Question 1

Which of the following best describes the difference between a quantitative variable and a categorical variable?

Accepted Answer

A quantitative variable represents numerical values that can be measured or counted, while a categorical variable represents characteristics or attributes that can be grouped but not measured numerically.

Question 2

A researcher wants to compare the average test scores of two different teaching methods. Which statistical test is most appropriate?

Accepted Answer

A two-sample t-test for means

Question 3

Which of the following equations correctly represents the calculation of the sample mean $$\bar{x}$$ for a data set with $$n$$ observations $$x_1$$, $$x_2$$, ..., x_n$$?

Accepted Answer

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

Question 4

A hospital administrator wants to know if there is an association between gender (male/female) and whether patients receive a flu shot (yes/no). Which statistical test should be used?

Accepted Answer

Chi-square test for independence

Question 5

Which of the following statements about the Normal model is correct?

Accepted Answer

The Normal model is symmetric and bell-shaped, with mean $$\mu$$ and standard deviation $$\sigma.$$

Question 6

Suppose you want to estimate the average height of all students at a university. Which of the following best describes the population and the sample?

Accepted Answer

The population is all students at the university; the sample is the group of students whose heights you actually measure.

Question 7

A pharmaceutical company tests a new drug and finds a $$p-$$value of 0.03 when comparing it to a placebo. If the significance level is $$\alpha = 0.05$$, what is the correct conclusion?

Accepted Answer

Reject the null hypothesis; there is statistically significant evidence that the drug is different from the placebo.

Question 8

Which formula is used to calculate the standard deviation $$s of a $$sample with $$n$$ observations $$x_1$$, $$x_2$$, ..., x_n$$ and sample mean $$\bar{x}$$?

Accepted Answer

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

Question 9

A city council wants to know if the proportion of residents who support a new park is different from 50%. Which hypothesis test should they use?

Accepted Answer

One-sample z-test for proportions

Question 10

Which of the following is a correct interpretation of a 95% confidence interval for the mean weight of apples in an orchard?

Accepted Answer

We are 95% confident that the true mean weight of all apples in the orchard lies within the interval.

Question 11

A biologist collects data on the number of eggs laid by two different species of birds. Which graphical display is most appropriate for comparing the distributions?

Accepted Answer

Side-by-side boxplots

Question 12

Which of the following is the correct formula for the correlation coefficient $$r$$ between two quantitative variables $$x$$ and $$y$$?

Accepted Answer

$$r = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{(n-1)s_x s_y}$$

Question 13

A company wants to test if the average commute time for its employees is greater than 30 minutes. Which are the correct null and alternative hypotheses?

Accepted Answer

$$H_0$$: \mu = 30$$; H_a$$: \mu \u003e 30$$

Question 14

Which of the following best describes the purpose of a goodness-of-fit test?

Accepted Answer

To determine whether the observed frequencies in categories match the expected frequencies under a specific distribution.

Question 15

A medical researcher is interested in whether a new treatment changes the mean recovery time compared to the standard treatment. Which type of study design is most appropriate?

Accepted Answer

Randomized controlled experiment

Question 16

Which of the following is a correct statement about the Central Limit Theorem?

Accepted Answer

As the sample size increases, the sampling distribution of the sample mean approaches a Normal distribution, regardless of the population's distribution.

Question 17

A student wants to test if the proportion of defective products in a factory is less than 5%. Which is the correct set of hypotheses?

Accepted Answer

$$H_0$$: p = 0.05$$; H_a$$: p \u003c 0.05$$

Question 18

Which of the following is an example of using a simulation in statistics?

Accepted Answer

Using a computer to repeatedly sample from a probability distribution to estimate the likelihood of a rare event.

Question 19

A researcher collects paired data on blood pressure before and after a new diet in the same group of patients. Which test should be used to analyze the data?

Accepted Answer

Paired t-test

Question 20

Which of the following is the correct formula for the standard error of the mean $$SE_{\bar{x}}$$ for a sample of size $$n$$ with standard deviation $$s$$?

Accepted Answer

$$SE_{\bar{x}} = \frac{s}{\sqrt{n}}$$

Introduction to Statistics – Syllabus and Study Guide

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