1. Midterm Review Session

2. Things to Review
Concepts
Basic formulae
Statistical tests

3. Things to Review
Concepts
Basic formulae
Statistical tests

4. Populations <-> Parameters; Samples <-> Estimates

5. Nomenclature Population Parameter Sample Statistics Mean  Variance 
Standard Deviation  s

6. In a random sample, each member of a population has an equal and independent chance of being selected.

7. Review - types of variables
Nominal
Categorical variables
Numerical variables
Ordinal
Discrete
Continuous

8. Reality
Ho true
Ho false
Result
Reject Ho
correct
Type I error
Do not reject Ho
correct
Type II error

9. Sampling distribution of the mean, n=10

11. Things to Review
Concepts
Basic formulae
Statistical tests

13. Things to Review
Concepts
Basic formulae
Statistical tests

14. 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

15. 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

16. 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

17. 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:
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

18. Binomial test
Null hypothesis
Pr[success]=po
Sample
Test statistic
x = number of successes
Null distribution
Binomial n, po
compare
How unusual is this test statistic?
P > 0.05
P < 0.05
Reject Ho
Fail to reject Ho

19. Binomial test

20. 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

21. 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:

22. Discrete distribution
2 goodness of fit test
Null hypothesis:
Data fit a particular
Discrete distribution
Sample
Calculate expected values
Test statistic
Null distribution:
2 With
N-1-param. d.f.
compare
How unusual is this test statistic?
P > 0.05
P < 0.05
Reject Ho
Fail to reject Ho

23. 2 Goodness-of-Fit test

24. Possible distributions
Pr[x] = n * frequency of occurrence

25. Proportional Binomial Poisson Given a number of categories
Probability proportional to number of opportunities
Days of the week, months of the year
Proportional
Number of successes in n trials
Have to know n, p under the null hypothesis
Punnett square, many p=0.5 examples
Binomial
Number of events in interval of space or time
n not fixed, not given p
Car wrecks, flowers in a field
Poisson

26. 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

27. 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:

28. 2 Contingency Test Null hypothesis: Sample No association
between variables
Sample
Calculate expected values
Test statistic
Null distribution:
2 With
(r-1)(c-1) d.f.
compare
How unusual is this test statistic?
P > 0.05
P < 0.05
Reject Ho
Fail to reject Ho

29. 2 Contingency test

30. 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

31. 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:

32. One-sample t-test
Null hypothesis
The population mean
is equal to o
Sample
Null distribution
t with n-1 df
Test statistic
compare
How unusual is this test statistic?
P > 0.05
P < 0.05
Reject Ho
Fail to reject Ho

33. One-sample t-test Ho: The population mean is equal to o
Ha: The population mean is not equal to o

34. Paired vs. 2 sample comparisons

35. 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:

36. *n is the number of pairs
Paired t-test
Null hypothesis
The mean difference
is equal to o
Sample
Null distribution
t with n-1 df
*n is the number of pairs
Test statistic
compare
How unusual is this test statistic?
P > 0.05
P < 0.05
Reject Ho
Fail to reject Ho

37. Paired t-test Ho: The mean difference is equal to 0
Ha: The mean difference is not equal 0

38. 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.
Formulae:

39. Two-sample t-test
Null hypothesis
The two populations
have the same mean
12
Sample
Null distribution
t with n1+n2-2 df
Test statistic
compare
How unusual is this test statistic?
P > 0.05
P < 0.05
Reject Ho
Fail to reject Ho

40. Two-sample t-test Ho: The means of the two populations are equal
Ha: The means of the two populations are not equal

41. Which test do I use?

42. Methods for a
single variable 1
How many variables
am I comparing? 2
Methods for
comparing two
variables

43. Methods for one variable
Is the variable
categorical
or numerical?
Categorical
Comparing to a
single proportion po
or to a distribution?
Numerical po
distribution
One-sample t-test
2 Goodness-
of-fit test
Binomial test

44. Methods for two variablesX Y

45. Methods for two variablesX
Contingency
analysis
Logistic
regression Y
t-test
Regression

46. Methods for two variables
Is the response variable
categorical or numerical?
Categorical
Numerical
Contingency
analysis
t-test

47. How many variables am I comparing? 2 1 t-test Is the response variable
Is the variable
categorical
or numerical?
Is the response variable
categorical or numerical?
Categorical
Comparing to a
single proportion po
or to a distribution?
Numerical
Numerical
Categorical po
distribution
Contingency
analysis
t-test
2 Goodness-
of-fit test
One-sample t-test
Binomial test

48. 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?

49. 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 Each of these seasons lasts one quarter of the year.
What is the probability of rain on a randomly-chosen day in Vancouver?

50. 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.

51. 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.

52. 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.

53. 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 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.