Statistics Knowledge Bank

1.1-Sampling

You draw a sample from a population to infer abundance of some trait amoung the pop.

Random sampling is best:

simple random sampling: a subset of $n$ members from a population.

1. each of the $n$ members is chosen from pop randomly

   -each of the $n$ members has equal probability of being chosen to be in subset.

2. each subset of $n$ members is equally likely to be chosen to be the sample

Example: Lottery, random # generator

Sample of convenience: Removing heavy blocks from the top of a pile for testing, rather than throughout the pile.

sample variation:

• -different samples vary
• -a sample doesn't perfectly reflect the population.

independent: items in sample are independnt if knowing their value does not help predict that of others.

• if sample <5% total pop (for cases of non-replacement)

1.2 - Summary Statistics

$X_1, X_2, ..., X_n$ is a sample.

Sample standard deviation: $s=\sqrt{s^2}$

*Purpose is to estimate spread of population about pop mean.

-so ideally computer deviation of all pop members

-dev abt $\bar{X} <$ dev abt $\mu$, dividing by $n-1$, instead of $n$ provides appropriate correction.

-median is often used in presence of outliers.

SE mean: Standard error of the mean: $\frac{s}{\sqrt{n}}$, $n=$ sample size.

frequency: # of times category type appears in sample.

sample proportion (relative freq): $\frac{\text{freq}}{n}$

Summaries like all of the above are called statistics when done on a sample. Are called population parameters when on an entire pop.

Histograms

subranges containing data are called class intervals

rel frequency: $\frac{\text{freq}}{n}$

density = $\frac{\text{rel freq}}{class width}$

for unequal width, the y-axis must be density.

Skew:

mean~center of mass of the histogram

Probability

experiment: process whose outcome cannot certainly be predicted.

example: coin toss, die roll, measure bolt diameter

-weight cereal box contents.

sample space: set of all possible outcomes of experiment.

Example:

• coin toss {head, tails}
• die roll: {1,2,3,4,5,6}
• hole punch in metal: hole punch is 10 mm diameter but variations in metal and angle cause the hole diameter to vary between (10, 10.2). $\{x|10.0<x<10.2\}$

!! Choice of sample space:

Example:

• item length should be 5 but varies between 4 and 6. Sample space 1) $\{x|4<x<6\}$
• if you only want to know whether it is good or not, consider: {too long, too short, just fine}

event: subset of a sample space.

Example:

• die roll has sample space {1,2,3,4,5,6}; evens has subset {2,4,6}.
• hole punch (above) with holes <10.1 has subset $\{x|10.0<x<10.1\}$

An event has happened if the experiment outcome is within the event's subset.

Empty set 0 and the entire sample space are events for all sample spaces.

Combine simple events to create more complex events:

• $A\cup B$ is the set of outcomes for which $A$ or $B$, or both $A$ and $B$ together, have ocurred. Is read as Union. Can be thought of as "or". Example: The event $A\cup B$ has happened if $A$ or $B$ has happened, or both.

  

• $A\cap B$ is the set of outcomes which belong to both $A$ and $B$. Is read as intersection. Can be thought of as "and". Example: The event $A\cap B$ has happened if both $A$ and $B$ has happened.

• $A^c$ complement of A is the set of outcomes which are not within $A$. So it is everything that isn't $A$.

• Mutually exclusive events are events which cannot happen together.

  Example: coin toss coming up heads or tails/

  

  In general, events $A_1,...A_n$ are mutually exclusive if no 2 have the events in common (no overlap).

Probability

• measure how likely an event is to ocurr.

• proportion of times an event would occur over many runs of the experiment.

• These are denoted, for some event $A$, as $P(A)$.