prelim 2 - stsci 2150

significance level, a

a probability used as a criterion either reject or accept the null hypothesis

if the p value is less than the significance value

you reject the null hypothesis

if the p value is greater than the significance value

you accept the null hypothesis

What is a Poisson distribution?

A distribution that models the probability of a certain number of events happening in a fixed interval, given a known average rate and independence of events.

Example: Counting the number of customers arriving at a coffee shop per hour, given an average rate of arrivals.

When would you use a Uniform distribution?

Use it when every outcome within a range is equally likely.

Example: Rolling a fair die (each side has a 1/6 chance) or selecting a random number between 1 and 10, where each number is equally likely.

What is a Proportional distribution?

Describes situations where probabilities are assigned proportionally based on certain weights or factors, though not a strict probability distribution on its own.

Example: Allocating budget based on the size of each department (larger departments receive a proportionally higher share).

What is a Binomial distribution?

A distribution that models the number of successes in a fixed number of trials, each with the same probability of success and two possible outcomes (success or failure).

Example: Counting the number of heads in 10 coin flips, where each flip has a 50% chance of landing heads.

standard deviation

how much do observations vary

• measures the amoubt of variability in the sample data

• unbiased estimator

how much sample means would vary if we repeatedly drew samples and found the mean repeatedly

• standard deviation of the sampling distribution

standard error

p-value

the probability under the null hypothesis of obtaining data as extreme or more extreme than the observed data

type 1 error

rejecting a true null hypothesis

• false positive

basically saying that the null is false when in reality the null is true

type 2 error

not rejecting a false null hypothesis

• saying that the null is true when in reality it is false

confidence interval

provide a plausible range of values for the parameter. this is more information than one get from a hypothesis test

hypothesis test

only tell us whether one particular value is plausible for the parameter or not

dbinom()

gives the probability of a specific number of successes

pbinom()

gives the probability of a number of successes or fewer

contingency analysis

2 categorical variables

• whats the relationshop

• are they independent or not

• how strong is the relationship

indenpendent

no association

relative risk

ratio of probabilities fo teh event for two groups

pr(worse outcomes in group 1)/pr (worse outcome in group 2)

odds ratio

ratio of odds for the event for two groups

o1/o2 = odds1)/odds(event in group 2)

how do you interpret relative risk

the probability of _____ is ___ times greater in _____ than in _____

odds

probability of success/probability of failure = p/1-p

odds interpretation

the odds of disease after being exposed are 1.5 times greater than the odds of disease if you were not exposed

null value for odds ration

1

odds in one group are similar to odds in another group

degrees of freedom for contingency test

(row-1)(column-1)

test of single proportion

• test statistic is x, number of successes out of n trials

• null distribution s the binomial distribution

test of association

• test statistic is x² =

• null distribution is chi-squre distribution with df =

p value in r

1 - pchisq(test statistic = 20, df = 1)

critical value in r

qchisqu(.99, df=1)

interpret the null value for a 95% confidence interval

0

if the nterval has 0, it is. not significant

degree of freedom for proportional and uniform distributions

df = #categories -1

degree of freedom for poisson distribution

df = #categories - 2

poisson distribution in biology

successes occur randomly and independently in time or space

clumped events

excess observed values compared to expected in the high counts

dispersed events

excess of observed in middle categories

binomial has a

fixed upper bound

poisson has

• no success or failure paradigm

• no upper bound

uniform

pr(ith category) = 1/k with k categories

goodness of fit test qualities

• one categorical variable

• does the data fit the probability model

• expected based on probability model

test of association qualities

• two categorical variables

• is there an association between two variables

r for finding areas under the curve

• greater than

1 - pnorm (x, mean, standard deviation)

r for finding areas under the curve

• less than

pnorm (x, mean, standard deviation)

r for finding a specific number that falls into a percentile

qnorm( q, mean, standard deviation)

sketching sampling distribution

standard deviation = initial standard deviation/ sqr(n)