# PowerPoint Presentation by OmGrbA

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```									Midterm Review Session
Things to Review
• Concepts
• Basic formulae
• Statistical tests
Things to Review
• Concepts
• Basic formulae
• Statistical tests
Populations <-> Parameters;
Samples <-> Estimates
Nomenclature
Population   Sample
Parameter    Statistics
Mean                       x
Variance                  s2
Standard                  s
Deviation
In a random sample, each
member of a population has
an equal and independent
chance of being selected.
Review - types of variables
Nominal
• Categorical variables
Ordinal

Discrete
• Numerical variables
Continuous
Reality

Ho true          Ho false

Result
Reject Ho    Type I error     correct

Do not reject Ho   correct         Type II error
Sampling distribution of the mean, n=10

Sampling distribution of the mean, n=100

Sampling distribution of the mean, n = 1000
Things to Review
• Concepts
• Basic formulae
• Statistical tests
Things to Review
• Concepts
• Basic formulae
• Statistical tests
Sample                         Null hypothesis

Test statistic                    Null distribution
compare

How unusual is this test statistic?

P < 0.05                 P > 0.05

Reject Ho                       Fail to reject Ho
Statistical tests
• Binomial test
• Chi-squared goodness-of-fit
– Proportional, binomial, poisson
• Chi-squared contingency test
• t-tests
– One-sample t-test
– Paired t-test
– Two-sample t-test
Statistical tests
• Binomial test
• Chi-squared goodness-of-fit
– Proportional, binomial, poisson
• Chi-squared contingency test
• t-tests
– One-sample t-test
– Paired t-test
– Two-sample t-test
Quick reference summary:
Binomial test
• What is it for? Compares the proportion of successes
in a sample to a hypothesized value, po
• What does it assume? Individual trials are randomly
sampled and independent
• Test statistic: X, the number of successes
• Distribution under Ho: binomial with parameters n and
po .
• Formula:
n  x
P(x)   p 1 p
nx
P = 2 * Pr[xX]
x 
P(x) = probability of a total of x successes
p = probability of success in each trial
n = total number of trials
Binomial test
Null hypothesis
Sample                          Pr[success]=po

Test statistic                      Null distribution
x = number of successes   compare
Binomial n, po

How unusual is this test statistic?

P < 0.05                   P > 0.05

Reject Ho                       Fail to reject Ho
Binomial test

H0: The relative frequency of successes in the population is p0

HA: The relative frequency of successes in the population is not p0
Statistical tests
• Binomial test
• Chi-squared goodness-of-fit
– Proportional, binomial, poisson
• Chi-squared contingency test
• t-tests
– One-sample t-test
– Paired t-test
– Two-sample t-test
Quick reference summary:
2 Goodness-of-Fit test
• What is it for? Compares observed frequencies in
categories of a single variable to the expected
frequencies under a random model
• What does it assume? Random samples; no expected
values < 1; no more than 20% of expected values < 5
• Test statistic: 2
• Distribution under Ho: 2 with
df=# categories - # parameters - 1
• Formula:
                   
2
Observed  Expected
 
2
                i

Expectedi
i

all classes
2 goodness of fit test
Sample                                 Null hypothesis:
Data fit a particular
Discrete distribution
Calculate expected values

Test statistic
Observedi  Expectedi                 Null distribution:
2

 
2
                                      compar
all classes          Expectedi
e                2 With
N-1-param. d.f.


How unusual is this test statistic?

P < 0.05                      P > 0.05

Reject Ho                                 Fail to reject Ho
2 Goodness-of-Fit test

H0: The data come from a certain distribution

HA: The data do not come from that distrubition
Possible distributions
n  x
Pr[ x]   p 1 p
nx

x 

e          X
PrX  
X!

Pr[x] = n * frequency of occurrence
Given a number of categories
Proportional   Probability proportional to number of opportunities
Days of the week, months of the year

Number of successes in n trials
Binomial      Have to know n, p under the null hypothesis
Punnett square, many p=0.5 examples

Number of events in interval of space or time
Poisson      n not fixed, not given p
Car wrecks, flowers in a field
Statistical tests
• Binomial test
• Chi-squared goodness-of-fit
– Proportional, binomial, poisson
• Chi-squared contingency test
• t-tests
– One-sample t-test
– Paired t-test
– Two-sample t-test
Quick reference summary:
2 Contingency Test
• What is it for? Tests the null hypothesis of no association
between two categorical variables
• What does it assume? Random samples; no expected
values < 1; no more than 20% of expected values < 5
• Test statistic: 2
• Distribution under Ho: 2 with
df=(r-1)(c-1) where r = # rows, c = # columns
• Formulae:
Observedi  Expectedi 
2

Expected 
RowTotal* ColTotal
GrandTotal
2                        Expectedi
all classes
2 Contingency Test
Sample                              Null hypothesis:
No association
between variables
Calculate expected values

Test statistic
Observedi  Expectedi               Null distribution:
2

 
2
                                      compar
all classes          Expectedi
e              2 With
(r-1)(c-1) d.f.


How unusual is this test statistic?

P < 0.05                    P > 0.05

Reject Ho                               Fail to reject Ho
2 Contingency test

H0: There is no association between these two variables

HA: There is an association between these two variables
Statistical tests
• Binomial test
• Chi-squared goodness-of-fit
– Proportional, binomial, poisson
• Chi-squared contingency test
• t-tests
– One-sample t-test
– Paired t-test
– Two-sample t-test
Quick reference summary:
One sample t-test
• What is it for? Compares the mean of a numerical
variable to a hypothesized value, μo
• What does it assume? Individuals are randomly
sampled from a population that is normally distributed.
• Test statistic: t
• Distribution under Ho: t-distribution with n-1 degrees of
freedom.
• Formula:
Y  o
t
SEY
One-sample t-test

Sample                          Null hypothesis
The population mean
is equal to o

Test statistic                       Null distribution
compare          t with n-1 df
Y  o
t
s/ n

How unusual is this test statistic?

         P < 0.05                  P > 0.05

Reject Ho                       Fail to reject Ho
One-sample t-test

Ho: The population mean is equal to o

Ha: The population mean is not equal to o
Paired vs. 2 sample
comparisons
Quick reference summary:
Paired t-test
• What is it for? To test whether the mean difference in a
population equals a null hypothesized value, μdo
• What does it assume? Pairs are randomly sampled
from a population. The differences are normally
distributed
• Test statistic: t
• Distribution under Ho: t-distribution with n-1 degrees of
freedom, where n is the number of pairs
• Formula:               d  do
t
SE d
Paired t-test

Sample                           Null hypothesis
The mean difference
is equal to o

Test statistic                      Null distribution
t with n-1 df
d  do           compare
t
*n is the number of pairs

SE d

How unusual is this test statistic?
P > 0.05
         P < 0.05

Reject Ho                         Fail to reject Ho
Paired t-test

Ho: The mean difference is equal to 0

Ha: The mean difference is not equal 0
Quick reference summary:
Two-sample t-test
• What is it for? Tests whether two groups have the
same mean
• What does it assume? Both samples are random
samples. The numerical variable is normally
distributed within both populations. The variance of
the distribution is the same in the two populations
• Test statistic: t
• Distribution under Ho: t-distribution with n1+n2-2
degrees of freedom.                                      1 
Y1  Y2                       2 1
SEY Y  sp   
• Formulae:         t                                  n1 n 2 
1   2

SE Y Y                 s 
2
p
df1s12  df2 s2
2

1   2
df1  df2

Two-sample t-test
Null hypothesis
Sample                        The two populations
have the same mean
12

Test statistic                        Null distribution
compare         t with n1+n2-2 df
Y1  Y2
t
SE Y Y
1   2

How unusual is this test statistic?
         P < 0.05                      P > 0.05

Reject Ho                           Fail to reject Ho
Two-sample t-test

Ho: The means of the two populations are
equal

Ha: The means of the two populations are
not equal
Which test do I use?
Methods for a
single variable
1

How many variables
am I comparing?
2    Methods for
comparing two
variables
Methods for one variable
Is the variable
categorical
Categorical       or numerical?

Comparing to a
single proportion po                        Numerical
or to a distribution?

po               distribution

2 Goodness-     One-sample t-test
Binomial test                 of-fit test
Methods for two variables

X
Explanatory variable
Response variable           Categorical                    Numerical
Contingency table
Categorical          Grouped bar graph
Y                               Mosaic plot
Multiple histograms
Scatter plot
Numerical      Cumulative frequency distributions
Methods for two variables

X
Explanatory variable
Response variable           Categorical                     Numerical
Contingency table
Contingency                        Logistic
Categorical          Grouped bar graph
analysis                    regression
Y                               Mosaic plot
Multiple histograms
Scatter plot
Numerical                   t-test distributions
Cumulative frequency                  Regression
Methods for two variables

Is the response variable
categorical or numerical?

Categorical                Numerical

Contingency
t-test
analysis
How many variables
am I comparing?

1                                 2

Is the variable
categorical                           Is the response variable
or numerical?
categorical or numerical?
Categorical

Comparing to a               Numerical                                 Numerical
single proportion po
Categorical
or to a distribution?

po
distribution

2 Goodness-                             Contingency
t-test
Binomial test                           One-sample t-test      analysis
of-fit test
Sample Problems
An experiment compared the testes sizes of four
experimental populations of monogamous flies to four
populations of polygamous flies:

a. What is the difference in mean testes size for males from monogamous populations
compared to males from polyandrous populations? What is the 95% confidence interval for
this estimate?
b. Carry out a hypothesis test to compare the means of these two groups. What conclusions
can you draw?
Sample Problems

In Vancouver, the probability of rain during a winter day
is 0.58, for a spring day 0.38, for a summer day 0.25,
and for a fall day 0.53. Each of these seasons lasts one
quarter of the year.

What is the probability of rain on a randomly-chosen
day in Vancouver?
Sample problems
A study by Doll et al. (1994) examined the relationship
between moderate intake of alcohol and the risk of heart
disease. 410 men (209 "abstainers" and 201 "moderate
drinkers") were observed over a period of 10 years, and the
number experiencing cardiac arrest over this period was
recorded and compared with drinking habits. All men were
40 years of age at the start of the experiment. By the end of
the experiment, 12 abstainers had experienced cardiac
arrest whereas 9 moderate drinkers had experienced
cardiac arrest.

Test whether or not relative frequency of cardiac arrest was
different in the two groups of men.
Sample Problems
An RSPCA survey of 200 randomly-chosen Australian
pet owners found that 10 said that they
had met their partner through owning the pet.

A. Find the 95% confidence interval for the proportion
of Australian pet owners who find love through their
pets.

B. What test would you use to test if the true proportion
is significantly different from 0.01? Write the formula
that you would use to calculate a P-value.
Sample Problems
One thousand coins were each flipped 8 times, and
the number of heads was recorded for each coin.
Here are the results:

Does the distribution of coin flips match the
distribution expected with fair coins? ("Fair coin"
means that the probability of heads per flip is 0.5.)
Carry out a hypothesis test.
Sample problems
Vertebrates are thought to be unidirectional in growth, with size either increasing or holding
steady throughout life. Marine iguanas from the Galápagos are unusual in a number of ways, and a
team of researchers has suggested that these iguanas might actually shrink during the low food
periods caused by El Niño events (Wikelski and Thom 2000). During these events, up to 90% of the
iguana population can die from starvation. Here is a plot of the changes in body length of 64
surviving iguanas during the 1992-1993 El Niño event.

The average change in length was −5.81mm, with standard deviation 19.50mm.
Test the hypothesis that length did not change on average during the El Niño event   .

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