Flashcards: Statistics Foundations and Probability — 94 cards

What is Statistics as a scientific method?

It is used to collect, present, analyze, interpret data, and draw conclusions.

What does theoretical statistics develop and prove?

Statistical theorems, methods, formulas, rules, and laws.

How does applied statistics use statistical methods?

It uses them to solve real problems.

What is the population in statistics?

The set of all observation units studied for their characteristics.

What is a sample in statistics?

A selected part of the population for analysis.

What distinguishes a census from a sample survey?

A census surveys every population element; a sample survey only part.

What defines a random sample?

Each population element has a known nonzero chance of selection.

What defines a non-random sample?

Some population elements have no chance of selection.

What defines a variable in data analysis?

A characteristic studied across units that can take different values.

How does a quantitative variable differ from a qualitative variable?

A quantitative variable is expressed numerically, a qualitative variable is categorized without numbers.

What distinguishes a discrete variable from a continuous variable?

A discrete variable has countable isolated values, a continuous variable takes any value within intervals.

What is the key feature of nominal measurement?

Numbers serve only as category labels without order or magnitude.

What does a ratio scale's zero represent compared to an interval scale's zero?

A ratio scale zero indicates absence of the phenomenon; an interval scale zero does not.

Which scale does weight use and which scale does temperature use?

Weight uses a ratio scale; temperature uses an interval scale.

What is the formula for relative frequency of a category?

$f_r=\frac{f}{\sum f}$

According to Sturges' rule, how is the number of grouped intervals estimated?

$C=1+3.3\log N$ where $N$ is the number of observations.

What is the arithmetic mean for a population denoted by?

It is denoted by μ.

How is the median defined for an even number of observations?

It is the arithmetic mean of the two middle values.

What does the mode represent in a data series?

The value occurring with the greatest frequency.

Can a data series have more than one mode?

Yes, it can have more than one mode or no mode.

What does a smaller standard deviation indicate about data dispersion?

It indicates less dispersion around the mean.

What is the formula for the coefficient of variation for a population?

$CV=\frac{\sigma}{\mu}\times100$

How is the coefficient of variation calculated for a sample?

$CV=\frac{s}{\bar{x}}\times100$

How is the interquartile range calculated?

It is the difference between the third and first quartiles: $IQR=Q_3-Q_1$.

What is the sample space in an experiment?

The set of all possible outcomes of the experiment.

What is the formula for the probability of an elementary outcome when all outcomes are equally likely?

$P(E_i)=\frac{1}{\text{total number of outcomes}}$

How is the probability of an event A calculated when all outcomes are equally likely?

$P(A)=\frac{\text{number of favorable outcomes for }A}{\text{total number of outcomes}}$

How does relative-frequency probability estimate event probability?

As $P(A)=\frac{f}{n}$ where f is realizations of A and n is trials.

What happens to the relative-frequency probability estimate as the sample size grows?

The estimate becomes more stable.

What defines subjective probability?

Probability assigned from individual opinion, experience, information, and belief.

What are the four conditions of a binomial experiment?

n identical trials, two outcomes per trial, constant probabilities, independent trials.

What is a random variable?

A variable determined by the outcome of a random experiment.

What distinguishes a discrete random variable from a continuous one?

Discrete variables take countably many values; continuous variables take any value in intervals.

What conditions must a discrete probability distribution satisfy?

Each probability is between 0 and 1, and all probabilities sum to 1.

What is the formula for the expected value of a discrete random variable?

$E(X)=\mu=\sum xP(x)$

How is the standard deviation of a discrete random variable calculated?

$\sigma=\sqrt{\sum x^2P(x)-\mu^2}$

What defines the shape and parameters of the normal distribution?

It is symmetric and bell-shaped with mean μ as center and standard deviation σ as dispersion.

What is the standardized normal distribution?

A normal distribution with mean 0 and standard deviation 1.

What are standardized values in the standardized normal distribution called?

They are called z-values or z-scores.

What percentage of area lies within one standard deviation in a normal distribution?

68.26% of the total area lies within one standard deviation.

What is the mean of the standardized normal distribution?

The mean is 0.

What is the standard deviation of the standardized normal distribution?

The standard deviation is 1.

What symbol denotes the random variable of the standardized normal distribution?

The random variable is denoted by Z.

How is the probability $P(Z \leq a)$ expressed using the cumulative distribution function F?

$P(Z \leq a) = F(a)$.

How is the probability $P(Z \geq a)$ expressed using the cumulative distribution function F?

$P(Z \geq a) = 1 - F(a)$.

How is the probability $P(a \leq Z \leq b)$ expressed using the cumulative distribution function F?

$P(a \leq Z \leq b) = F(b) - F(a)$.

What formula standardizes a normally distributed value x?

$z = \frac{x - \mu}{\sigma}$ standardizes x.

What is a sampling distribution?

It is the probability distribution of a sample statistic with its possible values and probabilities.

What causes the difference between a sample statistic and population parameter?

Random error when the sample is random and no nonsampling error occurred.

What is the expected value of the sampling distribution of the sample mean?

It equals the population mean, $\mu_{\bar X}=\mu$.

What is the standard error formula for the sample mean when sample ≤5% of population?

$\sigma_{\bar X}=\frac{\sigma}{\sqrt n}$.

When sample fraction exceeds 5%, how is the sample mean's standard error adjusted?

By using the finite population correction: $\sigma_{\bar X}=\frac{\sigma}{\sqrt n}\sqrt{\frac{N-n}{N-1}}$.

What does the central limit theorem say about sample means from large samples?

They are approximately normally distributed regardless of population distribution.

When is a sample generally considered large for the central limit theorem?

When $n\geq30$.

What is the standard error formula for a sample proportion when $n/N \leq 0.05$?

$\sigma_P=\sqrt{\frac{pq}{n}}$ where $q=1-p$.

What does statistical estimation assign to a population parameter?

Numerical value or values using sample information.

What is the first step in the estimation procedure?

Selecting a sample.

What distinguishes a point estimate from an interval estimate?

A point estimate is a single value; an interval estimate is a range.

How is a confidence interval constructed?

As point estimate plus or minus margin of error.

What are the common confidence levels and how is confidence level denoted?

90%, 95%, 99%; confidence level is 1−α, α is significance level.

What is the formula for a confidence interval when population standard deviation is known?

$\bar x\pm z\frac{\sigma}{\sqrt n}$.

What is the formula for a confidence interval when population standard deviation is unknown?

$\bar x\pm t\frac{s}{\sqrt n}$ with n−1 degrees of freedom.

What is the null hypothesis H₀ in hypothesis testing?

A statement about a population parameter treated as true until rejected.

When does a Type I error occur in hypothesis testing?

When the null hypothesis is true but incorrectly rejected.

What is the probability of a Type I error called?

Alpha (α).

When does a Type II error occur in hypothesis testing?

When a false null hypothesis is not rejected.

What is the probability of a Type II error called?

Beta (β).

How is test power defined in relation to Type II error?

Test power is 1 minus beta (1−β).

What happens to β when α decreases with fixed sample size?

Beta (β) increases.

Where are rejection regions located in a two-tailed test?

At both ends of the distribution.

What are the main steps in the critical-value approach to hypothesis testing?

Formulate H₀ and H₁, choose sampling distribution, determine rejection regions, calculate test statistic, make decision.

What is the formula for the test statistic when testing a population mean with a normal distribution?

$z=\frac{\bar x-\mu}{\sigma_{\bar X}}$

How is $\sigma_{\bar X}$ calculated in the test statistic formula for a population mean?

$\sigma_{\bar X}=\frac{\sigma}{\sqrt n}$

What does the p-value represent under the assumption that H₀ is true?

The probability the sample statistic deviates from the hypothesized parameter toward H₁ at least as much as observed.

When using the p-value approach, when is H₀ rejected?

When the p-value is less than α.

When using the p-value approach, when is H₀ not rejected?

When the p-value is greater than or equal to α.

What is the formula for required sample size when estimating a population mean with known σ and margin of error E?

$n=\frac{z^2\sigma^2}{E^2}$

What is the formula for required sample size when estimating a population proportion with margin of error E?

$n=\frac{z^2pq}{E^2}$

What are the four stages of the p-value testing procedure?

Formulate hypotheses, choose distribution, calculate p-value, make decision.

When is the normal distribution used for estimating a population mean with known standard deviation?

When the sample is small and population normal, or sample large with n ≥ 30.

What is the formula for a confidence interval for a population mean in normal cases?

$\bar{x} \pm z\sigma_{\bar{x}}$

How is the margin of error expressed in a confidence interval for a population mean?

$E = z\sigma_{\bar{x}}$

What is the critical value z for a 95% confidence interval?

z = 1.96

When is the t distribution used for a population mean?

When population standard deviation is unknown and population is normal for small samples.

What is the formula for the z statistic for difference of means with known standard deviations?

$Z=\frac{\bar{X}_1-\bar{X}_2-(\mu_1-\mu_2)}{\sigma_{\bar{X}_1-\bar{X}_2}}$

How is $\sigma_{\bar{X}_1-\bar{X}_2}$ calculated for two independent samples?

$\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}$

What does the goodness-of-fit test assess?

Whether observed frequencies follow a specified theoretical distribution.

What defines a multinomial experiment?

n identical trials with k>2 outcomes, independent trials, and constant outcome probabilities.

What is the formula for expected frequency in goodness-of-fit tests?

$E=np$

How are degrees of freedom calculated in goodness-of-fit tests?

$df=k-1$

What is the formula for the goodness-of-fit chi-square statistic?

$\chi^2=\sum\frac{(O-E)^2}{E}$

What are the conditions for the goodness-of-fit chi-square test?

It is right-tailed and requires expected frequencies of at least 5.

How are degrees of freedom calculated for a contingency table independence test?

$df=(R-1)(K-1)$

How is the expected frequency in a contingency table cell calculated?

$E=\frac{(\text{row total})(\text{column total})}{\text{sample size}}$