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DSST Principles of Statistics study materials
DSST Principles of Statistics study resources

DSST Principles of Statistics practice questions

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Question 1Principles and concepts underlying higher-order statistics

A researcher reports that the sampling distribution of the sample mean is approximately normal even though the population is strongly skewed. Which theorem justifies this result?

  1. The law of large numbers
  2. Chebyshev's inequality
  3. The central limit theorem Correct answer.
  4. The multiplication rule of probability

Explanation

The central limit theorem states that for sufficiently large sample sizes the sampling distribution of the mean approaches normality regardless of the population's shape. The other named results address convergence of averages, bounds on dispersion, and joint probability, not the shape of the sampling distribution.

Question 2Principles and concepts underlying higher-order statistics

In hypothesis testing, the probability of rejecting a true null hypothesis is denoted by which quantity?

  1. Power
  2. The p-value
  3. Beta
  4. Alpha Correct answer.

Explanation

Alpha, the significance level, is defined as the probability of a Type I error: rejecting the null hypothesis when it is actually true. Beta is the probability of a Type II error, power equals one minus beta, and the p-value is a computed summary of the data rather than a fixed error rate.

Question 3Principles and concepts underlying higher-order statistics

A distribution has a long right tail with most observations clustered at low values. Which description fits this distribution?

  1. Negatively skewed
  2. Bimodal and symmetric
  3. Positively skewed Correct answer.
  4. Symmetric with zero skewness

Explanation

A long tail extending toward high values is the defining feature of positive (right) skewness, and the mean typically exceeds the median in such data. Negative skewness would show a long left tail, while symmetry would show balanced tails.

Question 4Principles and concepts underlying higher-order statistics

A two-sample t-test is conducted at alpha equal to 0.05 and yields a p-value of 0.03. What is the correct decision and interpretation?

  1. Fail to reject the null hypothesis because 0.03 is below 0.95
  2. Reject the null hypothesis because 0.03 is less than 0.05 Correct answer.
  3. Accept the null hypothesis as proven true
  4. Increase the sample size until the p-value falls below 0.01

Explanation

The decision rule compares the p-value with the significance level: when the p-value is at or below alpha, the null hypothesis is rejected. A p-value of 0.03 is below 0.05, so the result is statistically significant at that level. Rejection does not prove the null false, only that the data are inconsistent with it at the chosen error rate.

Question 5Principles and concepts underlying higher-order statistics

Which quantity measures the average squared deviation of observations from their sample mean?

  1. Interquartile range
  2. Standard error of the mean
  3. Sample variance Correct answer.
  4. Coefficient of variation

Explanation

Sample variance is defined as the sum of squared deviations from the mean divided by n minus one, so it is literally an average squared deviation. The standard error describes variability of a sample mean across repeated samples, the interquartile range spans the middle half of the data, and the coefficient of variation is a relative dispersion ratio.

Question 6Principles and concepts underlying higher-order statistics

A test correctly identifies 90 of 100 truly diseased patients and correctly identifies 80 of 100 healthy patients. What is the sensitivity of the test?

  1. 0.47
  2. 0.80
  3. 0.90 Correct answer.
  4. 0.89

Explanation

Sensitivity is the proportion of truly diseased individuals who test positive, computed here as 90 divided by 100, which equals 0.90. Specificity would be the 80 divided by 100 figure for healthy individuals, and the other values mix the two groups.

Question 7Principles and concepts underlying higher-order statistics

In a simple linear regression, the coefficient of determination equals 0.64. How is this interpreted?

  1. The correlation between the variables is exactly 0.64
  2. 64 percent of predictions fall within one standard deviation
  3. The slope of the regression line equals 0.64
  4. 64 percent of the variation in the response is explained by the predictor Correct answer.

Explanation

R-squared is the proportion of total variation in the response variable accounted for by the regression model, so 0.64 means 64 percent is explained. The correlation coefficient would be the square root of 0.64 with a sign matching the slope, and the slope is a separate estimated quantity.

Question 8Principles and concepts underlying higher-order statistics

A random variable takes only the values 0 and 1 with probabilities 0.3 and 0.7 respectively. What is its expected value?

  1. 1.00
  2. 0.50
  3. 0.70 Correct answer.
  4. 0.30

Explanation

The expected value of a discrete random variable is the sum of each value times its probability, giving 0 times 0.3 plus 1 times 0.7, which equals 0.7. This matches the probability of the value 1 for a Bernoulli variable.

Question 9Principles and concepts underlying higher-order statistics

Which statement correctly distinguishes a parameter from a statistic?

  1. A parameter is computed from sample data, while a statistic describes the population
  2. A statistic is a constant, while a parameter varies from sample to sample
  3. A parameter is a fixed population characteristic, while a statistic is computed from a sample Correct answer.
  4. Parameters and statistics are both always unknown constants

Explanation

A parameter is a numerical characteristic of a population, usually unknown and treated as fixed, whereas a statistic is a value calculated from sample observations. Because samples vary, statistics are random variables with sampling distributions. This distinction underlies all inferential procedures.

Question 10Principles and concepts underlying higher-order statistics

A 95 percent confidence interval for a population mean is reported as 12.4 to 18.6. Which interpretation is correct?

  1. 95 percent of individual observations lie between 12.4 and 18.6
  2. The probability that the true mean lies in this specific interval is 0.95
  3. The method used produces intervals that capture the true mean in about 95 percent of repeated samples Correct answer.
  4. 95 percent of sample means equal the population mean

Explanation

Confidence level describes the long-run performance of the procedure: about 95 percent of intervals built this way contain the true parameter. For any single realized interval, the parameter either is or is not inside it, so the 0.95 probability statement applies to the method rather than to that one interval. The interval also concerns the mean, not individual observations.

Question 11Principles and concepts underlying higher-order statistics

A study compares three independent group means using one-way analysis of variance. What does the F statistic ratio represent?

  1. The pooled standard deviation divided by the grand mean
  2. Within-group variance divided by total variance
  3. The largest group mean divided by the smallest group mean
  4. Between-group variance divided by within-group variance Correct answer.

Explanation

The F statistic is the ratio of the mean square between groups to the mean square within groups. A large ratio indicates group means differ more than expected from random variation alone. Under the null hypothesis of equal population means, this ratio follows an F distribution.

Question 12Principles and concepts underlying higher-order statistics

A researcher wants to compare all pairwise means among five groups while controlling the familywise error rate. Which approach is most appropriate?

  1. Running five separate one-sample t-tests at alpha 0.05
  2. Reducing alpha to 0.05 divided by the number of observations
  3. Using a post hoc procedure such as Tukey's honestly significant difference test Correct answer.
  4. Reporting only the largest pairwise difference without adjustment

Explanation

Tukey's procedure is designed for all pairwise comparisons and holds the familywise error rate at the chosen level. Conducting many unadjusted tests inflates the chance of at least one false positive. Adjustments such as Bonferroni divide alpha by the number of tests, not by the number of observations.

Question 13Principles and concepts underlying higher-order statistics

In a chi-square test of independence for a contingency table, what does the null hypothesis state?

  1. The sample was drawn from a normal distribution
  2. The two categorical variables are independent in the population Correct answer.
  3. The observed frequencies equal the expected frequencies exactly
  4. The row totals equal the column totals

Explanation

The chi-square test of independence evaluates whether the two categorical variables are associated, with the null hypothesis asserting independence. Expected frequencies are estimated under that assumption and compared with observed counts. Exact equality of observed and expected counts would only occur by chance under independence.

Question 14Principles and concepts underlying higher-order statistics

A regression model predicting salary from years of experience reports a slope of 2.5 with a standard error of 1.0 and 20 degrees of freedom. What is the t statistic for testing whether the slope differs from zero?

  1. 0.40
  2. 5.00
  3. 2.50 Correct answer.
  4. 1.25

Explanation

The t statistic for a slope is the estimate divided by its standard error, so 2.5 divided by 1.0 equals 2.5. This value is then compared with a t distribution having the stated degrees of freedom. The other values misplace the division or multiply the quantities.

Question 15Principles and concepts underlying higher-order statistics

Which condition is required for the sampling distribution of the sample proportion to be well approximated by a normal distribution?

  1. Both the expected number of successes and failures must be reasonably large Correct answer.
  2. The population proportion must equal 0.5 exactly
  3. The sample size must be less than 10 percent of the population
  4. The observations must be measured on an interval scale

Explanation

The normal approximation for a sample proportion is adequate when both the expected number of successes and the expected number of failures are sufficiently large, often stated as at least 10 each. The population proportion need not equal 0.5, and the 10 percent condition concerns independence when sampling without replacement rather than normality.

Question 16Principles and concepts underlying higher-order statistics

A researcher computes a Pearson correlation of negative 0.85 between two variables. Which conclusion is best supported?

  1. About 85 percent of the variance is shared between the variables
  2. The regression slope of the response on the predictor must be positive
  3. There is a strong negative linear association between the variables Correct answer.
  4. Changes in one variable cause decreases in the other

Explanation

A correlation of negative 0.85 indicates a strong negative linear relationship, meaning that as one variable increases the other tends to decrease. Causation cannot be inferred from correlation alone. The shared variance equals r-squared, which is about 0.72 here.

Question 17Principles and concepts underlying higher-order statistics

A hypothesis test is conducted with alpha equal to 0.01 instead of 0.05 while all else stays the same. What is the effect on the two error probabilities?

  1. Both Type I and Type II error probabilities decrease
  2. Neither error probability changes because only the p-value matters
  3. The Type I error probability decreases and the Type II error probability increases Correct answer.
  4. The Type I error probability increases and the Type II error probability decreases

Explanation

Lowering the significance level makes it harder to reject the null hypothesis, which reduces the chance of a Type I error. The same stricter criterion raises the chance of failing to reject a false null, so the Type II error probability increases and power falls.

Question 18Principles and concepts underlying higher-order statistics

Which statement about the median compared with the mean is generally true for skewed data?

  1. The median can only be computed for symmetric distributions
  2. The median is more resistant to extreme outliers than the mean Correct answer.
  3. The median is affected more strongly by outliers than the mean
  4. The mean and median are always equal regardless of shape

Explanation

The median depends on the position of the middle observation, so extreme values have limited influence on it, whereas the mean incorporates every value and can be pulled strongly by outliers. For skewed distributions the median is often preferred as a measure of center. The two measures coincide only in special symmetric cases.

Question 19Principles and concepts underlying higher-order statistics

In a two-way factorial design with two factors each at two levels, what does a significant interaction effect indicate?

  1. Both main effects must also be significant
  2. The two factors are perfectly correlated
  3. The effect of one factor on the response depends on the level of the other factor Correct answer.
  4. The residual variance is larger than the total variance

Explanation

An interaction means the difference in response between levels of one factor is not constant across levels of the other factor, so the factors combine non-additively. Main effects can be significant or not independently of an interaction. Perfect correlation of factors would make their effects inseparable, which is a different design problem.

Question 20Principles and concepts underlying higher-order statistics

A researcher wants to estimate a population mean with a margin of error half as large as the current one, using the same confidence level. Assuming the standard deviation is unchanged, how must the sample size change?

  1. It must be multiplied by four Correct answer.
  2. It must remain the same
  3. It must be doubled
  4. It must be halved

Explanation

The margin of error is proportional to the standard deviation divided by the square root of the sample size. Halving the margin of error therefore requires doubling the square root of n, which quadruples n. Doubling the sample size only reduces the margin by a factor of about 1.41.

Question 21Principles and concepts underlying higher-order statistics

A test has power equal to 0.80 against a particular alternative. What does this value mean?

  1. There is an 80 percent chance of a Type I error
  2. Eighty percent of the sample values fall within the confidence interval
  3. There is an 80 percent chance the null hypothesis is true
  4. There is an 80 percent chance of rejecting the null hypothesis when that alternative holds Correct answer.

Explanation

Power is the probability of correctly rejecting a false null hypothesis, computed as one minus the Type II error rate. A power of 0.80 means that if the specified alternative is true, the test rejects the null in about 80 percent of repeated samples. Power says nothing about the probability that the null itself is true.

Question 22Principles and concepts underlying higher-order statistics

Two variables have a Pearson correlation of zero, yet a scatterplot shows a clear U-shaped pattern. What is the best explanation?

  1. A U-shaped pattern always produces a correlation near one
  2. The correlation measures only linear association, so a strong nonlinear relationship can produce zero correlation Correct answer.
  3. The variables must be categorical rather than quantitative
  4. The correlation was computed incorrectly because zero correlation implies independence

Explanation

Pearson correlation quantifies the strength of linear association, and symmetric nonlinear patterns such as a U shape can yield a correlation near zero despite a strong relationship. Zero correlation therefore does not imply independence in general. Inspecting a scatterplot before relying on correlation is essential.

Question 23Principles and concepts underlying higher-order statistics

A multiple regression includes a predictor whose variance inflation factor is 12. What is the most likely consequence for inference on that predictor's coefficient?

  1. The standard error is inflated and the coefficient may be unstable Correct answer.
  2. The coefficient estimate is biased toward zero
  3. The model's residuals become perfectly normal
  4. The intercept becomes unidentifiable

Explanation

A variance inflation factor of 12 signals strong collinearity with other predictors, which inflates the standard error of the affected coefficient and makes its estimate unstable across samples. Collinearity does not by itself bias the coefficient estimates under the standard assumptions. It also has no bearing on residual normality or intercept identifiability.

Question 24Principles and concepts underlying higher-order statistics

A hypothesis test yields a p-value of 0.049 with alpha equal to 0.05. Which statement is most defensible?

  1. The null hypothesis is false with probability 0.049
  2. Replication would certainly produce a significant result again
  3. The result is statistically significant but may not be practically important Correct answer.
  4. The effect size is necessarily large because the result is significant

Explanation

A p-value just below alpha indicates statistical significance at the chosen threshold, but significance does not measure the size or practical relevance of an effect. The p-value is not the probability that the null hypothesis is true. Near-threshold results are also especially likely to vary on replication.

Question 25Principles and concepts underlying higher-order statistics

A bootstrap procedure repeatedly resamples with replacement from the observed data to estimate the standard error of a statistic. What property of the bootstrap makes it useful in this setting?

  1. It approximates the sampling distribution using the empirical distribution as a stand-in for the population Correct answer.
  2. It eliminates the need for random sampling in the original study
  3. It guarantees exact coverage for any statistic with small samples
  4. It assumes the statistic is normally distributed and therefore avoids resampling

Explanation

The bootstrap treats the observed sample as an approximation of the population and simulates the sampling distribution by resampling with replacement. This makes it useful when theoretical standard errors are hard to derive or normality is doubtful. It is an approximation, however, and does not repair flaws in the original sampling design.

Question 26Principles and concepts underlying higher-order statistics

A researcher reports the standard error of the mean as 2.0 for a sample of size 36. What is the sample standard deviation?

  1. 2.0
  2. 12.0 Correct answer.
  3. 72.0
  4. 0.33

Explanation

The standard error of the mean equals the sample standard deviation divided by the square root of n. With n = 36, the square root is 6, so the standard deviation equals 2.0 times 6, which is 12.0.

Question 27Principles and concepts underlying higher-order statistics

A distribution of test scores is strongly skewed to the right with a few very high outliers. Which measure of central tendency is most resistant to those outliers?

  1. The mean
  2. The range
  3. The standard deviation
  4. The median Correct answer.

Explanation

The median depends on position in the ordered data, so extreme values do not pull it toward the tail. The mean incorporates every value, so a few very high scores shift it upward. The range and standard deviation describe spread rather than center.

Question 28Principles and concepts underlying higher-order statistics

A sample of 100 observations yields a sample mean of 50 and a sample variance of 25. What is the estimated standard error of the sample mean?

  1. 5.0
  2. 2.5
  3. 0.25
  4. 0.5 Correct answer.

Explanation

The standard error equals the sample standard deviation divided by the square root of n. The standard deviation is the square root of 25, which is 5, and the square root of 100 is 10, giving 5/10 = 0.5.

Question 29Principles and concepts underlying higher-order statistics

Which quantity describes the spread of the sampling distribution of a statistic across repeated samples?

  1. The population variance
  2. The standard error Correct answer.
  3. The confidence level
  4. The sample range

Explanation

The standard error is by definition the standard deviation of a statistic's sampling distribution. It quantifies how much a statistic such as a sample mean would vary from sample to sample. The other quantities describe the data or a decision rule, not the variability of the statistic.

Question 30Principles and concepts underlying higher-order statistics

A population is exactly normal with mean 70 and standard deviation 8. A single observation is drawn at random. What is the probability that it exceeds 78?

  1. About 0.500
  2. About 0.159 Correct answer.
  3. About 0.841
  4. About 0.023

Explanation

The value 78 is one standard deviation above the mean, since (78 - 70)/8 = 1. For a normal distribution, about 16 percent of the area lies above one standard deviation, so the probability is roughly 0.159.

Question 31Principles and concepts underlying higher-order statistics

Which statement best describes what a p-value measures?

  1. The probability that the null hypothesis is true given the observed data
  2. The probability the study will replicate if repeated exactly
  3. The probability that the alternative hypothesis is false
  4. The probability of data at least as extreme as observed, assuming the null hypothesis is true Correct answer.

Explanation

A p-value is computed under the assumption that the null hypothesis holds and expresses how unusual the observed result, or something more extreme, would be. It is a statement about the data conditional on the null, not about the truth of either hypothesis.

Question 32Principles and concepts underlying higher-order statistics

A statistician wants to describe the strength of linear association between two continuous variables measured on the same subjects. Which technique is designed for this purpose?

  1. Analysis of variance
  2. Chi-square goodness of fit
  3. Pearson correlation Correct answer.
  4. One-sample t test

Explanation

Pearson correlation summarizes both direction and strength of a linear relationship between two continuous variables. The other procedures compare means or frequencies and do not produce a measure of association strength.

Question 33Principles and concepts underlying higher-order statistics

Which symbol conventionally denotes the probability of a Type I error in a hypothesis test?

  1. Sigma
  2. Mu
  3. Alpha Correct answer.
  4. Beta

Explanation

Alpha is the chosen significance level, the probability of rejecting a true null hypothesis. Beta denotes the probability of a Type II error, while mu and sigma denote population mean and standard deviation.

Question 34Principles and concepts underlying higher-order statistics

A sample of 25 observations from a normal population yields a sample mean of 40 and a sample standard deviation of 5. A 95 percent confidence interval for the population mean is constructed using the t distribution. What is the margin of error, given that the critical t value with 24 degrees of freedom is 2.064?

  1. About 1.03
  2. About 10.32
  3. About 0.41
  4. About 2.06 Correct answer.

Explanation

The margin of error is the critical t value times the standard error, where the standard error is 5 divided by the square root of 25, or 1.0. Multiplying 2.064 by 1.0 gives about 2.06.

Question 35Principles and concepts underlying higher-order statistics

A researcher tests whether the mean weight loss differs among four diets using one-way analysis of variance. The between-group sum of squares is 90 and the within-group sum of squares is 240, with 3 and 20 degrees of freedom respectively. What is the F statistic?

  1. 2.50 Correct answer.
  2. 0.40
  3. 7.50
  4. 0.375

Explanation

The between-group mean square is 90/3 = 30 and the within-group mean square is 240/20 = 12. The F statistic is the ratio 30/12 = 2.50.

Question 36Principles and concepts underlying higher-order statistics

In a simple linear regression with 30 observations, the estimated slope is 3.2 and its standard error is 1.6. What is the value of the t statistic for testing whether the true slope is zero?

  1. 0.50
  2. 5.12
  3. 2.00 Correct answer.
  4. 1.60

Explanation

The t statistic for a slope is the estimate divided by its standard error, so 3.2/1.6 = 2.00. This value is compared with a t distribution having n minus 2 degrees of freedom.

Question 37Principles and concepts underlying higher-order statistics

A two-sided test of a population mean yields z equal to 2.10. Using a standard normal table, what is the approximate p-value?

  1. 0.018
  2. 0.036 Correct answer.
  3. 0.071
  4. 0.982

Explanation

The area above z = 2.10 in one tail is about 0.0179, and a two-sided test doubles that to about 0.036. The doubling accounts for extreme values in both directions.

Question 38Principles and concepts underlying higher-order statistics

A 90 percent confidence interval for a proportion is computed as 0.42 to 0.58. Which statement correctly interprets this interval?

  1. About 90 percent of sample proportions fall between 0.42 and 0.58
  2. The true proportion is within 0.08 of the sample proportion with probability 0.90
  3. If the study were repeated many times, about 90 percent of intervals built this way would contain the true proportion Correct answer.
  4. There is a 90 percent probability that the true proportion lies between 0.42 and 0.58

Explanation

Confidence refers to the long-run behavior of the procedure across repeated samples, not to a probability assigned to a fixed parameter. About 90 percent of intervals constructed by this method would capture the true proportion.

Question 39Principles and concepts underlying higher-order statistics

A test of independence is run on a 3 by 4 contingency table with a total sample size of 200. How many degrees of freedom does the chi-square statistic have?

  1. 12
  2. 6 Correct answer.
  3. 11
  4. 5

Explanation

For a contingency table the degrees of freedom equal (rows minus 1) times (columns minus 1). Here that is 2 times 3, which equals 6.

Question 40Principles and concepts underlying higher-order statistics

A regression of household income on years of education yields an intercept of 12 and a slope of 4, with income measured in thousands of dollars. What is the predicted income for a person with 15 years of education?

  1. 60 thousand dollars
  2. 72 thousand dollars Correct answer.
  3. 48 thousand dollars
  4. 27 thousand dollars

Explanation

The predicted value is the intercept plus the slope times the predictor, so 12 + 4 times 15 = 72. The slope applies to each additional year of education.

Question 41Principles and concepts underlying higher-order statistics

A quality engineer wants to detect a shift in a process mean quickly and uses a chart with a known in-control mean and standard deviation. Which quantity determines the probability that a single point falls beyond three standard errors of the mean when the process is in control?

  1. About 0.10, from the width of a 90 percent interval
  2. About 0.0027, from the normal tail area beyond three standard deviations Correct answer.
  3. About 0.5, since the process is centered
  4. About 0.05, from the conventional significance level

Explanation

Under a normal model, the two-sided tail area beyond three standard deviations is roughly 0.0027, which is the false-alarm rate for a single point on a three-sigma chart. This is a direct normal probability calculation.

Question 42Principles and concepts underlying higher-order statistics

A researcher computes a 95 percent confidence interval for a difference in means and obtains 0.4 to 3.8. What does this interval imply about a two-sided test of no difference at the 0.05 level?

  1. The null hypothesis of no difference would be rejected Correct answer.
  2. The null hypothesis would fail to be rejected
  3. The test result cannot be inferred from the interval
  4. The null hypothesis would be rejected only at the 0.01 level

Explanation

A confidence interval that excludes zero is consistent with rejecting the null hypothesis of no difference at the corresponding two-sided alpha. Because the entire interval lies above zero, the test at the 0.05 level rejects.

Question 43Principles and concepts underlying higher-order statistics

A survey of 400 voters finds 220 in favor of a proposal. What is the sample proportion in favor, and what is its estimated standard error?

  1. 0.55 with standard error about 0.025 Correct answer.
  2. 0.55 with standard error about 0.0006
  3. 0.55 with standard error about 0.50
  4. 0.45 with standard error about 0.025

Explanation

The proportion is 220/400 = 0.55. The standard error is the square root of (0.55 times 0.45 divided by 400), which is about 0.0249.

Question 44Principles and concepts underlying higher-order statistics

A study reports an odds ratio of 2.0 with a 95 percent confidence interval of 1.1 to 3.6. Which conclusion is best supported?

  1. The exposure has no association with the outcome because the interval is wide
  2. The result proves causation between exposure and outcome
  3. The exposure is associated with higher odds of the outcome, and the association is statistically significant at the 0.05 level Correct answer.
  4. The association is significant but the odds ratio is exactly 1.0

Explanation

An odds ratio above 1 suggests higher odds of the outcome with exposure, and an interval excluding 1.0 indicates statistical significance at the 0.05 level. Wide intervals signal imprecision, not absence of association. Observational data alone do not establish causation.

Question 45Principles and concepts underlying higher-order statistics

A random variable has a mean of 20 and a variance of 16. A sample of 64 independent observations is drawn. What are the mean and standard deviation of the sample mean?

  1. Mean 20 and standard deviation 0.5 Correct answer.
  2. Mean 20 and standard deviation 2.0
  3. Mean 2.5 and standard deviation 0.5
  4. Mean 20 and standard deviation 0.25

Explanation

The sample mean is unbiased, so its mean equals 20. Its standard deviation is the population standard deviation 4 divided by the square root of 64, which is 8, giving 0.5.

Question 46Principles and concepts underlying higher-order statistics

A dataset of 11 ordered observations has a median of 34. A twelfth observation of 500 is added. How do the median and mean change?

  1. The median shifts slightly while the mean increases substantially Correct answer.
  2. The median increases substantially while the mean is unchanged
  3. Both the median and the mean increase by the same amount
  4. Neither the median nor the mean changes

Explanation

Adding one extreme value pulls the mean strongly upward because the mean uses every value. The median depends on position, so it moves only to the average of the two middle values of the enlarged sample.

Question 47Principles and concepts underlying higher-order statistics

A researcher plans a two-sided test and wants power of 0.90 for a specified effect size at alpha 0.05. Which change will increase the required sample size?

  1. Reducing the population standard deviation
  2. Accepting a larger effect size as the target
  3. Planning for power of 0.95 instead of 0.90 Correct answer.
  4. Using alpha 0.10 instead of 0.05

Explanation

Required sample size grows as the target power rises, because detecting an effect more reliably requires more information. Raising alpha, targeting a larger effect, or reducing variability all reduce the sample size needed.

Question 48Principles and concepts underlying higher-order statistics

In a multiple regression with several predictors, the coefficient on a predictor is positive in the full model but negative in a simple regression using only that predictor. Which explanation is most plausible?

  1. Heteroscedasticity, which reverses the direction of partial effects
  2. A coding error, since coefficient signs cannot differ between models
  3. Suppression, in which other predictors are correlated with the predictor and alter the sign of its partial relationship Correct answer.
  4. Multicollinearity, which always forces coefficients toward zero but never changes sign

Explanation

A sign reversal can occur when the other predictors are correlated with the predictor of interest, so the partial coefficient reflects the relationship after adjusting for them. This is a form of suppression and does not imply an error. Multicollinearity inflates standard errors but does not by itself guarantee a sign flip.

Question 49Principles and concepts underlying higher-order statistics

A researcher uses maximum likelihood to fit a logistic regression and compares two nested models. Which test is appropriate for the contribution of the added predictors?

  1. A Pearson correlation between predicted and observed values
  2. A likelihood ratio test comparing the deviances of the two models Correct answer.
  3. A paired t test on the fitted probabilities
  4. A chi-square goodness-of-fit test on the response categories alone

Explanation

For nested logistic models fitted by maximum likelihood, the difference in deviances follows an approximate chi-square distribution with degrees of freedom equal to the number of added parameters. This likelihood ratio test directly assesses whether the extra predictors improve fit.

Question 50Principles and concepts underlying higher-order statistics

A statistician applies a log transformation to a right-skewed response variable before fitting a linear model. Which consequence is most directly expected?

  1. The response variable becomes exactly normally distributed
  2. The regression slope becomes invariant to the transformation
  3. The coefficient of determination is guaranteed to increase
  4. The model's errors may better satisfy normality and constant-variance assumptions Correct answer.

Explanation

A log transformation compresses the upper tail and can stabilize variance, making residual assumptions of normality and homoscedasticity more plausible. It does not guarantee a higher R-squared, and the slope is expressed on the log scale rather than being invariant.