Stat 151
what is a census?
special sample that includes everyone in the entire population
what are the problems with a census (3)
too expensive, undercoverage, time consuming
what is a population parameter, how do can they get it? what do they use instead to estimate the population parameter
numerical summary of a population , with a census but they use sample statisitcs instead
there are two inferences made from sample statistics how do they differ?
population inference: results from the sample can be generalized to an entrie population (as estimate)
causal inference: the difference in the responses is caused by the difference in treatments when comparing the results from two treatment groups
when only should we make pop inf's
when we have random sampling
what does randomizing help with
eliminates the effect of unknown extraneous factors, makes sure that on avg the sample looks like rest of the pop
non random sampling leads to
biased results
what are the 4 random sampling methods
1. simple random samples
2. stratified random sampling
3. systematic random sampling
4. cluster random sampling
how does a simple random sample work
each sample of size n in the pop has the same chance of being selected
what do we call sample-to- sample differences
sampling variability
how does stratified random sampling work? what does it reduce?
population is divided into homogenous groups called strata then take an SRS within each stratum before the results are combined. can reduce bias and variabilty of results
how does systematic random sampling work
starts with a random indivudal then sample every x^th person, make sure the order of the list is not associated with the responses sought
how does cluster random sampling work
split pop into clusters (similar groups), then select 1 or a few clusters at random and perform a census
bias means
the tendency for a sample to differ from the corresponding population in some systematic way
what are the 4 sources of bias?
1) selection bias (undercoverage)
2) response bias
3) voluntary response bias
4) nonresponse bias
selection bias (undercoverage) is when
some portion of the population is not sampled at all or has a smaller representation in the sample than it has in the population (usually these poeple differ from the rest of the pop)
response bias means
anything in the survery design that influences the responses (ex. being asked about illegal or unpopular behavior)
voluntary response bias
when indivudals can choose on their own wheter to participate in the sample
nonresponse bias
when a large proportion of those sampled fail to respond
when should we make causal (cause and effect) inference
we should only make causal inf when we have random allocation, when there is no random allocation the difference in responses could be caused by lurking variables
what are lurking variables
variables that are related to both group memberships and to the responses
what are two types of study designs
1) observational studies
2) randomized experiment
an observational study is when , what are the two types
the investigator observes indiviudals and measures variables of interest but does NOT attempt to influence the response, good for trends and possible relationships
1) retrospective
2) prospective
retrospective - observational study
identify subjects and collect data at that moment in time of caused effects
prospective- observational study
identify subjects in advance and collect data as events unfolded over the next period of time (future)
a randomized comparative experiment
allows us to prove a cause and effect relationship by doing the following
a) manipulates factor levels to create treatments
b) randomly assigns subjects to these treatment levels
c) compares the responses of the subject groups across treatment levels
can we make causal inf from observational studies
no
random allocation allows for what inf
causal inf are allowed
random selection of individuals allows for what inferences
population inf are allowed
categorical variables get what type of graph
bar chart and pie chart
numerical variables get what type of graphs
dot plots, stem plots, histograms, time blots, box plots, scatterplots
after categorical data has been sample it should be summarized to provide what following information
what values have been observed? how often did every value occur?
distribution of acategorical variable is given in form a table poviding what following info
each possible category, frequency of individuals who fall into each category or relative frequency of individuals who fall into each category
how do frequency and relative frequency differ
freuqency- number, relative frequency- percentage
how to calculate relative frequency
frequency/ number of observations
what should a relative freuqency table add up to
100%
what will the bar chart effectively show
the frequencies or percent in different categories
what will the pie chart effectively show
the relationship between parts and the whole
the shape of frequency bar chart and a realtive freq bar chart will be the same, true or false?
true
how to calculate slice size for a pie chart
category relative freq X 360 degrees
how can we explore the relationship between TWO categorical values
contingency table
what is a margin in a contingency table
the margins of the table give totals and the frequency distributions for each of the variables
each frequency distribution in a contingency table is called what
a marginal distribution of its respective variable
how to calculate marginal distribution
total for the variable/ total of the sample
what is a conditional distribution
shows the distribution of one variable for just the observations that satisy a condition on another variable. for example instead of looking at the total of two variables you only look at one variable ex. instead of arts AND science students in total for full time and part time you look at art and science alone
what does a conditional distribution tell us
if two variables are dependent because theres a difference in their distributions between the variables
dependent variables are
if the conditional distribution of one variable is not the same for each category for another, there is an association between these variables
indepedent variables are
if the conditonal distribution of one variable is the same for each category of another, no association between these variables
n is ?
sample size, number of observations of the variable y
variable y is
the variable of interest, what we have sample data of
y1 vs y2
y1 is the first sample observation of the variable y where as y2 is the second sample observation of the variable y
the three most common measures for centre are
the mean (center), the median and the mode
y bar is what
sample mean
how to calculate sample mean
sum of all observations/ number of observations
when there is outliers is the mean a good measure of centre
no because it is not resistant to outliers
what is the alternative value to measure the centre when there is outliers
median
what is the median (M)
the value that divides the ordered sample into two sets of the same size, one hald below M and the other half above M
how to find the median, if n is odd? if n is even?
order data, smallest to largest. If n is odd use the single middle value is n is even use the avg of the middle 2 values
what is mode
an alternative measure for the centre, it is the value that occurs with the highest frequency in a data set
pro and con of mode
pro- easy to locate, con- data may have none or more than 1 mode whereas it will only have one mean and one median
on a graph the mode is the value where the distribution is the
tallest
on a graph the median is the point where the distribution is
cut into two parts of the same area
on a graph the mean is
the balance point of the distribution
in a symmetric distribution what do the measures of centre equal relative to each other.
mean= median= mode
in a positively (right) skewed distribution what do the measures of centre equal relative to each other.
mean > median> mode
in a negatively (left) skewed distribution what do the measures of centre equal relative to each other.
mean < median < mode
what is range
measure of spread. max- min
what are the 3 measures of spread
range, SD, IQR
what stands true about the sum of standard deviations
sum of deviations is equal to 0 always
hwo to calculate IQR
Q3 (75th percentile) - Q1 (25th percentile)
what are upper and lower fences
they are used to determine outliers.
upper fence = Q3 + 1.5 X IQR , every measure above is an outlier
lower fence = Q1- 1.5 X IQR, every measure below is an outlier
symmetric distribution
median line in center of box and whiskers of equal length
skewed right
median line left of center and long right whisker
skewed left
median line right of center and long left whisker