Q: A company wants to estimate the proportion of customers who are satisfied with their service. They survey 400 customers and find that 320 are satisfied. Construct a 95\% confidence interval for the population proportion. (Use $$z_{0.025} = 1.96.)$$

The confidence interval is $$0.8 \pm 1.96 \times \sqrt{\frac{0.8 \times 0.2}{400}}$$, which is $$(0.76, 0.84).$$

Q: A random variable $$X$$ has the following probability mass function: $$P(0) = 0.2$$, $$P(1) = 0.5$$, $$P(2) = 0.3. $$What is the expected value $$\mu_X$$?

$$\mu_X = 0 \times 0.2 + 1 \times 0.5 + 2 \times 0.3 = 1.1$$

Q: A sample of 25 batteries has a mean lifetime of 100 hours and a sample standard deviation of 10 hours. What is the 90\% confidence interval for the mean battery lifetime? (Use $$t_{0.05, 24} = 1.711.)$$

$$100 \pm 1.711 \times \frac{10}{5}$$, which is $$(96.578, 103.422)$$

Q: If $$P(A) = 0.3$$, $$P(B) = 0.5$$, and $$P(A \text{ and } B) = 0.2$$, what is $$P(A \text{ or } B)$$?

$$P(A \text{ or } B) = 0.3 + 0.5 - 0.2 = 0.6$$

Question 1

Suppose you are analyzing the relationship between hours studied and exam scores using a least-squares regression line. The correlation coefficient is $$r = 0.8. $$What is the coefficient of determination $$R^2$$ for this model, and what does it represent?

Accepted Answer

$$R^2 = 0.64$, which means 64\% of the variance in exam scores is explained by hours studied.

Question 2

A researcher collects data on the number of defective products in 10 random samples from a factory. The mean number of defectives per sample is 2, and the standard deviation is 0.5. If the population distribution is unknown but the sample size is 10, which distribution should be used to construct a confidence interval for the mean number of defectives?

Accepted Answer

Student's t-distribution with 9 degrees of freedom, because the sample size is less than 30 and the population distribution is unknown.

Question 3

A fair die is rolled once. What is the probability of rolling a number greater than 4 using the classical method?

Accepted Answer

$$P(E) = \frac{2}{6} = \frac{1}{3}$$

Question 4

Which of the following is NOT a property of a probability density function (pdf) for a continuous random variable?

Accepted Answer

The sum of the probabilities for all possible values equals 1.

Question 5

A company wants to estimate the proportion of customers who are satisfied with their service. They survey 400 customers and find that 320 are satisfied. Construct a 95\% confidence interval for the population proportion. (Use $$z_{0.025} = 1.96.)$$

Accepted Answer

The confidence interval is $$0.8 \pm 1.96 	imes \sqrt{\frac{0.8 	imes 0.2}{400}}$$, which is $$(0.76, 0.84).$$

Question 6

Which of the following scenarios describes a binomial experiment?

Accepted Answer

Counting the number of heads in 20 flips of a fair coin.

Question 7

A random variable $$X$$ has the following probability mass function: $$P(0) = 0.2$$, $$P(1) = 0.5$$, $$P(2) = 0.3. $$What is the expected value $$\mu_X$$?

Accepted Answer

$$\mu_X = 0 	imes 0.2 + 1 	imes 0.5 + 2 	imes 0.3 = 1.1$$

Question 8

A sample of 25 batteries has a mean lifetime of 100 hours and a sample standard deviation of 10 hours. What is the 90\% confidence interval for the mean battery lifetime? (Use $$t_{0.05, 24} = 1.711.)$$

Accepted Answer

$$100 \pm 1.711 	imes \frac{10}{5}$$, which is $$(96.578, 103.422)$$

Question 9

If $$P(A) = 0.3$$, $$P(B) = 0.5$$, and $$P(A 	ext{ and } B) = 0.2$$, what is $$P(A 	ext{ or } B)$$?

Accepted Answer

$$P(A 	ext{ or } B) = 0.3 + 0.5 - 0.2 = 0.6$$

Question 10

A continuous random variable $$X is $$uniformly distributed on the interval $$[2, 6]. $$What is the value of the probability density function $$f(x)$$ for $$2 \leq x \leq 6$$?

Accepted Answer

$$f(x) = \frac{1}{6-2} = 0.25$$

Question 11

A sample proportion $$\hat{p} is $$calculated from a sample of size $$n = 100$$ with $$p = 0.4. $$What is the standard deviation $$\sigma_{\hat{p}} of $$the sampling distribution of $$\hat{p}$$?

Accepted Answer

$$\sigma_{\hat{p}} = \sqrt{\frac{0.4 	imes 0.6}{100}} = 0.049$$

Question 12

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

Accepted Answer

For large sample sizes ($$n \geq 30$$), the sampling distribution of the sample mean is approximately normal, regardless of the population distribution.

Question 13

A probability model assigns $$P(A) = 0.7$$, $$P(B) = 0.4$$, and $$P(A 	ext{ and } B) = 0.2. $$Does this model satisfy the rules of probability?

Accepted Answer

Yes, because $$P(A 	ext{ and } B) \leq P(A)$$ and $$P(A 	ext{ and } B) \leq P(B)$$, and all probabilities are between 0 and 1.

Question 14

A student calculates a Z-score for a test score as $$Z = 2.0. $$What does this mean?

Accepted Answer

The test score is 2 standard deviations above the mean.

Question 15

Which of the following is a correct interpretation of a 95\% confidence interval for a population mean?

Accepted Answer

If we repeated the sampling process many times, about 95\% of the calculated intervals would contain the true population mean.

Question 16

A probability distribution for a discrete random variable $$X is $$given by $$P(1) = 0.3$$, $$P(2) = 0.4$$, $$P(3) = 0.2. $$What value must $$P(4)$$ take for this to be a valid probability distribution?

Accepted Answer

$$P(4) = 0.1$$

Question 17

A medical test for a rare disease has a 99\% chance of correctly identifying a diseased person (true positive) and a 95\% chance of correctly identifying a healthy person (true negative). If 1\% of the population has the disease, what is the probability that a randomly selected person tests positive?

Accepted Answer

$$P(	ext{positive}) = 0.01 	imes 0.99 + 0.99 	imes 0.05 = 0.0594$$

Question 18

Which of the following is an example of two independent events?

Accepted Answer

Flipping a coin and rolling a die.

MA 113: Midterm 1 & 2 Study Guide – Regression, Probability, Discrete & Continuous Distributions, Sampling, and Confidence Intervals

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