Biostatistics: Exponential and Inference Flashcards (96 cards)

What methods does biostatistics use to study health data?

It collects, describes, analyzes, and interprets health data while accounting for variability.

What does descriptive statistics summarize, and what does inference generalize to?

Descriptive statistics summarizes observed data; inference generalizes to a population using probability.

What is a population, and what is each of its elements called?

It is the set studied; each element is a statistical unit, or individual.

How can random sampling affect selection bias?

It limits bias caused by choosing the units.

What are the possible values of a statistical variable called?

Its modalities.

What distinguishes nominal qualitative data from ordinal qualitative data?

Nominal categories have no order; ordinal categories are ordered.

How is the relative frequency of a category with count $n_i$ in a dataset of size $n$ calculated?

$f_i = n_i/n$.

What is the cumulative frequency through ordered category $x_i$?

$F_i = N_i/n$, where $N_i$ counts observations with values at most $x_i$.

How many contiguous classes are typically chosen when grouping a continuous variable?

From 5 to 10 classes.

How is the center of a class with endpoints $e_{i-1}$ and $e_i$ calculated?

$c_i = (e_{i-1}+e_i)/2$.

What must histogram bar height represent when class widths differ?

The frequency density.

What is the mode of a dataset?

The value or category with the greatest frequency.

What does the median do to an ordered series?

It divides the series so that at least half the values are at or below it and at least half are at or above it.

How is the mean calculated from frequencies and observed values?

$\bar{x}=\frac{1}{n}\sum_i n_i x_i=\sum_i f_i x_i$.

In a right-skewed distribution, how do the mean and median usually compare?

The mean is usually greater than the median.

How is the interquartile range calculated?

$Q_3-Q_1$.

What portion of the data does the interquartile range span?

The central half of the data.

What is the variance of a data series?

$V=\frac{1}{n}\sum_i n_i(x_i-\bar{x})^2=\frac{1}{n}\sum_i n_i x_i^2-\bar{x}^2$.

What does the coefficient of variation compare across different characters?

Their dispersions, even when their units or levels differ.

For successive choices with $n_1,\ldots,n_k$ options per step, how many total choices are there?

$n_1\times\cdots\times n_k$

For disjoint families with sizes $m_1,\ldots,m_k$, how many cases are there in total?

$m_1+\cdots+m_k$

What is the factorial of an integer $n\geq1$?

$n!=n(n-1)\cdots2\times1$

How many orders can arrange $n$ distinct objects in a line?

$n!$ orders.

What does a permutation without repetition arrange?

All $n$ distinct objects.

What is the formula for permutations of $n$ distinct objects?

$P_n=A_n^n=n!$

What does a permutation with repetition arrange?

Objects divided into groups of identical objects.

What is the formula for permutations with group sizes $n_1,\ldots,n_k$?

$\frac{n!}{n_1!n_2!\cdots n_k!}$

What does a combination without replacement of p elements from n represent?

An unordered subset of p distinct elements chosen from n, where $0\le p\le n$.

What is the formula for the number of combinations without replacement?

$C_n^p=\binom{n}{p}=\frac{n!}{p!(n-p)!}=\frac{A_n^p}{p!}$.

What does a combination with repetition of p objects from n types represent?

An unordered choice where a type may be selected multiple times, and only each type’s count matters.

What is the formula for the number of combinations with repetition?

$\binom{n+p-1}{p}=C_{n+p-1}^p$.

What symmetry identity relates the combination coefficients for choosing p and n−p items?

$\binom{n}{p}=\binom{n}{n-p}$

What is Pascal’s relation for $1\le p\le n-1$?

$\binom{n}{p}=\binom{n-1}{p-1}+\binom{n-1}{p}$

What is the binomial theorem for real a and b and integer $n\ge1$?

$(a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^k b^{n-k}$

How should a compound counting problem be split before choosing a formula?

Into successive stages or disjoint cases.

What formula counts a simultaneous selection?

Combinations.

What formula counts ordered draws without replacement?

Arrangements without repetition.

What formula counts ordered draws with replacement?

Arrangements with repetition.

What does a discrete random variable assign to each experiment outcome?

A real number.

What does a discrete variable’s probability law list for each possible value?

Its probability, $p_i=P(X=x_i)$.

What is the expected-value formula for a discrete variable?

$E(X)=\sum_i x_i p_i$.

In a Bernoulli trial, what value does the indicator take for success?

$1$.

What is the probability formula for exactly $k$ successes in a binomial law?

$P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$.

What is the variance of $X\sim B(n;p)$?

$V(X)=np(1-p)$.

What probability mass function defines a Poisson variable with parameter $\lambda$?

$P(X=k)=e^{-\lambda}\frac{\lambda^k}{k!}$.

What is the probability mass function of a geometric variable?

$P(X=k)=(1-p)^{k-1}p$.

What probability mass function defines the hypergeometric carrier count?

$P(X=k)=\frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}$.

What is the variance of a hypergeometric variable with $p=K/N$?

$V(X)=np(1-p)\frac{N-n}{N-1}$.

How are probabilities represented for a continuous random variable?

By areas under its density.

What is the probability of any exact value for a continuous variable?

Zero.

What two conditions must a probability density satisfy?

It is nonnegative and has total area 1.

How is an interval probability calculated from a density?

By integrating the density over the interval.

How is the continuous cumulative distribution function defined?

$F(x)=P(X\leq x)=\int_{-\infty}^{x}f(t)\,dt$

What is the constant density of a variable uniform on $[a,b]$?

$f(x)=\frac{1}{b-a}$ on the interval.

What data issues should be checked before calculations begin?

Impossible values, duplicates, and missing data.

Does rounding a continuous measurement make its variable discrete?

No, the variable remains continuous.

Why are cumulative frequencies unsuitable for nominal variables?

Their categories have no natural order.

What percentage of 200 adults were in the normal BMI category?

56%.

How is a sector angle calculated from category frequency $f_i$?

$\alpha_i = 360^\circ \times f_i$.

How is the median rank found when a raw ordered series has an odd number of values?

The median is the value at rank $(n+1)/2$.

What formula gives the interpolated median in grouped data?

$Me=e_{i-1}+a_i\frac{0.5-F_{i-1}}{F_i-F_{i-1}}$, in the class where cumulative frequency crosses 0.5.

Which measure of central tendency suits nominal characteristics?

The mode.

How is the mean of two combined groups calculated?

$\bar{x}=\frac{n_1\bar{x}_1+n_2\bar{x}_2}{n_1+n_2}$

How is the range of a data series calculated?

$x_{\max}-x_{\min}$.

Which values determine the range of a data series?

Only the two extreme values.

For ordered raw data, what is the rank of $Q_1$?

$\lceil n/4\rceil$.

When is the variance of a data series zero?

Only when all values in the series are equal.

With equally likely outcomes, how is probability calculated by counting cases?

Favorable cases divided by possible cases.

How many blood groups result from four ABO types and two Rh types?

$4\times2=8$ blood groups.

How is the size of the union of two overlapping sets calculated?

$\operatorname{card}(A\cup B)=\operatorname{card}(A)+\operatorname{card}(B)-\operatorname{card}(A\cap B)$

How many orders result from arranging three distinct patients in a line?

$3!=6$ orders.

How many orders remain for five patients when the most urgent goes first?

$4!=24$ orders.

How many orders are possible when a mother and child stay consecutive among five patients?

$2\times4!=48$ orders.

How many distinct sequences can be formed from two A, two T, one C, and one G?

180 sequences.

How many ways can nine specified medicine boxes be shelved?

1,260 ways.

How are combinations of p elements from n written and read?

$C_n^p$ or $\binom{n}{p}$, read as “p among n.”

What are the boundary values of $C_n^p$?

$C_n^0=C_n^n=1$ and $C_n^1=n$.

How many unordered groups of three can be chosen from ten people?

120 groups.

How does the separator model represent combinations with repetition?

Choose the p object positions among the p object-symbols and n−1 separators.

How are p-element subsets divided when proving Pascal’s relation?

Into subsets containing a fixed element and subsets not containing it.

What does position p in row n of Pascal’s triangle represent?

$\binom{n}{p}$

What is the binomial expansion of $(a+b)^3$?

$a^3+3a^2b+3ab^2+b^3$

What sum results from setting a=b=1 in the binomial theorem?

$\sum_{k=0}^{n}\binom{n}{k}=2^n$

How many subsets does a set with $n$ elements have?

$2^n$ subsets.

What formula arranges all distinct objects?

$n!$

How do you count an event requiring at least one occurrence?

Subtract the count with none from the total.

Which principle counts successive choices in a compound problem?

The product principle.

Which principle counts disjoint cases in a compound problem?

The sum principle.

For two children of AS × AS parents, what is the probability exactly one is SS?

$P(X=1)=\frac{6}{16}$.

How does adding a constant affect the expectation of $aX+b$?

$E(aX+b)=aE(X)+b$.

What conditions characterize events modeled by a Poisson law?

Events occur independently, one at a time, at a constant average rate.

When can a binomial law be approximated by a Poisson law?

When $n\geq30$ and $p\leq0.1$, using parameter $np$.

What is the geometric law’s probability that $X>k$?

$P(X>k)=(1-p)^k$.

How does without-replacement sampling affect variance compared with replacement?

It has less variance at the same $n$ and $p$.

What is the expected value formula for a continuous variable?

$E(X)=\int x f(x)\,dx$

What is the variance of a uniform variable on $[a,b]$?

$\frac{(b-a)^2}{12}$