midterm review session. things to review concepts basic formulae statistical tests

Midterm Review Session

Things to Review

• Concepts

• Basic formulae

• Statistical tests

Populations <-> Parameters;Samples <-> Estimates

Nomenclature

Population

Parameter

Sample

Statistics

Mean

Variance s2

Standard Deviation

s€

x

In a random sample, each member of a population has an

equal and independent chance of being

selected.

Review - types of variables

• Categorical variables

• Numerical variablesDiscrete

Continuous

Nominal

Ordinal

Reality

Result

Ho true Ho false

Reject Ho

Do not reject Ho correct

correctType I error

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

Sample

Test statistic

Null hypothesis

Null distributioncompare

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

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 Ho: binomial with parameters n and

po.

• Formula:

P(x) = probability of a total of x successesp = probability of success in each trialn = total number of trials

€

P(x) =n

x

⎛

⎝ ⎜

⎞

⎠ ⎟px 1− p( )

n−x

P = 2 * Pr[xX]

Sample

Test statisticx = number of successes

Null hypothesisPr[success]=po

Null distributionBinomial n, po

compare


P < 0.05 P > 0.05


Binomial test

Binomial test

Statistical tests





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 =Observedi − Expectedi( )

2

Expectediall classes

∑

Sample Null hypothesis:Data fit a particular

Discrete distribution

Null distribution:2 With

N-1-param. d.f.

compare


P < 0.05 P > 0.05


2 goodness of fit test

Calculate expected values

Test statistic

€


2


∑

2 Goodness-of-Fit test

H0: The data come from a certain distribution

HA: The data do not come from that distrubition

Possible distributions

€

Pr X[ ] =e−μ μ X

X!€

Pr[x] =n

x

⎛

⎝ ⎜

⎞

⎠ ⎟px 1− p( )

n−x

Pr[x] = n * frequency of occurrence

Proportional

Binomial

Poisson

Given a number of categories Probability proportional to number of opportunitiesDays of the week, months of the year

Number of successes in n trialsHave to know n, p under the null hypothesisPunnett square, many p=0.5 examples

Number of events in interval of space or timen not fixed, not given pCar wrecks, flowers in a field

Statistical tests





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:

€


2


∑

€

Expected =RowTotal *ColTotal

GrandTotal

Sample Null hypothesis:No association

between variables

Null distribution:2 With

(r-1)(c-1) d.f.

compare


P < 0.05 P > 0.05


2 Contingency Test

Calculate expected values

Test statistic

€


2


∑

2 Contingency test

H0: There is no association between these two variables

HA: There is an association between these two variables

Statistical tests





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 Ho: t-distribution with n-1 degrees of

freedom.• Formula:

€

t =Y − μo

SEY

SampleNull hypothesis

The population mean is equal to

o

One-sample t-test

Test statistic Null distributiont with n-1 dfcompare


P < 0.05 P > 0.05


€

t =Y − μo

s / n

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, μ

do

• What does it assume? Pairs are randomly sampled from a population. The differences are normally distributed


• Distribution under Ho: t-distribution with n-1 degrees of

freedom, where n is the number of pairs• Formula:

€

t =d − μdo

SEd


The mean differenceis equal to

o

Paired t-test

Test statistic Null distributiont with n-1 df

*n is the number of pairscompare


P < 0.05 P > 0.05


€

t =d − μdo

SEd

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


• Distribution under Ho: t-distribution with n1+n2-2

degrees of freedom.• Formulae:

€

t =Y 1 −Y 2SE

Y 1−Y 2

€

SEY 1 −Y 2

= sp2 1

n1

+1

n2

⎛

⎝ ⎜

⎞

⎠ ⎟

€

sp2 =

df1s12 + df2s2

2

df1 + df2


The two populations have the same mean

1

2

Two-sample t-test

Test statistic Null distributiont with n1+n2-2 dfcompare


P < 0.05 P > 0.05


€

t =Y 1 −Y 2SE

Y 1−Y 2

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?

How many variablesam I comparing?

1

2

Methods for a single variable

Methods forcomparing two

variables

Methods for one variableIs the variable

categoricalor numerical?

Comparing to a single proportion po

or to a distribution?

Binomial test2 Goodness-

of-fit testOne-sample t-test

Categorical

Numerical

po distribution

Methods for two variables

Explanatory variableResponse variable Categorical Numerical

CategoricalContingency tableGrouped bar graph

Mosaic plot

NumericalMultiple histograms

Cumulative frequency distributionsScatter plot

X

Y


Explanatory variableResponse variable Categorical Numerical

CategoricalContingency tableGrouped bar graph

Mosaic plot

NumericalMultiple histograms

Cumulative frequency distributionsScatter plot

X

Y

Contingencyanalysis

t-test

Logistic regression

Regression


Is the response variable categorical or numerical?

Contingencyanalysis

t-test

Categorical Numerical

1 2

How many variablesam I comparing?

Is the variable categorical

or numerical?

Comparing to a single proportion po

or to a distribution?

Binomial test2 Goodness-

of-fit testOne-sample t-test

Categorical

Numerical

po distribution

Is the response variable categorical or numerical?

t-test

Categorical

Contingencyanalysis

Numerical

Sample ProblemsAn 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 populationscompared to males from polyandrous populations? What is the 95% confidence interval forthis estimate?b. Carry out a hypothesis test to compare the means of these two groups. What conclusionscan 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 problemsA 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 theyhad met their partner through owning the pet.

A. Find the 95% confidence interval for the proportionof 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 problemsVertebrates are thought to be unidirectional in growth, with size either increasing or holdingsteady throughout life. Marine iguanas from the Galápagos are unusual in a number of ways, and ateam of researchers has suggested that these iguanas might actually shrink during the low foodperiods caused by El Niño events (Wikelski and Thom 2000). During these events, up to 90% of theiguana population can die from starvation. Here is a plot of the changes in body length of 64surviving 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.

midterm review session. things to review concepts basic formulae statistical tests

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