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

$$E[X] = 1 \times 0.2 + 2 \times 0.5 + 3 \times 0.3 = 2.1$$

Question 1

A pharmaceutical company claims that a new drug cures 80% of patients. In a clinical trial, 60 out of 75 patients were cured. What is the empirical probability that a patient is cured based on this trial?

Accepted Answer

$$\frac{60}{75} = 0.8$$

Question 2

Which of the following events are disjoint (mutually exclusive)?

Accepted Answer

Rolling a 2 and rolling a 5 on a single die.

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

$$0.2$$

Question 4

A weather forecaster estimates a 70% chance of rain tomorrow. What type of probability is this?

Accepted Answer

Subjective probability

Question 5

Which formula represents the complement rule for an event $$A$$?

Accepted Answer

$$P(A^c) = 1 - P(A)$$

Question 6

A random variable $$X$$ has the following probability model: $$P(X=1)=0.2$$, $$P(X=2)=0.5$$, $$P(X=3)=0.3. $$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 7

If a continuous random variable $$X is $$uniformly distributed between 0 and 10, what is the probability that $$X is $$less than 4?

Accepted Answer

$$P(X \u003c 4) = \frac{4}{10} = 0.4$$

Question 8

A factory produces light bulbs with a 2% defect rate. If you randomly select 100 bulbs, which probability model is most appropriate to model the number of defective bulbs?

Accepted Answer

Binomial probability model

Question 9

Which condition must be met to use the binomial probability model?

Accepted Answer

Each trial is independent and has only two possible outcomes.

Question 10

A sample of 50 students is taken from a university of 10,000 students. The sample mean GPA is 3.2 with a standard deviation of 0.4. What is the standard error of the mean?

Accepted Answer

$$SE = \frac{0.4}{\sqrt{50}} \approx 0.0566$$

Question 11

Which statement best describes the Central Limit Theorem?

Accepted Answer

The sampling distribution of the sample mean approaches normality as sample size increases, regardless of the population's distribution.

Question 12

A researcher wants to estimate the proportion of voters who support a new policy. She finds that 120 out of 200 sampled voters support it. What is the point estimate for the proportion?

Accepted Answer

$$\hat{p} = \frac{120}{200} = 0.6$$

Question 13

Which formula is used to calculate the margin of error for a one-proportion $$z-$$interval?

Accepted Answer

$$ME = z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Question 14

In hypothesis testing, what does the $$p-$$value represent?

Accepted Answer

The probability of observing data as extreme or more extreme than the sample, assuming the null hypothesis is true.

Question 15

A hospital tests whether a new treatment reduces recovery time. The null hypothesis is $$H_0$$: \mu = 10$$ days. The sample mean is 8 days, with p$-value = 0.03$. At $$\alpha = 0.05$$, what is the correct conclusion?

Accepted Answer

Reject the null hypothesis; the treatment significantly reduces recovery time.

Question 16

Which of the following best describes a Type I error?

Accepted Answer

Rejecting the null hypothesis when it is actually true.

Question 17

A company wants to increase the power of its hypothesis test. Which action will most effectively increase power?

Accepted Answer

Increase the sample size.

Question 18

If $$X$$ and $$Y$$ are independent random variables with $$E[X]=3$$, $$E[Y]=5$$, what is $$E[X+Y]$$?

Accepted Answer

$$E[X+Y] = 3 + 5 = 8$$

Question 19

A Poisson probability model is most appropriate for which scenario?

Accepted Answer

Counting the number of emails received per hour.

Question 20

Which formula gives the standard deviation of a discrete random variable $$X$$ with possible values $$x_i$$ and probabilities $$p_i$$?

Accepted Answer

$$\sigma_X = \sqrt{\sum (x_i - \mu_X)^2$ p_i}$$

Probability, Random Variables, and Statistical Inference: Study Notes for Elementary Statistics

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