Flashcards: Statistics for Management — 88 cards

What is statistics as a way of reasoning and tools?

Statistics is a way of reasoning and a collection of tools to analyze data.

What are statistics in the plural?

Statistics in plural are results of calculations made with data.

What are data?

Data are collections of numbers, characters, images, or items providing information.

What does doing statistics involve before analysis?

Doing statistics involves thinking before the analysis.

How are results shown in doing statistics?

Results are shown with mathematics and charts.

What is the final step in doing statistics?

Telling others the results so they understand them correctly.

How many contextual questions do data require?

Data require six contextual questions.

What distinguishes descriptive from inferential statistics?

Descriptive statistics describe sample data; inferential statistics draw conclusions about a population.

How do quantitative data differ from qualitative data?

Quantitative data are measured by numbers with units, qualitative data identify categories.

When is a numerical variable considered quantitative?

When calculations with its values make sense.

What type of variable is an identifier variable?

A categorical variable with unique cases per category.

What is a simple random sample?

A sample where every size-n sample has equal selection chance.

Name common sampling techniques.

Systematic, convenience, cluster, and stratified sampling.

What are the four sources of data?

Published sources, designed experiments, surveys, and observational studies.

Which charts suit qualitative variables better, bar or histogram?

Bar charts and pie charts suit qualitative variables better.

What does a histogram use to represent frequency?

A bar over each equal-width class interval represents frequency.

What does a frequency table list for each category?

It lists the number of observations and may show percentages.

How are categories arranged in a Pareto diagram?

Categories are arranged by decreasing height from left to right.

What were the bin frequencies for [20–30) in the European teenage-smoking data?

The bin frequency for [20–30) was 15.

Which data type is better represented by stem-and-leaf plots?

Quantitative data are better represented by stem-and-leaf plots.

What intervals does a histogram partition quantitative values into?

Equal-width class intervals.

Which survey provided the 2007 European teenage-smoking data?

European School Survey Project on Alcohol and Other Drugs, 2007.

What is the formula for the sample mean?

$\bar{x}=\frac{\sum_{i=1}^{n}x_i}{n}$

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

It is the average of the two middle values in the ordered sequence.

What does the standard deviation measure in a data set?

It measures variation around the mean indicating data spread.

How is the range of a data set calculated?

By subtracting the smallest observation from the largest.

What is the interquartile range (IQR)?

The difference between the third and first quartiles, $Q_3 - Q_1$.

When are median and IQR preferred over mean and standard deviation?

When the distribution is not unimodal or symmetric or has outliers, gaps, or clusters.

What is the formula for the coefficient of variation (CV)?

$\mathrm{CV} = \frac{s}{\bar{x}}$ comparing standard deviation to the mean.

What does a sample z-score represent?

The number of sample standard deviations a value is from the sample mean.

What does a contingency table organize?

Counts for combinations of two categorical variables.

When are two categorical variables independent?

When one variable's distribution is the same across all categories of the other.

What is the formula for expected count under independence in a contingency table?

$E_{ij}=\frac{R_iC_j}{n}$ where $R_i$ is row total, $C_j$ is column total, and $n$ is sample size.

How is the chi-squared statistic calculated?

$\chi^2=\sum_i\sum_j\frac{(O_{ij}-E_{ij})^2}{E_{ij}}$ comparing observed and expected counts.

What does Cramer's V measure?

The strength of association between categorical variables standardized from chi-squared.

What is the range of Cramer's V?

From 0 for no association to 1 for perfect association.

What is Simpson's paradox?

A relationship seen within groups disappears when groups are combined by a lurking variable.

What does a scatterplot represent in data analysis?

Each case as a point (Xi,Yi) showing association between two quantitative variables.

What does the correlation coefficient r measure?

The direction and strength of a linear association between two quantitative variables.

Why does correlation not prove causation?

Because association may result from coincidence, causality, or a lurking variable.

What is the formula of a linear regression model?

$\hat{y}=b_0+b_1x$ where $b_0$ is y-intercept and $b_1$ is slope.

How is a residual calculated in regression analysis?

Residual equals observed response minus predicted response: $e=y-\hat{y}$.

What defines the least-squares regression line?

It uniquely minimizes the sum of squared residuals.

What is the difference between interpolation and extrapolation?

Interpolation predicts within the model's domain; extrapolation predicts outside it and needs caution.

What does the coefficient of determination $r^2$ represent?

The percentage of variation in Y explained by the linear regression model.

What is the formula for the coefficient of determination $r^2$?

$r^2=\frac{\text{variation accounted for by the model}}{\text{total variation}}$

What does $r^2=0.895$ mean in the manatee deaths example?

The model accounts for 89.5% of the variation in manatee deaths.

What residual plot features indicate a useful regression fit?

Roughly constant horizontal scatter, no bends, and very few outliers.

What does the standard error of regression measure?

How much the y-values vary around the fitted line.

What is another name for the standard error of regression?

The root mean squared error.

What is the standard error value in the manatee deaths example?

7.663 manatees killed.

According to the empirical rule, how close are 95% of manatee deaths to the regression line?

Within 15.3 manatees of the regression line.

What does the law of large numbers state about repeated random trials?

The proportion of times an event occurs approaches its empirical probability.

What is the formula for the empirical probability of an event A?

$P(A)=\frac{\#\text{ times A occurs}}{\#\text{ trials}}$

What is the complement rule formula for any event A?

$P(A^C)=1-P(A)$

What is the addition rule formula for disjoint events A and B?

$P(A\text{ or }B)=P(A)+P(B)$

What is the general addition rule formula for any two events A and B?

$P(A\cup B)=P(A)+P(B)-P(A\cap B)$

What is the formula for conditional probability of B given A, assuming $P(A)\ne0$?

$P(B\mid A)=\frac{P(A\cap B)}{P(A)}$

What is the general multiplication rule formula for any two events A and B?

$P(A\cap B)=P(A)P(B\mid A)=P(B)P(A\mid B)$

When are events A and B independent in terms of conditional probability and intersection?

When $P(B\mid A)=P(B)$ or equivalently $P(A\cap B)=P(A)P(B)$.

What does a random variable assign in a random event?

Numerical values according to the outcomes.

What distinguishes a discrete random variable from a continuous one?

Discrete has countable outcomes; continuous takes any value in a range.

What conditions must a probability model satisfy for all values?

Probabilities must be nonnegative and sum to 1.

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

$\mu=E(X)=\sum x\,p(x)$

What is the formula for the variance of a random variable?

$\sigma^2=\sum(x-\mu)^2p(x)$

How is the standard deviation of a random variable calculated?

As the square root of the variance, $\sigma=\sqrt{\sigma^2}$.

What outcomes do Bernoulli trials have?

Only success and failure outcomes.

What is the formula for combinations of r successes in n trials?

$\binom nr=\frac{n!}{r!(n-r)!}$

What is the expected value of a binomial random variable with n trials and success probability p?

$\mu = np$

What is the standard deviation of a binomial random variable with success probability p?

$\sigma = \sqrt{npq}$ where $q=1-p$

What does the Poisson distribution model?

The number of rare events in a measurement interval such as time, length, or area.

When is the Poisson distribution a special case of the binomial model?

When $n \to \infty$ and $p \to 0$.

In the typing-errors example, what is the Poisson mean errors per transaction?

0.34 errors per transaction.

How long is a transaction of 255 words in the typing-errors example?

3.4 minutes.

What does a probability density function represent for a continuous variable?

The probability that the variable falls within an interval by the area under its curve.

What are the characteristics of a Normal model distribution?

It is approximately unimodal and symmetric with mean μ and standard deviation σ.

How is a Normal model denoted with mean μ and standard deviation σ?

As N(μ,σ).

What does the 68-95-99.7 rule state about Normal-model values?

Approximately 68%, 95%, and 99.7% lie within one, two, and three standard deviations of the mean.

What formula standardizes a value to a z-score?

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

To which model does standardizing a value with a z-score convert it?

The standard Normal model N(0,1).

When is the Normal approximation to a binomial model appropriate?

When both $np\ge15$ and $nq\ge15$ are satisfied.

What does a population parameter describe?

An entire population.

What does a sample statistic describe?

A sample and is calculated from its observations.

What is a sampling distribution of a statistic?

The probability distribution of that statistic across all samples of size n.

What does the Central Limit Theorem state about sample means?

Their sampling distribution becomes approximately Normal as sample size increases.

What sample size is usually sufficient for the Central Limit Theorem to apply?

A sample size of n≥30, depending on the original distribution.

What was the mean weight reported for the sample of 200 college women with low BMI?

140 pounds.

What are the population parameters for all 18-year-old women in the example?

Mean μ=143.74 pounds and standard deviation σ=51.54 pounds.