Q: A bag contains 5 red balls and 7 blue balls. If two balls are drawn at random without replacement, what is the probability that both are red?

$$P = \frac{5}{12} \times \frac{4}{11} = \frac{20}{132} = \frac{5}{33}$$

Question 1

A pharmaceutical company wants to test whether a new drug lowers blood pressure more effectively than the current standard. They randomly assign 100 patients to receive either the new drug or the standard drug and measure their blood pressure after 8 weeks. What type of study is this?

Accepted Answer

An experimental study

Question 2

Which of the following is a population parameter?

Accepted Answer

The mean cholesterol level of all adults in the United States

Question 3

A researcher records the eye color (blue, brown, green, hazel) of 500 students. What type of data is this?

Accepted Answer

Categorical, nominal

Question 4

Which of the following best describes the median?

Accepted Answer

The value that splits the ordered data into two equal halves

Question 5

Given the following data set: 2, 4, 4, 6, 8, what is the sample variance?

Accepted Answer

$$s^2$$ = \frac{$$(2-4.8)^2$$ + $$(4-4.8)^2$$ + $$(4-4.8)^2$$ + $$(6-4.8)^2$$ + $$(8-4.8)^2$$}{4} = 5.2$$

Question 6

A boxplot of exam scores shows a lower whisker extending to 45, a lower quartile at 60, a median at 75, an upper quartile at 85, and an upper whisker at 100. Which of the following statements is true?

Accepted Answer

The interquartile range is $$IQR = 85 - 60 = 25$$

Question 7

A study finds a correlation coefficient $$r = -0.85$$ between hours spent watching TV and GPA among college students. Which of the following is the best interpretation?

Accepted Answer

There is a strong negative linear relationship between TV watching and GPA

Question 8

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

Accepted Answer

$$P = \frac{2}{6} = \frac{1}{3}$$

Question 9

A bag contains 5 red balls and 7 blue balls. If two balls are drawn at random without replacement, what is the probability that both are red?

Accepted Answer

$$P = \frac{5}{12} 	imes \frac{4}{11} = \frac{20}{132} = \frac{5}{33}$$

Question 10

Which of the following is NOT a property of a probability distribution for a discrete random variable?

Accepted Answer

Each outcome must have a unique probability

Question 11

A binomial experiment consists of 8 trials with probability of success $$p = 0.3. $$What is the probability of exactly 2 successes? (Use the binomial formula.)

Accepted Answer

$$P(X=2) = \binom{8}{2} (0.3)^2$ (0.7)^6$

Question 12

For a normal distribution with mean $$\mu = 100$$ and standard deviation $$\sigma = 15$$, what is the $$z-$$score for a value $$x = 130$$?

Accepted Answer

$$z = \frac{130 - 100}{15} = 2$$

Question 13

A 95% confidence interval for the mean weight of apples in an orchard is (150g, 170g). Which of the following is a correct interpretation?

Accepted Answer

We are 95% confident that the true mean weight of apples is between 150g and 170g

Question 14

Which of the following changes will result in a narrower confidence interval for the mean, assuming all else is constant?

Accepted Answer

Increasing the sample size

Question 15

A researcher is testing $$H_0$$: \mu = 50$$ vs. H_a$$: \mu 
eq 50$$ at $$\alpha = 0.05$$. The sample mean is 54, the sample standard deviation is 8, and the sample size is 16. What is the value of the test statistic?

Accepted Answer

$$t = \frac{54 - 50}{8/\sqrt{16}} = 2$$

Question 16

Which of the following is an example of a Type I error in hypothesis testing?

Accepted Answer

Rejecting a true null hypothesis

Question 17

The Central Limit Theorem states that:

Accepted Answer

The sampling distribution of the sample mean approaches a normal distribution as the sample size increases

Question 18

A hospital administrator wants to estimate the average length of stay for patients. She takes a random sample of 64 patients and finds a mean of 5 days with a standard deviation of 2 days. What is the 95% confidence interval for the mean length of stay? (Use $$z^* = 1.96)$$

Accepted Answer

$$5 \pm 1.96 	imes \frac{2}{8}$$

Question 19

Which of the following is true about the standard deviation?

Accepted Answer

It measures the average spread of data values around the mean

Comprehensive Study Notes for Introductory Statistics

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