All nine units, weighted exactly like the real exam multiple-choice section. Work top to bottom — every unit opens easy and climbs to exam-level difficulty.
Which measure of center is resistant to outliers?
The median only depends on the middle value's position, not the actual magnitude of extreme values, so it is resistant. The mean is pulled toward outliers.
A distribution with a long tail extending to the right is described as:
Skew is named for the direction the tail points. A long right tail means the distribution is skewed right, and typically mean > median.
For a data set: 4, 7, 7, 9, 10, 12, 15 what is the median?
The data are already ordered: 4, 7, 7, 9, 10, 12, 15. With n = 7, the median is the 4th value: 9.
If Q1 = 12 and Q3 = 19, what is the interquartile range (IQR)?
IQR = Q3 − Q1 = 19 − 12 = 7. The IQR measures the spread of the middle 50% of the data.
A test has mean 68 and standard deviation 3. A student scores 74. What is the z-score?
z = (74 − 68) / 3 = 6 / 3 = 2. The score is 2 standard deviations above the mean.
In a box plot, if the median line is closer to Q1 than to Q3, the middle 50% of data is:
If the median sits closer to Q1, then the lower quarter is compressed and the upper quarter is stretched out, suggesting the middle 50% leans toward the right (higher values are more spread out).
Using the 1.5×IQR rule, with Q1 = 20 and Q3 = 30, which value would be flagged as an outlier?
IQR = 10, so 1.5×IQR = 15. Lower fence = 20 − 15 = 5, upper fence = 30 + 15 = 45. Since 47 > 45, it falls beyond the upper fence and is flagged as an outlier. The other values (35, 25, 10) all fall within [5, 45].
Adding a constant of 5 to every value in a data set will:
Adding a constant shifts every value equally, which shifts the mean by that constant but does not change how spread out the values are relative to each other, so the standard deviation is unchanged.
Multiplying every value in a data set by 2 will cause the standard deviation to:
Multiplying each value by a constant k multiplies the standard deviation by |k|. Here k = 2, so the standard deviation doubles. (Variance would quadruple since it uses squared units.)
Heights of adult men are approximately Normal with mean 70 inches and SD 3 inches. Using the Empirical Rule, about what percent of men are between 64 and 76 inches?
64 = 70 − 2(3) and 76 = 70 + 2(3), so this range covers 2 standard deviations on each side of the mean. The Empirical Rule states about 95% of values fall within 2 SDs of the mean.
A distribution is strongly skewed with several outliers. Which pair of summary statistics is most appropriate to report?
Median and IQR are resistant to outliers and skew, making them the appropriate choice when a distribution is not symmetric or contains extreme values. Mean and SD are pulled by outliers.
Two classes take the same test. Class A has mean 80 and SD 5. Class B has mean 80 and SD 12. Which statement is true?
Standard deviation measures spread. Since 12 > 5, Class B's scores are more variable/spread out than Class A's, even though both classes share the same mean.
SAT Math scores are approximately Normal with mean 520 and SD 100. What proportion of test-takers score above 650?
z = (650 − 520)/100 = 1.30. From the Normal table, P(Z < 1.30) = 0.9032, so P(Z > 1.30) = 1 − 0.9032 = 0.097.
A distribution of exam scores is Normal with mean 75 and SD 8. What score corresponds to the 90th percentile?
The z-score for the 90th percentile is about 1.28. Raw score = 75 + 1.28(8) = 75 + 10.24 = 85.24 ≈ 85.2.
In Data Set A (mean 50, SD 10), a value of 65 has the same z-score as which value in Data Set B (mean 100, SD 20)?
In Set A, z = (65−50)/10 = 1.5. Applying z = 1.5 to Set B: x = 100 + 1.5(20) = 100 + 30 = 130.
Removing the single largest value from a right-skewed data set will most likely:
Because the mean incorporates the magnitude of every value, removing an extreme high value pulls the mean down noticeably. The median, based on position/rank, changes very little or not at all.
Test scores are Normal with mean 82, SD 6. What is the probability a randomly selected score is between 75 and 90?
z1 = (75−82)/6 = −1.17, z2 = (90−82)/6 = 1.33. P(Z<1.33) − P(Z<−1.17) ≈ 0.9082 − 0.1210 = 0.787.
A right-skewed distribution has mean 45 and median 38. Which best explains the relationship?
In a right-skewed distribution, a small number of large values in the long right tail pull the mean upward while barely affecting the median's rank-based position, so mean > median.
A university reports that among applicants, SAT scores are Normal with mean 1200, SD 120. If the admissions cutoff is the 95th percentile, approximately what score is required?
The z-score for the 95th percentile is about 1.645. Raw score = 1200 + 1.645(120) ≈ 1200 + 197.4 = 1397.4 ≈ 1397.
Two data sets have the same range but Data Set X has a much larger IQR than Data Set Y. This suggests:
Since both share the same range (overall spread of min to max) but X has a larger IQR, the middle half of X's data is more spread out, while Y's data is more tightly clustered in the middle 50% despite covering the same total range.
A scatterplot shows points clustering tightly around a line that rises from left to right. This describes:
Points clustering tightly around an upward-sloping line indicate a strong, positive, linear association between the two variables.
The correlation coefficient r measures:
r quantifies both the direction (sign) and strength (closeness to -1 or 1) of a linear relationship between two quantitative variables. It says nothing about causation, and it is not the same as the slope.
A least-squares regression line minimizes:
The least-squares regression line is defined as the line that minimizes the sum of the squared residuals (vertical distances between observed and predicted y-values).
A regression of house price on square footage gives R-squared = 0.81. This means:
R-squared = 0.81 means 81% of the variability in house price is accounted for by the linear regression model using square footage; r itself would be the square root, 0.9.
A residual plot shows a clear curved (U-shaped) pattern. This suggests:
A curved pattern in the residual plot indicates that the relationship between the variables is not actually linear, so a linear regression model is not appropriate.
Two variables have a strong correlation of r = 0.92. Which statement is correct?
A high correlation only describes the strength of a linear association; it never by itself establishes causation, since confounding or lurking variables could explain the relationship.
A sample selected so that every member of the population has an equal chance of being chosen is called a:
A simple random sample (SRS) gives every possible sample of the given size an equal chance of being selected, which also means every individual has an equal chance of inclusion.
Which of the following is an example of an observational study rather than an experiment?
In an observational study, researchers simply observe and record variables without imposing any treatment. Recording diets and outcomes without intervention is observational; the other three examples all involve randomly assigning treatments, making them experiments.
A voluntary response sample is generally considered biased because:
People who choose to respond voluntarily often have strong or unusual opinions about the topic, so the sample is not representative of the broader population, leading to bias.
In an experiment, the group that does not receive the treatment of interest but instead receives an inactive substance is called the:
The control group receives no active treatment (often a placebo) and serves as the baseline for comparison against the treatment group(s).
A researcher divides a population into age groups, then takes a simple random sample from each age group. This sampling method is called:
Stratified sampling divides the population into homogeneous subgroups (strata) — here, age groups — and then takes a separate random sample from each stratum.
A researcher randomly selects 10 entire classrooms out of 200 in a school district and surveys every student in those 10 classrooms. This is:
Cluster sampling selects entire naturally occurring groups (clusters) at random and then includes every member of the chosen clusters, unlike stratified sampling, which samples from every group.
The main purpose of random assignment in an experiment is to:
Random assignment helps ensure that treatment groups are similar with respect to lurking variables (both known and unknown), so any observed difference can more confidently be attributed to the treatment itself.
A study finds that ice cream sales and drowning deaths are strongly correlated. The most likely explanation is:
Hot weather increases both ice cream sales and swimming (and therefore drowning risk), making it a lurking variable that creates an association between the two variables without either causing the other.
In a double-blind experiment:
In a double-blind experiment, both the subjects and the people measuring/evaluating the outcomes are unaware of who received which treatment, which helps prevent bias from expectations on either side.
A statistics teacher wants to compare two teaching methods but suspects that prior math ability could affect results. To account for this, the teacher should use:
Blocking on prior math ability groups students with similar ability together, then randomly assigns the teaching method within each block. This reduces variability due to ability differences, isolating the effect of the teaching method more clearly.
Which of the following would introduce undercoverage bias into a phone survey about internet usage?
Restricting calls to landlines systematically excludes households that only use cell phones (often younger or lower-income individuals), leading to undercoverage bias since part of the population has no chance of selection.
A wording issue where a question is phrased in a way that leads respondents toward a particular answer is called:
Response bias occurs when the way a question is worded, or the setting in which it's asked, systematically influences respondents toward a certain answer, distorting the results.
A survey is mailed to 1000 people, but only 150 return it. The main concern with this situation is:
When a large fraction of the selected sample fails to respond, the responders may differ in meaningful ways (e.g., stronger opinions, more free time) from non-responders, creating nonresponse bias even though everyone had a chance to be selected initially.
A pharmaceutical company wants to test a new drug's effect on blood pressure. Which design provides the strongest basis for a cause-and-effect conclusion?
Only a randomized, controlled experiment allows researchers to isolate the effect of the treatment from confounding variables, and blinding further removes bias from expectations, together providing the strongest basis for causal conclusions.
If P(A) = 0.4 and P(B) = 0.3, and A and B are mutually exclusive, what is P(A or B)?
For mutually exclusive events, P(A or B) = P(A) + P(B) = 0.4 + 0.3 = 0.7, since there is no overlap to subtract.
Two events A and B are independent if:
Two events are independent exactly when P(A and B) = P(A) x P(B); knowing one event occurred does not change the probability of the other.
A discrete random variable can take on:
A discrete random variable takes on a countable (often finite) set of distinct values, such as the number of heads in 10 coin flips, in contrast to continuous variables which take any value in an interval.
A fair six-sided die is rolled once. What is P(rolling a number greater than 4)?
The outcomes greater than 4 are 5 and 6, giving 2 favorable outcomes out of 6 total, so P = 2/6 = 1/3.
In a group, P(likes coffee) = 0.6, P(likes tea) = 0.5, and P(likes both) = 0.3. What is P(likes coffee or tea)?
P(coffee or tea) = P(coffee) + P(tea) - P(both) = 0.6 + 0.5 - 0.3 = 0.8.
A probability distribution table must satisfy which condition?
Every valid probability distribution requires each probability to be non-negative (between 0 and 1) and the sum of all probabilities across outcomes to equal exactly 1.
A binomial random variable X has n = 8 trials and probability of success p = 0.3. What is P(X = 3)?
Using the binomial probability formula with n = 8, k = 3, p = 0.3: C(8,3)(0.3)^3(0.7)^5 = 56(0.027)(0.16807) ≈ 0.254.
A random variable X has mean 10 and standard deviation 4. If Y = 3X + 5, what is the mean of Y?
E(Y) = 3*E(X) + 5 = 3(10) + 5 = 35.
Using the same random variable X (SD = 4) and Y = 3X + 5, what is the standard deviation of Y?
SD(Y) = |3| * SD(X) = 3(4) = 12. The added constant (+5) shifts the distribution but does not affect its spread.
If X and Y are independent random variables with Var(X) = 9 and Var(Y) = 16, what is Var(X + Y)?
For independent random variables, Var(X + Y) = Var(X) + Var(Y) = 9 + 16 = 25. (Standard deviations do not simply add, but variances do.)
A binomial variable has n = 20 and p = 0.6. What is the mean and standard deviation of X?
Mean = np = 20(0.6) = 12. SD = sqrt(np(1-p)) = sqrt(20 x 0.6 x 0.4) = sqrt(4.8) ≈ 2.19.
A geometric random variable models the number of trials until the first success, with p = 0.2 for each trial. What is P(first success occurs on the 4th trial)?
P(X = 4) = (0.8)^3 (0.2) = 0.512 x 0.2 = 0.1024 ≈ 0.105.
For a binomial setting, which condition is NOT required?
The binomial setting requires a fixed number of independent trials, each with two outcomes (success/failure) and constant probability of success. Normality of an underlying population is not one of the requirements (in fact X itself is discrete, not continuous).
A geometric random variable has p = 0.2. What is the expected number of trials until the first success?
For a geometric random variable, the expected value is E(X) = 1/p = 1/0.2 = 5 trials.
A quality control process finds that 10% of items are defective. In a random sample of 10 items, what is P(at most 2 defective)?
Using the binomial CDF with n = 10, p = 0.1: P(X <= 2) = P(0) + P(1) + P(2) ≈ 0.349 + 0.387 + 0.194 = 0.678.
The sampling distribution of a statistic describes:
A sampling distribution shows how a statistic (like a sample mean or sample proportion) would vary if you repeatedly took samples of the same size from the population and calculated the statistic each time.
As sample size increases, the standard deviation of the sampling distribution of the sample mean will:
The standard deviation of the sampling distribution of the sample mean (standard error) is sigma/sqrt(n), which decreases as n increases -- larger samples give more precise estimates.
A population proportion is p = 0.4. For a sample of size n = 100, what is the standard deviation of the sampling distribution of the sample proportion?
SD(p-hat) = sqrt[p(1-p)/n] = sqrt[(0.4)(0.6)/100] = sqrt(0.0024) ≈ 0.049.
Which condition is needed to treat the sampling distribution of p-hat as approximately Normal?
The Normal approximation for the sampling distribution of a sample proportion requires the large counts condition: np >= 10 and n(1-p) >= 10, ensuring enough expected successes and failures.
The Central Limit Theorem states that, for a sufficiently large sample size:
The Central Limit Theorem guarantees that as sample size grows large, the sampling distribution of the sample mean becomes approximately Normal, even if the underlying population distribution is not Normal.
A population has mean 50 and SD 12. For samples of size n = 36, what is the standard deviation of the sampling distribution of the sample mean?
Standard error = sigma/sqrt(n) = 12/sqrt(36) = 12/6 = 2.
Using the population above (mean 50, SD 12, n = 36), assuming conditions are met, what is P(sample mean > 53)?
Standard error = 12/sqrt(36) = 2. z = (53-50)/2 = 1.5. P(Z > 1.5) ≈ 1 - 0.9332 = 0.067.
A sample proportion is unbiased as an estimator of the population proportion because:
An estimator is unbiased when the mean of its sampling distribution equals the true population parameter. For p-hat, the mean of the sampling distribution is exactly p, though any single sample's p-hat may differ from p.
Increasing the sample size from n = 50 to n = 200 (a 4-fold increase) will change the standard error of the sample mean by a factor of:
Since SE = sigma/sqrt(n), increasing n by a factor of 4 increases sqrt(n) by a factor of 2, so the standard error is divided by 2 (halved).
A population is strongly skewed. For the sampling distribution of the sample mean to be approximately Normal, which is generally sufficient?
Even when the population is skewed, the Central Limit Theorem tells us that a sample size of roughly 30 or more is generally sufficient for the sampling distribution of the mean to be approximately Normal.
In a confidence interval for a proportion, the margin of error accounts for:
The margin of error reflects chance variability due to random sampling. It does NOT correct for bias introduced by poor sampling methods, nonresponse, or flawed study design.
Which condition is required before constructing a one-sample z-interval for a proportion?
A one-sample z-interval for a proportion requires a random sample, independence (10% condition), and the large counts condition: n*p-hat >= 10 and n*(1-p-hat) >= 10 to justify the Normal approximation.
A poll of 200 randomly selected voters finds 65% support a measure. What is the standard error of p-hat?
SE = sqrt[(0.65)(0.35)/200] = sqrt(0.0011375) ≈ 0.034.
Using the same poll (p-hat = 0.65, n = 200), what is the 95% confidence interval margin of error?
ME = 1.96 x 0.034 = 0.066 (rounded).
A one-proportion z-test is used to test a claim about a single population proportion. Which of these is the correct null hypothesis format?
In a one-proportion z-test, the null hypothesis states that the population proportion p equals some specific hypothesized value p0, written H0: p = p0.
A 95% confidence interval for a proportion is (0.584, 0.716). Which interpretation is correct?
The correct interpretation is that we are 95% confident that the interval (0.584, 0.716) captures the true population proportion -- this describes our confidence in the method applied to this sample, not a probability statement about the parameter itself.
A company claims 50% of customers prefer their new product. A sample of 150 customers finds 58% preferring it. Testing H0: p = 0.5 vs Ha: p != 0.5, what is the z test statistic?
SE under H0 = sqrt[(0.5)(0.5)/150] ≈ 0.0408. z = (0.58 - 0.50)/0.0408 ≈ 1.96.
Using the test statistic from the previous question (z ≈ 1.96), what is the approximate two-sided p-value?
For a two-sided test, p-value = 2 x P(Z > 1.96) ≈ 2(0.025) = 0.05.
Two independent samples are compared: Sample 1 (n=120, p-hat=0.45) and Sample 2 (n=130, p-hat=0.32), testing H0: p1 = p2. What is the pooled proportion used in the test?
Pooled p-hat = [(0.45)(120) + (0.32)(130)] / (120+130) = (54 + 41.6)/250 = 95.6/250 ≈ 0.382.
Using the pooled proportion (≈0.382) from the previous question, what is the approximate z test statistic for comparing the two proportions?
Pooled SE = sqrt[(0.382)(0.618)(1/120+1/130)] ≈ 0.0615. z = (0.45-0.32)/0.0615 ≈ 2.11.
When comparing two proportions with a confidence interval (not a hypothesis test), which standard error formula is used?
For a confidence interval, we do not assume the two proportions are equal, so we use the unpooled standard error: sqrt[p1-hat(1-p1-hat)/n1 + p2-hat(1-p2-hat)/n2].
A researcher tests H0: p = 0.3 vs Ha: p > 0.3 and obtains a p-value of 0.02. At alpha = 0.05, the correct conclusion is:
Since the p-value (0.02) is less than alpha (0.05), we reject H0 and conclude there is convincing evidence that the true proportion exceeds 0.3.
A Type I error in a hypothesis test about a proportion occurs when:
A Type I error occurs when the null hypothesis is actually true, but the test procedure leads us to (incorrectly) reject it. Its probability is controlled by the significance level alpha.
A confidence interval for the difference in two proportions is entirely above zero: (0.03, 0.18). This suggests:
Since the entire interval for p1 - p2 is positive (does not include 0), this provides convincing evidence that p1 is genuinely greater than p2 in the population.
Why do we use a t-distribution instead of a Normal distribution when constructing a confidence interval for a population mean?
When the population standard deviation sigma is unknown (the usual case), we estimate it with the sample standard deviation s. This added uncertainty means we use the t-distribution, which has slightly heavier tails than the Normal distribution.
The degrees of freedom for a one-sample t-procedure with sample size n is:
For a one-sample t-procedure, degrees of freedom = n - 1, reflecting the one parameter (the mean) estimated from the data.
A sample of size 25 has mean 52 and standard deviation 8. What is the standard error of the mean?
SE = s/sqrt(n) = 8/sqrt(25) = 8/5 = 1.6.
Using the sample above (mean 52, s=8, n=25), what is the critical value t* for a 95% confidence interval?
With df = 24, the t critical value for a 95% confidence interval is t* ≈ 2.064 (slightly larger than the Normal z* = 1.96 because of the extra uncertainty from estimating sigma).
Which of the following is a condition for a one-sample t-test for a mean?
A one-sample t-test requires a random sample and either an approximately Normal population (checked via graph) or a large enough sample size for the sampling distribution to be approximately Normal, along with independence (10% condition).
Using the same sample (mean 52, SE=1.6, t*=2.064), what is the margin of error for the 95% confidence interval?
ME = t* x SE = 2.064 x 1.6 ≈ 3.30.
A researcher tests H0: mu = 100 vs Ha: mu > 100 using a sample of n = 20 with mean 105 and s = 15. What is the t test statistic?
SE = 15/sqrt(20) ≈ 3.354. t = (105-100)/3.354 ≈ 1.49.
Using the t statistic from the previous question (t ≈ 1.49, df = 19), what is the approximate one-sided p-value?
Using a t-distribution with df = 19, P(T > 1.49) ≈ 0.076.
For a two-sample t-test comparing two independent means, which condition is NOT required?
Unlike the older pooled-variance approach, the standard (Welch) two-sample t-procedure does NOT require the two population standard deviations to be equal -- it estimates the standard error using each sample's own standard deviation.
Two independent samples are compared: Sample 1 (n=15, mean=78, s=6) and Sample 2 (n=18, mean=82, s=7). What is the standard error of the difference in means?
SE = sqrt(6^2/15 + 7^2/18) = sqrt(2.4 + 2.722) = sqrt(5.122) ≈ 2.26.
Using the samples above, what is the t test statistic for testing H0: mu1 = mu2 (mean1 = 78, mean2 = 82, SE ≈ 2.26)?
t = (78 - 82)/2.26 ≈ -1.77.
A study measures the same 12 subjects before and after a treatment. This calls for which type of t-procedure?
Because the same subjects are measured twice, the before/after observations are dependent (paired), so we compute the differences for each subject and use a one-sample t-procedure on those differences.
In a paired design, 12 differences (after minus before) have mean 4.2 and standard deviation 3.1. What is the t test statistic for testing H0: mu_diff = 0?
SE = 3.1/sqrt(12) ≈ 0.895. t = 4.2/0.895 ≈ 4.69.
Using the paired data above (n=12, mean diff=4.2, s diff=3.1), what is the 95% confidence interval margin of error for the mean difference?
With df = 11, t* ≈ 2.201. ME = t* x SE = 2.201 x 0.895 ≈ 1.97.
A 90% confidence interval for a population mean is narrower than a 95% confidence interval from the same data because:
Reducing the confidence level (e.g., from 95% to 90%) decreases the critical value t* (or z*), which directly shrinks the margin of error and produces a narrower interval, at the cost of less confidence that it captures the true parameter.
A chi-square goodness-of-fit test is used to determine whether:
A chi-square goodness-of-fit test compares the observed distribution of a single categorical variable across several categories to a hypothesized (expected) distribution.
A die is rolled 100 times, expecting each of 3 outcome groups equally often (expected count 33.3 each). Observed counts are 45, 30, 25. What is the chi-square test statistic?
Chi-square = (45-33.3)^2/33.3 + (30-33.3)^2/33.3 + (25-33.3)^2/33.3 ≈ 4.11 + 0.33 + 2.07 ≈ 6.50.
A chi-square test of independence uses a 3-row by 2-column table. What are the degrees of freedom?
df = (rows - 1)(columns - 1) = (3-1)(2-1) = 2 x 1 = 2.
In a regression setting, the null hypothesis for a t-test on the slope is typically:
The standard test for the significance of a linear relationship tests H0: beta1 = 0 (the population slope is zero, meaning no linear relationship) against Ha: beta1 != 0 (or a one-sided alternative).
A regression output shows slope b1 = 2.5 with standard error SE(b1) = 0.6, based on n = 20 data points. What is the t test statistic for H0: beta1 = 0?
t = b1/SE(b1) = 2.5/0.6 ≈ 4.17, with df = n - 2 = 18.
Using the regression above (b1=2.5, SE=0.6, n=20, df=18), what is the 95% confidence interval margin of error for the true slope?
With df = 18, t* ≈ 2.101. ME = t* x SE(b1) = 2.101 x 0.6 ≈ 1.26.
— End of 100 questions. Review any flagged explanations, then use the print button for a clean answer sheet. —
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