6-6 Normal as Approximation to Binomial
This section describes another interesting application of the normal distribution. In section 5-3, you learned how to find binomial probabilities. For instance, if a surgical procedure has an 85% chance of success a
Lecture #7 Chapter 7: Estimates and sample sizes In this chapter, we will learn an important technique of statistical inference to use sample statistics to estimate the value of an unknown population parameter. 7-2 Estimating a population proportion Recal
7.5 Estimating a Population Variance
We have seen how confidence intervals can be used to estimate the unknown value of a population mean or a proportion. We used the normal and student t distributions for developing these estimates. However, the variabil
8-6: Testing a claim about a standard deviation or variance In section 7.5 we used the chi-square distribution to help us construct confidence intervals about the population variance and standard deviation. Here, we will use the chi-square distribution to
The Poisson Probability Distribution
If In a binomial experiment you are interested in finding the probability of a specific number of occurrences takes place within a given specified interval (usually time or space), you can use the Poisson distribution.
BIOL 300: Biostatistics
Course web address: http:/www.zoology.ubc.ca/! ~whitlock/bio300/
Professor:! Dr. Michael Whitlock Professor Department of Zoology Office: 216 Biodiversity e-mail: [email protected]
Textbook Office hours: Mon. 1:30-3:00 and af
Sir Francis Galton The history of statistics has its roots in biology
Inventor of fingerprints, study of heredity of quantitative traits Regression & correlation
Karl Pearson
PolymathStudied genetics Correlation coefficient ! 2 test Standard deviation
Sir
Two common descriptions of data
! Location (or central tendency)
Describing data
! Width (or spread)
Measures of location
Mean Median Mode
Mean
Y=
!
n
Yi n
i= 1
n is the size of the sample!
Mean
Y1=56, Y2=72, Y3=18, Y4=42
Median
! The median is the middle
Sample size 10 from Normal distribution with =13 and ! 2=16
2
Sample size 10 from Normal distribution with =13 and ! 2=16
Frequency!
1.5
Frequency!
5 10 15 20 25
2
1.5
1
1
0.5
0.5
X = 13.5 s 2 = 12.1
5 10 15 20 25
X = 13.3 s 2 = 13.0
_
X
_
X
Sample size 1
Probability
The probability of an event is its true relative frequency, the proportion of times the event would occur if we repeated the same process over and over again.!
A and B are mutually exclusive
Two events are mutually exclusive if they cannot bot
Hypothesis testing
Hypothesis testing asks how unusual it is to get data that differHypothesisthe nullnutshell from testing in a hypothesis.
about this population, say, are If the data would be quite unlikely under H0, Population men and women the same he
Proportions
A proportion is the fraction of individuals having a particular attribute.
Example: 2092 adult passengers on the Titanic; 654 survived Proportion of survivors = 654/2092 ! 0.3
Probability that two out of three randomly chosen passengers surviv
Discrete distribution Fitting discrete distributions
A probability distribution describing a discrete numerical random variable For example, ! Number of heads from 10 flips of a coin ! Number of flowers in a square meter ! Number of disease outbreaks in a
Contingency analysis
! Test the independence of two or more categorical variables ! Well learn one kind: ! 2 contingency analysis
Music and wine buying
OBSERVED French music playing 40 German music playing 12 Totals
Bottles of French wine sold Bottles of
Normal distribution
0.4
The normal distribution is very common in nature
f ( x) =
1 2!
2
e
#
( x # )
2
2
2
0.3 0.2 0.1 -2 -1 0 1 2 3
Human body temperature Human birth weight Number of bristles on a Drosophila abdomen
Measurement
A normal distribution is
Inference about means
Because Y is normally distributed:
But. We dont know !
A good approximation to the standard normal is then:
Z=
Y ! Y ! = "Y "n
t=
Y! s/ n
t has a Students t distribution
Degrees of freedom
df = n - 1
Confidence interval for a mean
Y
Comparing means
! Tests with one categorical and one numerical variable ! Goal: to compare the mean of a numerical variable for different groups.
Paired vs. 2 sample comparisons
Paired comparisons allow us to account for a lot of extraneous variation 2-sa
Assumptions of t-tests
! Random sample(s) ! Populations are normally distributed ! (for 2-sample t) Populations have equal variances
Detecting deviations from normality
! Previous data/ theory ! Histograms ! Quantile plots ! Shapiro-Wilk test
Detecting de
Goals of experiments
! Eliminate bias ! Reduce sampling error (increase precision and power)
Design features that reduce bias
! Controls ! Random assignment to treatments ! Blinding
Controls
! A group which is identical to the experimental treatment in al
Analysis of variance (ANOVA) ANOVA
Comparing the means of more than two groups ! Like a t-test, but can compare more than two groups ! Asks whether any of two or more means is different from any other. ! In other words, is the variance among groups greate
Regression
! Predicts Y from X ! Linear regression assumes that the relationship between X and Y can be described by a line
Correlation vs. regression
Regression assumes.
! Random sample ! Y is normally distributed with equal variance for all values of X
Researcher and statistician error
Publication bias
Papers are more likely to be published if P<0.05 This causes a bias in the science reported in the literature.
~8% of biomedical papers have substantial statistical flaws
1
2
Computer-intensive methods
!
Descriptive Statistics
Sampling and Statistics
Statistics
We start the discussion in the natural way. We all have a general feeling about what statistics is. In the course of these lecture notes, we will lay out the detail about what statistics is and how
Estimation
Sample Proportions and Point Estimation
Sample Proportions
Let p be the proportion of successes of a sample from a population whose total proportion of successes is and let p be the mean of p and p be its standard deviation. Then
The Central Li
Hypothesis Testing
Hypothesis Testing
Whenever we have a decision to make about a population characteristic, we make a hypothesis. Some examples are:
>3
or
5.
5. Then we can think of our opponent suggesting
Suppose that we want to test the hypothesis tha
Lecture Notes For Most Math Classes Taught at Lake Tahoe Community College
http:/ltcconline.net/greenl/courses/LectureNotes.htm Example Course Syllabus (From Fall 2008) Index of Terms Videos on Projects and Examples
Descriptive Statistics Probability Dist
Math 201 Practice Final
Please work out each of the given problems. Credit will be based on the steps that you show towards the final answer. Show your work. Printable Key Problem 1 Match the following hypotheses and estimates with the appropriate test st
Math 201 Practice Midterm 3
Please work out each of the given problems. Credit will be based on the steps that you show towards the final answer. Show your work. Printable Key Problem 1 A study was done to determine whether LTCC transfer students had a lo
M ath 201 P ractice M idterm I
Please work out each of the given problems. Credit will be based on the steps that you show towards the final answer. Show your work. Printable Key Problem 1 Categorize these measurements associated with Lake Tahoe according