Question 1

Which of the following best describes the difference between a parameter and a statistic?

Accepted Answer

A parameter is a numerical summary of a population, while a statistic is a numerical summary of a sample.

Question 2

A researcher wants to estimate the average height of all students at a university. She randomly selects 100 students and calculates their mean height. Which of the following is the best description of the mean height she calculates?

Accepted Answer

It is a statistic because it is calculated from a sample.

Question 3

Which of the following is the correct formula for the sample mean $$\bar{x} of$$ $$n$$ observations $$x_1$$, $$x_2$$, ..., x_n$$?

Accepted Answer

$$\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$$

Question 4

A pharmaceutical company is testing a new drug. They want to know if the drug is more effective than the current standard. Which type of hypothesis test should they use?

Accepted Answer

A two-sample hypothesis test comparing the means of two populations.

Question 5

Which of the following statements about Type I and Type II errors is correct?

Accepted Answer

A Type I error occurs when a true null hypothesis is rejected.

Question 6

Suppose you are given the following data set: 2, 4, 4, 6, 8. What is the median of the data set?

Accepted Answer

4

Question 7

Which of the following best describes the Central Limit Theorem?

Accepted Answer

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

Question 8

A study finds that the mean test score for a group of students is 75 with a standard deviation of 10. If the scores are normally distributed, what percentage of students scored between 65 and 85?

Accepted Answer

Approximately 68%

Question 9

Which of the following is the correct formula for the sample variance $$s^2 of$$ $$n$$ observations $$x_1$$, $$x_2$$, ..., x_n$$ with sample mean $$\bar{x}$$?

Accepted Answer

$$s^2$$ = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x}$$)^2$$

Question 10

A hospital administrator wants to compare the average length of stay for patients in two different wards. Which statistical method is most appropriate?

Accepted Answer

Two-sample t-test for means

Question 11

Which of the following is NOT a requirement for using the Central Limit Theorem?

Accepted Answer

The population must be normally distributed for any sample size.

Question 12

A survey is conducted to estimate the proportion of voters who support a new policy. Which formula would you use to construct a 95% confidence interval for the population proportion $$p$$ based on a sample proportion $$\hat{p}$$ and sample size $$n$$?

Accepted Answer

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

Question 13

Which of the following best describes a Type II error in hypothesis testing?

Accepted Answer

Failing to reject a false null hypothesis.

Question 14

A teacher wants to know if a new teaching method improves student performance. She randomly assigns students to two groups: one uses the new method, the other uses the traditional method. Which statistical test should she use to compare the average scores of the two groups?

Accepted Answer

Independent two-sample t-test

Question 15

Which of the following is the correct formula for the z-score of a value $$x in a $$population with mean $$\mu$$ and standard deviation $$\sigma$$?

Accepted Answer

$$z = \frac{x - \mu}{\sigma}$$

Question 16

A clinical trial tests whether a new medication lowers blood pressure more than a placebo. The null hypothesis is that there is no difference. If the p-value is 0.03 and the significance level is 0.05, what should the researchers conclude?

Accepted Answer

Reject the null hypothesis; there is evidence the medication is effective.

Question 17

Which of the following is an example of a discrete random variable?

Accepted Answer

The number of cars passing through an intersection in an hour.

Question 18

A biologist is interested in the mean weight of a species of fish in a lake. She collects a random sample of 40 fish and finds a sample mean of 2.5 kg and a sample standard deviation of 0.4 kg. What is the 95% confidence interval for the mean weight? (Use $$t^* = 2.02$$ for 39 degrees of freedom.)

Accepted Answer

$$2.5 \pm 2.02 	imes \frac{0.4}{\sqrt{40}}$$

Question 19

Which of the following is the best definition of statistical inference?

Accepted Answer

Drawing conclusions about a population based on sample data.

Question 20

A company wants to know if there is a relationship between gender and preference for a new product. Which statistical test should they use?

Accepted Answer

Chi-square test for independence

Statistics 1000 & 54C: Course Syllabus and Study Guide

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