{ and } B)$$?

$$P(A \text{ and } B) = P(A) \times P(B|A) = 0.7 \times 0.5 = 0.35$$

Q: A die is rolled twice. What is the probability that both rolls show a 6?

$$P(6 \text{ and } 6) = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$$

Q: A medical test for a disease has a 95% chance of correctly identifying a diseased person (sensitivity) and a 90% chance of correctly identifying a healthy person (specificity). If 1% of the population has the disease, what is the probability that a randomly selected person tests positive?

$$P(\text{Positive}) = 0.01 \times 0.95 + 0.99 \times 0.10 = 0.0095 + 0.099 = 0.1085$$

Question 1

Suppose the heights of adult men in a city are normally distributed with a mean of 70 inches and a standard deviation of 3 inches. What is the probability that a randomly selected man is taller than 73 inches?

Accepted Answer

$$P(Z \u003e 1) = 1 - 0.8413 = 0.1587$$

Question 2

A fair coin is flipped 10 times. What is the probability of getting exactly 6 heads?

Accepted Answer

$$P = \frac{10!}{6!4!} 	imes (0.5)^6$ 	imes (0.5)^4$ = 210 	imes (0.5$$)^{10}$$ = 0.205$$

Question 3

If $$P(A) = 0.4$$ and $$P(B) = 0.5$$, and $$A$$ and $$B$$ are independent, what is $$P(A 	ext{ and } B)$$?

Accepted Answer

$$P(A 	ext{ and } B) = P(A) 	imes P(B) = 0.4 	imes 0.5 = 0.2$$

Question 4

Given the regression equation $$\hat{Y} = 2 + 3X$$, what is the predicted value of $$Y$$ when $$X = 4$$?

Accepted Answer

$$\hat{Y} = 2 + 3 	imes 4 = 14$$

Question 5

A random variable $$X$$ has the following probability distribution: $$P(X=1)=0.2$$, $$P(X=2)=0.5$$, $$P(X=3)=0.3. $$What is the expected value $$E(X)$$?

Accepted Answer

$$E(X) = 1 	imes 0.2 + 2 	imes 0.5 + 3 	imes 0.3 = 2.1$$

Question 6

If the correlation coefficient $$r$$ between two variables is $$-0.8$$, what does this indicate?

Accepted Answer

There is a strong negative linear relationship between the two variables.

Question 7

A student scores 85 on a test where the mean is 80 and the standard deviation is 5. What is the student's z-score?

Accepted Answer

$$z = \frac{85 - 80}{5} = 1$$

Question 8

If $$P(A) = 0.7$$ and $$P(B|A) = 0.5$$, what is $$P(A 	ext{ and } B)$$?

Accepted Answer

$$P(A 	ext{ and } B) = P(A) 	imes P(B|A) = 0.7 	imes 0.5 = 0.35$$

Question 9

A die is rolled twice. What is the probability that both rolls show a 6?

Accepted Answer

$$P(6 	ext{ and } 6) = \frac{1}{6} 	imes \frac{1}{6} = \frac{1}{36}$$

Question 10

If $$X$$ and $$Y$$ are independent random variables with $$SD(X) = 2$$ and $$SD(Y) = 3$$, what is $$SD(X + Y)$$?

Accepted Answer

$$SD(X + Y) = \sqrt{2^2$ + 3^2$} = \sqrt{13} \approx 3.61$$

Question 11

A medical test for a disease has a 95% chance of correctly identifying a diseased person (sensitivity) and a 90% chance of correctly identifying a healthy person (specificity). If 1% of the population has the disease, what is the probability that a randomly selected person tests positive?

Accepted Answer

$$P(	ext{Positive}) = 0.01 	imes 0.95 + 0.99 	imes 0.10 = 0.0095 + 0.099 = 0.1085$$

Question 12

If the variance of a random variable $$X is 16$$, what is the standard deviation?

Accepted Answer

$$SD(X) = \sqrt{16} = 4$$

Question 13

If $$P(A) = 0.3$$ and $$P(B) = 0.4$$, and $$P(A 	ext{ and } B) = 0.1$$, what is $$P(A 	ext{ or } B)$$?

Accepted Answer

$$P(A 	ext{ or } B) = 0.3 + 0.4 - 0.1 = 0.6$$

Question 14

A student’s test score is at the 90th percentile of a normal distribution. What is the approximate z-score for this student?

Accepted Answer

$$z \approx 1.28$$

Question 15

If $$X is a $$binomial random variable with $$n = 20$$ and $$p = 0.3$$, what is the expected value $$E(X)$$?

Accepted Answer

$$E(X) = np = 20 	imes 0.3 = 6$$

Question 16

If the correlation coefficient $$r$$ between $$X$$ and $$Y is 0$$, what does this mean?

Accepted Answer

There is no linear relationship between $$X$$ and $$Y.$$

Question 17

A random variable $$X is $$transformed by $$Y = 5X + 2. If$$ $$SD(X) = 4$$, what is $$SD(Y)$$?

Accepted Answer

$$SD(Y) = |5| 	imes SD(X) = 5 	imes 4 = 20$$

Question 18

If the mean of a data set is 50 and the standard deviation is 10, what value is two standard deviations above the mean?

Accepted Answer

$$50 + 2 	imes 10 = 70$$

Question 19

A researcher finds that the correlation between hours studied and exam score is $$r = 0.85. $$What does this imply?

Accepted Answer

There is a strong positive linear relationship between hours studied and exam score.

Statistics Final Exam Study Notes: Probability, Random Variables, and Regression

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