Statistical Analysis Methods Flashcards

Question 1

What is the difference between a population and a sample?

Accepted Answer

A population is the complete set of individuals or measurements that a study is interested in, while a sample is a subset of that population selected for observation. Researchers use samples because measuring every member is often too costly, slow, or impractical. Conclusions drawn from a well-chosen sample can be generalized to the population.

Question 2

What are the four main levels of measurement for variables?

Accepted Answer

The four levels are nominal, ordinal, interval, and ratio. Nominal data name categories with no inherent order, such as eye color. Ordinal data have a meaningful order but unknown gaps. Interval data have equal gaps but no true zero, such as Celsius temperature, while ratio data also have a true zero, such as height.

Question 3

How does the mean differ from the median as a measure of center?

Accepted Answer

The mean is the arithmetic average, found by summing all values and dividing by the count, so every value influences it. The median is the middle value when data are ordered, so it resists extreme values. For skewed data or outliers, the median often describes a typical value better than the mean does.

Question 4

What does standard deviation measure, and how does it differ from the range?

Accepted Answer

Standard deviation measures the typical distance of values from the mean, expressed in the same units as the data. The range is only the difference between the largest and smallest values, so it uses just two observations. Standard deviation uses every value, making it more informative about spread, though it remains sensitive to outliers.

Question 5

What is a histogram, and when is it useful?

Accepted Answer

A histogram is a bar chart that groups a numerical variable into adjacent intervals and shows how many observations fall in each interval. It is useful for seeing the shape of a distribution, including whether it is symmetric, skewed, unimodal, or has multiple peaks. Its bars touch because the intervals are continuous.

Question 6

What is a box plot, and what five numbers does it display?

Accepted Answer

A box plot summarizes the minimum, first quartile, median, third quartile, and maximum of a dataset. The box spans the middle half of the data, from the first to third quartile, and a line marks the median. Whiskers extend toward the extremes, and points beyond them are often plotted as outliers.

Question 7

What is an outlier, and why should analysts check for one before summarizing data?

Accepted Answer

An outlier is an observation that lies far from the bulk of the data. Analysts check for outliers because a single extreme value can distort the mean, inflate the standard deviation, and mislead conclusions. An outlier may reflect a recording error, a measurement fault, or a real rare case, so investigate it before removing it.

Question 8

What does the coefficient of variation express, and when is it helpful?

Accepted Answer

The coefficient of variation expresses the standard deviation as a percentage of the mean, giving a unitless measure of relative variability. It is helpful when comparing spread across datasets measured in different units or with very different averages. It is not meaningful when the mean is near zero.

Question 9

What is the difference between a sample space and an event?

Accepted Answer

The sample space is the set of all possible outcomes of a random experiment, while an event is any subset of that sample space. For example, when rolling a die, the sample space is one through six, and rolling an even number is an event. It occurs when the outcome lands in it.

Question 10

What does it mean for two events to be mutually exclusive?

Accepted Answer

Two events are mutually exclusive when they cannot both occur in the same trial, meaning their intersection is empty. For example, drawing a single card cannot produce both a heart and a spade. For mutually exclusive events, the probability that either occurs equals the sum of their individual probabilities.

Question 11

What is the complement rule, and how is it used?

Accepted Answer

The complement rule states that the probability of an event not occurring equals one minus the probability that it occurs. It is useful when the opposite event is easier to count. For example, the probability of at least one head in three coin flips equals one minus the probability of no heads.

Question 12

How does conditional probability differ from unconditional probability?

Accepted Answer

Conditional probability is the probability of an event given that another event has already occurred, written P(A given B). Unconditional probability considers the event without any additional information. Knowing that a condition holds can change the probability, so the two values are often different.

Question 13

What does it mean for two events to be independent?

Accepted Answer

Two events are independent when the occurrence of one does not change the probability of the other, so P(A and B) equals P(A) times P(B). Independence is a property of the probability model, not something to assume casually. Rolling two dice gives independent outcomes, while drawing cards without replacement does not.

Question 14

What is Bayes' theorem, and what does it allow you to do?

Accepted Answer

Bayes' theorem is a formula that relates a conditional probability to its reverse, so P(A given B) equals P(B given A) times P(A), divided by P(B). It lets you update a prior belief about an event after observing new evidence. It is widely used in diagnostic testing and spam filtering.

Question 15

What is a random variable, and how do discrete and continuous variables differ?

Accepted Answer

A random variable is a numerical quantity whose value depends on the outcome of a random process. A discrete random variable takes countable values, such as the number of defective items in a batch. A continuous random variable can take any value within an interval, such as the time until a machine fails.

Question 16

What do the expected value and variance of a random variable describe?

Accepted Answer

The expected value is the long-run average of a random variable over many repetitions, representing its center. The variance measures how spread out its values are around that expected value, as the average squared deviation. Standard deviation is the square root of the variance, returning the spread to the original units.

Question 17

Why does random sampling matter for drawing valid conclusions?

Accepted Answer

Random sampling gives every member of the population a known, nonzero chance of selection, which reduces selection bias. Under this design, the sample tends to resemble the population, so conclusions can be generalized with quantifiable uncertainty. Convenience samples, such as surveying whoever is easiest to reach, often produce systematically misleading results.

Question 18

What is a sampling distribution, and what does it describe?

Accepted Answer

A sampling distribution is the probability distribution of a sample statistic, such as the sample mean, across all possible samples of a fixed size from the same population. It describes how much the statistic would vary from sample to sample. Its spread is called the standard error, which shrinks as sample size grows.

Question 19

What is the central limit theorem, and why is it important?

Accepted Answer

The central limit theorem states that the distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's shape, provided the variance is finite. This is important because it justifies using normal-based methods for means even when the underlying data are not normal.

Question 20

What is a standard error, and how does it differ from a standard deviation?

Accepted Answer

The standard error is the standard deviation of a sampling distribution, usually of a sample statistic such as the mean. The standard deviation describes variability among individual observations in a single sample. The standard error describes how precisely the sample statistic estimates the population parameter, and it decreases as the sample size grows.

Question 21

What is a point estimate, and why is an interval estimate often reported alongside it?

Accepted Answer

A point estimate is a single value, such as a sample mean, used to approximate an unknown population parameter. Because any single value will differ from the truth, an interval estimate gives a range of plausible values. A confidence interval pairs the estimate with a margin of error, communicating the estimate and its uncertainty.

Question 22

How should you interpret a 95 percent confidence interval?

Accepted Answer

A 95 percent confidence interval comes from a procedure that, over many repeated random samples, would contain the true parameter about 95 percent of the time. It does not mean there is a 95 percent chance that the interval you computed contains the parameter, since the parameter is fixed. Wider intervals mean less precision.

Question 23

What is bias in an estimator, and what is a consistent estimator?

Accepted Answer

An estimator is biased when its expected value differs from the parameter it estimates, so it systematically over- or underestimates on average. The sample mean is unbiased for the population mean, while the sample variance divided by n is biased downward. An estimator is consistent when it converges to the true parameter as sample size grows.

Question 24

What is the margin of error, and what affects its size?

Accepted Answer

The margin of error is the maximum expected distance between a sample estimate and the population value, at a stated confidence level. It equals a critical value multiplied by the standard error. Larger samples and lower variability shrink the margin of error, while higher confidence levels widen it.

Question 25

What are the null and alternative hypotheses in a test?

Accepted Answer

The null hypothesis is the default claim, typically stating no effect or no difference, that the test assumes true until evidence argues against it. The alternative hypothesis states the claim the researcher wants to support, such as a difference or an effect in a specified direction. Hypotheses concern population parameters, not the sample data.

Question 26

What is a p-value, and how should it be interpreted?

Accepted Answer

A p-value is the probability of observing a result at least as extreme as the sample result, assuming the null hypothesis is true. A small p-value indicates that such data would be unlikely under the null hypothesis, which is evidence against it. It is not the probability that the null hypothesis is true.

Question 27

What is a Type I error, and how does it differ from a Type II error?

Accepted Answer

A Type I error occurs when a true null hypothesis is rejected, a false positive. A Type II error occurs when a false null hypothesis is not rejected, a false negative. The significance level alpha is the probability of a Type I error, while power is one minus the probability of a Type II error.

Question 28

How do you decide whether to reject the null hypothesis?

Accepted Answer

You compare the p-value to a predetermined significance level, usually written alpha. If the p-value is less than or equal to alpha, you reject the null hypothesis and report the result as statistically significant. Otherwise, you fail to reject it, which is not the same as proving it true.

Question 29

What is statistical power, and which factors increase it?

Accepted Answer

Statistical power is the probability that a test correctly rejects a false null hypothesis when a specified true effect exists. Power increases with a larger sample size, a larger true effect, a higher significance level, and lower variability in the data. Researchers often plan sample sizes to reach a desired power, such as 80 percent.

Question 30

What is the difference between one-tailed and two-tailed tests?

Accepted Answer

A two-tailed test checks for a difference in either direction, so the rejection region is split between both tails of the sampling distribution. A one-tailed test checks for a difference in one direction, concentrating the full rejection region in a single tail. The direction should be chosen before seeing the data, since choosing it afterward inflates false positives.

Question 31

Why might a result be statistically significant but practically unimportant?

Accepted Answer

With a very large sample, even a tiny difference can produce a small p-value, so the result is statistically significant. However, the actual effect may be too small to matter in practice. Reporting an effect size and a confidence interval alongside the p-value helps readers judge practical importance.

Question 32

What is a t test used for, and when is it appropriate?

Accepted Answer

A t test assesses whether a sample mean differs from a hypothesized value, or whether two group means differ, when the population standard deviation is unknown and estimated from the sample. It relies on the t distribution, which has heavier tails than the normal distribution for small samples.

Question 33

What does the Pearson correlation coefficient measure, and what range does it take?

Accepted Answer

The Pearson correlation coefficient measures the strength and direction of a linear relationship between two quantitative variables. It ranges from negative one to positive one, where values near the extremes indicate strong linear association and zero indicates no linear association. It does not establish causation, and it can miss nonlinear patterns.

Question 34

What is the difference between correlation and causation?

Accepted Answer

Correlation describes an association in which two variables tend to move together, while causation means that changing one variable actually produces a change in the other. Correlation can arise from a confounding variable, coincidence, or reverse causation. Establishing causation typically requires controlled experiments or careful study design beyond a correlation coefficient alone.

Question 35

What does a simple linear regression model estimate?

Accepted Answer

A simple linear regression model estimates a straight-line relationship between one predictor variable and one outcome variable, written as an intercept plus a slope times the predictor. The slope gives the average change in the outcome for a one-unit increase in the predictor. The intercept gives the predicted outcome when the predictor equals zero.

Question 36

What is the least squares method, and what does it minimize?

Accepted Answer

The least squares method chooses the regression line that minimizes the sum of squared vertical distances between observed outcomes and the values predicted by the line. Squaring prevents positive and negative deviations from canceling and penalizes large errors more heavily. Under standard assumptions, this approach yields estimates with desirable statistical properties.

Question 37

What is R squared, and what does it tell you about a regression model?

Accepted Answer

R squared is the proportion of outcome variation that the model explains, ranging from zero to one. A value of 0.70 means that about 70 percent of the variability in the outcome is accounted for by the predictors. A high value does not prove the model is correct or that predictors cause the outcome.

Question 38

What are the main assumptions behind ordinary least squares regression?

Accepted Answer

The main assumptions are linearity between predictors and the mean outcome, independent errors, constant error variance across fitted values, and approximately normal errors for valid inference. Violations can bias estimates or distort standard errors and p-values. Residual plots are a common tool for checking these assumptions.

Question 39

What does multiple regression add compared with simple linear regression?

Accepted Answer

Multiple regression includes two or more predictors in one model, so each coefficient estimates the association between that predictor and the outcome while holding the other predictors constant. This helps separate overlapping influences and reduces confounding by measured variables. Interpreting coefficients still requires care, since predictors can be correlated with one another.

Question 40

What is overfitting, and how can it be detected in a model?

Accepted Answer

Overfitting occurs when a model captures random noise in the training data rather than the underlying pattern, so it predicts new data poorly. It can be detected when performance is better on the training data than on held-out data. Comparing results on a separate validation or test set, or using cross-validation, helps reveal it.

Statistical Analysis Methods Flashcards

1Describing Data and Variables

2Probability Foundations and Rules

3Sampling and Estimation Concepts

4Hypothesis Testing Fundamentals

5Correlation and Regression Modeling

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