Statistical Data Analysis Practice Quiz (25 questions)

1. A researcher is studying the heights of students at a university. What is the population in this study?

A. The set of height values recorded in the study

B. All students at the university who are being studied

C. The height recorded for each selected student

D. The students selected to have their heights measured

Answer: (B)

Explanation

The population is the full set of individuals or objects under study, here the university students. The selected students form a sample, which is a subset of that population.

2. A survey measures the commuting times of 200 employees chosen from a company of 3,000 employees. What are the 200 chosen employees?

A. The complete population of employees being studied

B. A categorical variable describing commuting behavior

C. A numerical summary of the company’s commuting times

D. A sample drawn from the company’s employee population

Answer: (D)

Explanation

A sample is a subset of the population, so the 200 chosen employees are a sample of the company’s employees. The full set of 3,000 employees is the population, not the selected group.

3. In a study of household electricity use, what is the statistical variable if the researcher records each household’s monthly consumption?

A. The set of households included in the study

B. The procedure used to select households for measurement

C. The households selected from the larger community

D. Monthly electricity consumption, a characteristic of interest

Answer: (D)

Explanation

A statistical variable is a characteristic or property of the population that is of interest, such as monthly electricity consumption. The households themselves are individuals being studied, not the variable.

4. A dataset records each customer’s satisfaction as dissatisfied, neutral, or satisfied. How should this variable be classified?

A. Categorical and ordinal

B. Categorical and nominal

C. Numerical and discrete

D. Numerical and continuous

Answer: (A)

Explanation

Satisfaction levels are categories with a meaningful order, so the variable is categorical and ordinal. A nominal variable has categories without an inherent order.

5. After sorting a dataset from smallest to largest, which value is its median?

A. The middle observation in the ordered data

B. The arithmetic average of all observations

C. The observation that occurs most frequently

D. The maximum value minus the minimum value

Answer: (A)

Explanation

The median is the middle data point after the observations are ordered. The arithmetic average is the mean, which is a different measure of central tendency.

6. What does the variance of a dataset measure?

A. The expected squared deviation of values from their mean

B. The value occurring most frequently in the dataset

C. The difference between the largest and smallest values

D. The middle observation after values are ordered

Answer: (A)

Explanation

Variance is the expected value of squared deviations from the mean, so it measures dispersion around that mean. The largest-minus-smallest difference is the range, not the variance.

7. A random experiment consists of rolling a standard six-sided die once. What is the sample space?

A. The probability assigned to each possible outcome

B. The single outcome recorded in one particular roll

C. A numerical function that assigns values to outcomes

D. The set of possible outcomes: 1, 2, 3, 4, 5, and 6

Answer: (D)

Explanation

The sample space contains all possible outcomes of the random experiment, which for one die roll are 1 through 6. A particular roll produces just one outcome, not the full sample space.

8. For a coin-toss experiment, a function assigns 1 to heads and 0 to tails. What makes this function a random variable?

A. It identifies the outcome that occurs most frequently

B. It lists the probabilities of all possible outcomes

C. It assigns a numerical value to each possible outcome

D. It describes the full set of possible outcomes

Answer: (C)

Explanation

A random variable is a numerical function that assigns a number to each possible outcome of a random experiment. A list of outcomes is a sample space, while probabilities are described by a distribution.

9. Which function gives the probability that a discrete random variable takes each possible value in its range?

A. A cumulative distribution function

B. A probability mass function

C. A sample space function

D. A probability density function

Answer: (B)

Explanation

A probability mass function gives the probability of each possible value of a discrete random variable. A probability density function is associated with describing a continuous variable rather than assigning discrete point probabilities.

10. A discrete random variable can take values 0, 1, and 2 with probabilities 0.2, 0.5, and 0.3, respectively. How is its expected value obtained?

A. By adding the possible values without using their probabilities

B. By averaging the possible values with weights equal to their probabilities

C. By selecting the middle value after ordering the possible values

D. By selecting the value with the highest probability

Answer: (B)

Explanation

The expected value is the probability-weighted average of the possible values, so each value contributes according to its probability. Choosing the value with the highest probability gives the mode, not the expected value.

11. A normal curve is shifted horizontally while retaining the same spread. Which parameter determines its horizontal position?

A. The standard deviation

B. The mean

C. The variance

D. The z-score

Answer: (B)

Explanation

The mean determines where the bell-shaped curve is positioned on the horizontal axis. Variance controls the curve’s spread rather than its position.

12. If a normal distribution’s mean stays fixed while its variance increases, how does its curve change?

A. It becomes flatter and more spread out.

B. It becomes taller and more concentrated.

C. It shifts horizontally but keeps the same spread.

D. It becomes asymmetric while keeping its spread.

Answer: (A)

Explanation

Increasing variance spreads the distribution over a wider range and makes its curve flatter. A horizontal shift would result from changing the mean, not increasing variance.

13. What does a z-score tell you about an observation relative to its distribution?

A. Its distance from the mean in standard deviations

B. The observation’s rank among all values in the sample

C. The distribution’s variance in squared measurement units

D. Its distance from the mean in the original measurement units

Answer: (A)

Explanation

A z-score expresses how many standard deviations an observation lies above or below the mean. An observation stated in its original units has not been standardized.

14. How is the standard deviation related to variance, and what does it measure?

A. It is the square root of the mean and measures the expected value’s location.

B. It is the square of variance and measures the distribution’s horizontal position.

C. It is the reciprocal of variance and measures dispersion around the sample median.

D. It is the square root of variance and measures dispersion around the expected value.

Answer: (D)

Explanation

Standard deviation is the square root of variance and describes dispersion around the expected value. The mean, not the standard deviation, determines the distribution’s horizontal position.

15. What does a 95% confidence procedure mean when repeated samples are taken?

A. About 95% of sample estimates equal the true parameter.

B. About 95% of the intervals produced by the procedure contain the true parameter.

C. A particular interval has a 95% probability of containing the true parameter.

D. The true parameter falls within 95% of the observed sample values.

Answer: (B)

Explanation

The 95% rate describes the long-run proportion of intervals from repeated samples that contain the true parameter. It does not assign a 95% probability to a particular computed interval containing that fixed parameter.

16. If a sample size is multiplied by four while the population proportion remains unchanged, what happens to the standard error of the sample proportion?

A. It is multiplied by two.

B. It remains unchanged.

C. It is divided by four.

D. It is divided by two.

Answer: (D)

Explanation

Quadrupling the sample size reduces the standard error of the sample proportion by a factor of two when the population proportion is unchanged. Dividing the standard error by four confuses its change with the change in sample size.

17. When calculating the sample variance from squared deviations about the sample mean, what divisor gives the stated unbiased estimate of population variance?

A. The sample size

B. The square root of the sample size

C. The sample size minus one

D. The sample size plus one

Answer: (C)

Explanation

The sample variance divides the sum of squared deviations by $n-1$ to provide the stated unbiased estimate of population variance. Dividing by $n$ does not apply this correction.

18. Compared with the standard normal distribution, what feature of Student’s t-distribution makes its central 95% interval wider?

A. Its lighter tails

B. Its positive center

C. Its narrower peak

D. Its heavier tails

Answer: (D)

Explanation

Student’s t-distribution is bell-shaped around zero and has heavier tails than the standard normal, producing a wider central 95% interval. Lighter tails would not account for that wider interval.

19. Under which conditions is the classical t-interval exact?

A. A large sample comes from a population with any distribution.

B. An independent random sample comes from a normally distributed population.

C. A convenience sample comes from a symmetric population.

D. An independent random sample comes from a population with a known variance.

Answer: (B)

Explanation

The classical t-interval is exact when observations form an independent random sample from a normally distributed population. A large sample from an arbitrary population does not satisfy this exactness condition by itself.

20. In a hypothesis test, what role does the null hypothesis $H_0$ play?

A. It is the competing research claim, accepted when the sample evidence is weak.

B. It describes the observed result that the test statistic measures.

C. It is the default statement, rejected when sample evidence against it is sufficient.

D. It states the maximum probability of making a false-positive decision.

Answer: (C)

Explanation

The null hypothesis is the default statement, and rejection requires sufficient evidence against it. The competing research claim is the role of the alternative hypothesis, not the null hypothesis.

21. A researcher specifies a claim that competes with the default hypothesis and bears the burden of proof. What is this claim called?

A. The alternative hypothesis, $H_a$.

B. The null hypothesis, $H_0$.

C. The significance level, $\alpha$.

D. The p-value of the test.

Answer: (A)

Explanation

The alternative hypothesis is the competing research claim to which the burden of proof is assigned. The null hypothesis is the default statement against which the evidence is assessed.

22. A test fails to reject a null hypothesis that is actually false. Which type of error has occurred?

A. A correct failure to reject the null hypothesis.

B. A Type II error, or false negative.

C. A correct rejection of the null hypothesis.

D. A Type I error, or false positive.

Answer: (B)

Explanation

Failing to reject a false null hypothesis is a Type II error, also called a false negative. A Type I error instead occurs when a true null hypothesis is rejected.

23. A test has significance level $\alpha=0.05$ and produces a p-value of $0.04$. What decision rule applies?

A. Fail to reject $H_0$ because the p-value is less than $\alpha$.

B. Reject $H_0$ because the p-value is less than or equal to $\alpha$.

C. Fail to reject $H_0$ because the p-value equals the probability that $H_0$ is true.

D. Reject $H_0$ because the p-value is greater than $\alpha$.

Answer: (B)

Explanation

A test rejects the null hypothesis when its p-value is less than or equal to the significance level, and here $0.04 \leq 0.05$. A p-value below the significance level does not call for failing to reject the null.

24. A 95% confidence interval for $\mu_1-\mu_2$ is $[0.34,3.71]$. What does this interval indicate about the two means?

A. It establishes that the two population means are equal.

B. It provides evidence that $\mu_1>\mu_2$ because the interval lies above zero.

C. It provides evidence that $\mu_1<\mu_2$ because the interval lies above zero.

D. It gives no evidence about the direction of the difference because the interval excludes zero.

Answer: (B)

Explanation

Because the entire interval for $\mu_1-\mu_2$ is positive, it provides evidence that $\mu_1>\mu_2$. An interval containing zero, unlike this one, would not establish that direction.

25. In a randomized controlled experiment, what is the quantity $\mu_{treatment}-\mu_{control}$ called?

A. The average treatment effect.

B. The standard error of the difference.

C. The pooled population mean.

D. The treatment-group sample mean.

Answer: (A)

Explanation

The difference between the treatment and control population means is called the average treatment effect. The standard error describes uncertainty in estimating a difference rather than naming this population-mean contrast.