BIO 211: Statistics and Data Analysis

lectures 1-7 (quiz 1), Lectures 9-13 (quiz 2), Lectures 15-17( quiz 3), lecture 19 (quiz 4)

Science

A way of understanding the natural world by testing explanations

t-statistic for one sample test gets larger when

the difference between null and actual increases

null hypothesis of single sample t-test

actual mean is near the given mean

5 steps of single sample t-test

1. calculate sample mean

2. calculate standard deviation

3. calculate standard error of the mean

4. calculate t statistic

5. compare to t-distribution with degrees of freedom

3 types of t-tests

• compare data to a fixed number

• compare two groups of paired data

• compare two groups of unpaired data

z value’s 0.05

1.96

95% confidence interval

2 SE above and below the mean

applications of t

• confidence interval of the mean

• compare estimate of the mean to given value

• compare paired groups

• compare unpaired groups

does z or t reject the null more?

t

degrees of freedom is related to

sample size

the t distribution

• always centered at zero

• has a single parameter, degrees of freedom

• as degrees of freedom gets bigger, it gets closer to a normal distribution

z =

observation - mean /SD

whats the problem if we try to find the Z of the sampling distribution of the means?

the denominator, we dont know the standard deviation of the sampling distribution of the mean

how is the t distribution different

it has a bell shape but the tails are thicker, making it more likely for observations to fall beyond two standard deviations from the mean

t distribution is not

normal because it has thicker tails and the standard error is not constant and it depends on the sample, adding error to our estimate of t. however, as sample size increases SE gets better and t gets closer to z

thick tails of students t

helpful for resolving problem with a less reliable estimate of standard error when n is small

anatomy of a test statistic

point estimate - null value / SE of point estimate

SE =

s / sqrt n

students t has its own

t distribution

how do we fix the z equation for sampling distributions?

use students t = Y-u / SE

z score

the number of standard deviations it falls above or below the mean

observations more than 2 SD away from the mean

|z| > 2

A p value is not…

• the probability that the null hypothesis is true

• the probability that the alternative hypothesis is false

• not the probability of random chance

• does not indicate the magnitude of importance

Type 1 error

rejecting the null hypothesis when it is true, false positive

if we dont want a type 1 error

decrease a

statistical power

the probability that a random sample will lead to a rejection of the false null hypothesis (type 2 error); increases with sample size; complement of beta

fixing probability of type 1 error

• make alpha smaller, divide by number of tests, bonferroni, can be too restrictive

• accept a certain number of errors

The probability of making at lease 1 type 1 error in N tests

1- (1-a)^N

The probability of making no errors in N tests

(1-a)^N

problem with multiple tests

by chance, 5% of data will give you a false positive

type 2 error increases with

overlap between null and alt

if we dont want a type 2 error

increase a

area to the right of the critical value of the alt graph

reject null, when you are supposed to, or statistical power

if the test statistic is to the left of the critical value

the p value is larger than the alpha value

p value on graph

to the right of the test statistic

increasing a increases

error rate

p(type 1 error | H0 true) =

5%

to the left of the critical value on the alt hypothesis curve

the probability for a type 2 error or beta

when using a 5 % significance level there is a

5% change of making a type one error

we dont want to incorrectly reject the null more than…

5% of the time

Type 2 error

not rejecting the null hypothesis when we should have, false negative

normal distribution

natural, and artifically made, most widely used assumption in statistics; gaussian distribution

standard normal distribution

mean is 0 and sd is 1; x axis = z values; area under curve is probability; infinite tails

low p-value

unlikely, reject

high p-value

likely, accept

chi squared does not work with…

proportions

chi square distribution

distribution of the statistic under the null hypothesis

steps of chi squared

1. question, anything interesting in the data?

2. null hypothesis

3. find chi squared

4. is the test statistic grater than the critical value?

5. make conclusion

why do we square in chi square?

• negatives will become positive

• unusual differences will be exaggerated

df =

number of categories - 1

x² =

sum of (observed-expected)² / expected

Null hypothesis (H0)

you want to reject, what you would expect if the world was boring

hypothesis testing framework

• null hypothesis

• alternative hypothesis

• conduct hypothesis under assumption that null hypothesis is true

• use test results to determine if null hypothesis is true or not

P-value

probability of obtaining the observation or more extreme value if the null hypothesis is true, probability measured as area under the curve to the right of the chi squared

null distribution

sampling distribution of the test statistic assuming H0, describes the probability distribution of expected outcomes under H0

Alternative hypothesis (HA or H1)

the interesting hypothesis you want to support; cannot be proven by null hypothesis testing

How is null hypothesis testing used in science?

to see if we have sufficient evidence to support a scientific claim

The standard deviation of the sampling distribution for proportion is the same as…

the standard error of the estimated proportion

example of multinomial distribution

chances of something happen on days of the week

discrete data

usually finite countable numbers, but can be infinite

what plots are used for discrete probability distributions?

bar plots

What rule can we use in discrete probability distributions and why?

because the data is mutually exclusive, you can use the sum rule

probability density function of continuous probability distributions

a smooth curve

How to find the probability of something using a continuous distribution?

the proportion of data in the shaded area

since continuous probabilities are estimated as the area under the curve, the probability of an exact value is….

0

N(u, o)

normal with mean u and standard deviation o

not all bell shaped curves are

normal distributions

Pr [a<x<b]

integral between a and b; generalized sum rule

binomial distribution is…

a function over each possible outcome

probability of proportion calculated using..?

binomial probability formula

proportion =

successes / trials

binomial distribution depends on..

p and n

Pascals triangle

• sides are all ones

• second row diagonal: 1,2,3,4…

• each number is the sum of the ones on top

• 1’s are zeros

• n = 5, k = 3. row 5 column 3

simple binomial distribution formula

number of scenarios x probability of single scenario

k

number of successes

# of scenarios

(n/k) = n! / k! (n-k)!

0!

1

(n/k)

n! / k! (n-k!)

(n/n-1) =

n

n!

n x n-1 x n-2 …

1!

1

choose function

calculates the number of ways to choose k successes in n trials

P(single scenario)

p^k (1-p)^n-k

n

number of trials

if we selected four individuals to participate in the experiment, how would we find the probability of a certain number of individuals succeeding or failing?

Since they are independent joint (and) variables, multiply each persons probability together, and then multiply by the amount of ways it could possibly happen, aka which person does what

Bernoulli random variable

when an individual trial has only two outcomes

since 35% of people refused to administer a shock, the probability of SUCCESS is

0.35

failure milgrams

administers shock

Success milgrams

refuses to administer shock

Trials in Milgram’s experiment

each person

Milgram Experiment

• Stanley Milgram

• Experimenter orders teacher to shock the learner each time they answer a question incorrectly

• shocks are not real

• 65% of people would give the shocks

What did the Milgram experiment measure?

the willingness of participants to obey an authority figure who instructed them to perform acts that conflicted with their personal conscience

Sensitivity

High probability of true positives, fewer false negatives; Pr (+|has disease) = TP / TP + FN

Specificity

High probability of True negatives, fewer false positives; Pr(-|doesn’t have disease) = TN / TN + FP

Is it better to know if the test will be positive given you have the disease, or that you have the disease given the test is positive.

knowing if you have the disease given the test is positive.

Precision

Pr [has disease | positive result] = TP / TP + FP

False negative / Type 2 error

test says negative when truly positive

False positive / Type 1 error

tests says positive when really negative

True Negative

Test predicts negative when truly negative.