Q: Given the regression line $$\hat{y} = 2 + 3x$$, what is the predicted value of $$y$$ when $$x = 4$$?

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

Question 1

A researcher wants to estimate the average height of all university students. She selects 50 students at random and finds their mean height is 170 cm. In this context, which of the following statements is correct?

Accepted Answer

The value 50 is the sample size $$n$$, and 170 is the sample mean $$\bar{x}$$, both statistics.

Question 2

Which of the following best describes the difference between an observational study and an experiment?

Accepted Answer

An experiment involves manipulating variables to observe effects, while an observational study does not manipulate variables.

Question 3

Suppose a discrete random variable $$X$$ has the following probability distribution: $$P(X=1)=0.2$$, $$P(X=2)=0.5$$, $$P(X=3)=?$$ What is the missing probability value?

Accepted Answer

$$0.3$$

Question 4

A pharmaceutical company tests a new drug on 100 patients and finds the mean recovery time is 8 days with a standard deviation of 2 days. What is the $$z-$$score for a patient who recovers in 10 days?

Accepted Answer

$$z = \frac{10 - 8}{2} = 1$$

Question 5

Which of the following statistics is an unbiased estimator of its corresponding parameter?

Accepted Answer

Sample mean $$\bar{x}$$ for population mean $$\mu$$

Question 6

A fair die is rolled. What is the sample space $$S$$ for this experiment?

Accepted Answer

$$S = \{1, 2, 3, 4, 5, 6\}$$

Question 7

Which of the following is a property of the standard normal distribution?

Accepted Answer

It has a mean of $$0$$ and a standard deviation of $$1.$$

Question 8

A 95% confidence interval for a population mean $$\mu is $$constructed as $$\bar{x} \pm E$$ with $$E = z_{\alpha/2} \frac{\sigma}{\sqrt{n}}. If$$ $$\sigma is $$unknown, which value replaces $$z_{\alpha/2} in $$the formula?

Accepted Answer

$$t_{\alpha/2, n-1}$$

Question 9

A random variable $$X is $$defined as the time (in hours) it takes for a chemical reaction to complete, and it can take any value between 2 and 5. Which type of random variable is $$X$$?

Accepted Answer

Continuous

Question 10

A uniform random variable $$X is $$defined on the interval $$[0, 4]. $$What is the height of the probability density function $$f(x)$$?

Accepted Answer

$$f(x) = \frac{1}{4}$$

Question 11

A sample of size $$n=40 is $$drawn from a population with mean $$\mu=100$$ and standard deviation $$\sigma=20. $$According to the Central Limit Theorem, what is the standard deviation of the sampling distribution of $$\bar{x}$$?

Accepted Answer

$$\frac{20}{\sqrt{40}}$$

Question 12

A hospital administrator wants to estimate the proportion $$p of $$patients who recover within 3 days. In a sample of 200 patients, 120 recover within 3 days. What is the point estimate $$\hat{p}$$?

Accepted Answer

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

Question 13

Which of the following statements about the $$t-$$distribution is true?

Accepted Answer

It is symmetric and bell-shaped, and approaches the standard normal distribution as $$n$$ increases.

Question 14

A hypothesis test is conducted at $$\alpha = 0.05. $$The $$P-$$value is found to be $$0.03. $$What is the correct decision?

Accepted Answer

Reject the null hypothesis.

Question 15

In the context of hypothesis testing, what is a Type I error?

Accepted Answer

Rejecting a true null hypothesis.

Question 16

Given the regression line $$\hat{y} = 2 + 3x$$, what is the predicted value of $$y$$ when $$x = 4$$?

Accepted Answer

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

Question 17

A scatterplot shows that as the number of hours studied increases, exam scores tend to increase. What type of correlation is this?

Accepted Answer

Positive correlation

Question 18

Which of the following is NOT a property of probability?

Accepted Answer

The probability of the empty set is 1.

Question 19

A sample of 25 patients has a mean blood pressure of 120 mmHg and a sample standard deviation of 10 mmHg. Construct a 95% confidence interval for the population mean blood pressure, assuming normality. Which formula should you use for the margin of error $$E$$?

Accepted Answer

$$E = t_{\alpha/2, 24} \frac{10}{\sqrt{25}}$$

Question 20

Which of the following best describes the linear correlation coefficient $$r$$?

Accepted Answer

$$r$$ measures the strength and direction of a linear relationship between two quantitative variables.

Comprehensive Study Notes for Introductory Statistics (Chapters 1–8, 10)

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