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 Parameter Sample Statistics Mean  Variance  Standard Deviation  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 Numerical variables Ordinal Discrete Continuous

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

Sampling distribution of the mean, n=10

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: 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

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

Binomial 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: 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:

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

2 Goodness-of-Fit test

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

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

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:

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

2 Contingency 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: 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:

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

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:

*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

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. Formulae:

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

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 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

Methods for two variables X Y

Methods for two variables X Contingency analysis Logistic regression Y t-test Regression

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

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

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.