Transcript Document
Midterm Review Session
Things to Review
• Concepts • Basic formulae • Statistical tests
Things to Review
• Concepts • Basic formulae • Statistical tests
P
opulations <->
P
arameters;
S
amples <-> E
s
timates
Nomenclature
Mean Variance Standard Deviation Population Parameter Sample Statistics
x
s 2 s
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
H o true
Result
Reject H o Type I error H o false correct Do not reject H o 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 compare Null distribution How unusual is this test statistic?
P < 0.05
P > 0.05
Reject H o Fail to reject H o
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, p o
• What does it assume?
Individual trials are randomly sampled and independent
• Test statistic:
X, the number of successes
• Distribution under H o :
binomial with parameters n and p o .
• Formula:
P
(
x
)
n x
p x
1
p
n
x
P = 2 * Pr[x X] P(x) = probability of a total of x successes p = probability of success in each trial n = total number of trials
Sample Binomial test Null hypothesis Pr[success]=p o Test statistic x = number of successes compare Null distribution Binomial n, p o How unusual is this test statistic?
P < 0.05
P > 0.05
Reject H o Fail to reject H o
Binomial test
H 0 : The rela tive frequency of succ esses in the population is
p
0 H A : The rela tive frequency of succ esses in the population is not
p
0
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 H o : 2 with df=# categories - # parameters - 1 • Formula: 2
all classes
Observed i
Expected i
2
Expected i
2 goodness of fit test Sample Null hypothesis: Data fit a particular Discrete distribution Calculate expected values Test statistic 2
all classes
Observed i
Expected i
2
Expected i
compar e Null distribution: 2 With N-1-param. d.f.
How unusual is this test statistic?
P < 0.05
P > 0.05
Reject H o Fail to reject H o
2
Goodness-of-Fit test
H 0 : T he data come from a certain distribution H A : T he data do not come from that distrubition
Possible distributions
Pr[
x
]
n
x
p x
1
p
n
x e
X
Pr
X
!
Pr[x] = n * frequency of occurrence
Proportional Given a number of categories Probability proportional to number of opportunities Days of the week, months of the year Binomial Number of successes in n trials Have to know n, p under the null hypothesis Punnett square, many p=0.5 examples Poisson Number of events in interval of space or time 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 H o : 2 with df=(r-1)(c-1) where r = # rows, c = # columns • Formulae:
Expected
RowTotal
*
ColTotal GrandTotal
2
all classes
Observed i
Expected i
2
Expected i
2 Contingency Test Sample Null hypothesis: No association between variables Calculate expected values Test statistic 2
all classes
Observed i
Expected i
2
Expected i
compar e Null distribution: 2 With (r-1)(c-1) d.f.
How unusual is this test statistic?
P < 0.05
P > 0.05
Reject H o Fail to reject H o
2
Contingency test
H 0 : T here is no association between these two variables H A : T here 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, μ o
• What does it assume?
Individuals are randomly sampled from a population that is normally distributed.
• Test statistic:
t
• Distribution under H o :
t-distribution with n-1 degrees of freedom.
• Formula:
t
Y
SE Y o
Sample One-sample t-test Null hypothesis
The population mean is equal to
o
Test statistic
t
Y
o s
/
n
compare Null distribution
t with n-1 df
How unusual is this test statistic?
P < 0.05
P > 0.05
Reject H o Fail to reject H o
One-sample t-test
H o : The population mean is equal to o H a : 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, μ do
• What does it assume?
Pairs are randomly sampled from a population. The differences are normally distributed
• Test statistic:
t
• Distribution under H o :
t-distribution with n-1 degrees of freedom, where n is the number of pairs
• Formula:
t
d
SE d do
Sample Paired t-test Null hypothesis
The mean difference is equal to
o t
Test statistic
d
SE d do
compare Null distribution
t with n-1 df *n is the number of pairs
How unusual is this test statistic?
P < 0.05
P > 0.05
Reject H o Fail to reject H o
Paired t-test
H o : The mean difference is equal to 0 H a : 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 H o :
t-distribution with n 1 +n 2 -2 degrees of freedom.
• Formulae:
t
Y
1
SE
Y
2
Y
1
Y
2
SE Y
1
s p
2
Y
2
df
1
s
1 2
s p
2 1
n
1
df
1
df
2
s
2 2
df
2 1
n
2
Sample Two-sample t-test Null hypothesis
The two populations have the same mean
1
2
Test statistic
t
Y
1
Y
2
SE Y
Y
2 compare Null distribution
t with n
How unusual is this test statistic?
P < 0.05
P > 0.05
1 +n 2 -2 df
Reject H o Fail to reject H o
Two-sample t-test
H o : The means of the two populations are equal H a : 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
Categorical Is the variable categorical or numerical?
Comparing to a single proportion p o or to a distribution?
p o Binomial test distribution 2 Goodness of-fit test Numerical One-sample t-test
Methods for two variables
Y
Response variable Categorical Numerical
X
Explanatory var iable Categorical
Contingen cy table Grouped b ar graph Mosaic plot Multiple histograms Cumulative frequency distributions
Numerical
Scatter plot
Methods for two variables
Y
Response variable Categorical Numerical
X
Explanatory var iable Categorical
Contingen cy table Contingency Grouped b ar graph analysis Mosaic plot Multiple histograms
Numerical
Logistic regression
Methods for two variables
Is the response variable categorical or numerical?
Categorical Numerical Contingency analysis t-test
1 How many variables am I comparing?
2 Is the variable categorical or numerical?
Is the response variable categorical or numerical?
Categorical Comparing to a single proportion p o or to a distribution?
Numerical Categorical Numerical p o distribution Binomial test 2 Goodness of-fit test One-sample t-test Contingency analysis t-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 .