Ch 15, Ch 16, Ch 17 and Cumulative FINAL

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If SStotal = 20 and SSbetween = 14, SSwithin =_________?

6 SStotal = SSbetween + SSwithin So SStotal - SS between = SS within. 20-14 = 6.

How to find f e for: Chi-Square experiment with 2 categorical variables

f e = (estimated H0 population for particular category) * (total number of group i.e. republicans)

Estimate based on between groups variability:

MS between

2 X 2 design

2 factor A 2 factor B

Correlation implies causation. a. True b. false

If tobt = 1.50 and df = 8, what would you conclude using α = 0.052tail? a. reject H0 b. retain H0

B *Find df & use table to find tcrit.

Factorial Experiment

Effects of two or more factors are assessed.

Estimate based on with in groups variability:

MS within

What kind of test is Chi-Square?

NON directional. --- Square is ALWAYS positive.

Interaction effect of 2 X 2:

Occurs when the effect of one factor is not the same at all levels of other factor.

If F obt is < 1 you will always _______ H0.

Retain H0.

The F distribution has no negative values. T or F:

True

T test:

Use MEAN as basic statistic for evaluating null hypothesis.

Evaluate Chi-Square

X2obt > or = to X2crit = REJECT H0 X2obt < X2crit = RETAIN H0

F e =

expected frequency

Non parametric tests:

sign test, minimal requirements aka distribution "free" tests.

You conduct a study with two independent groups. You analyze the data using an independent samples t test and get a t obt value = 3.50. If you had conducted a one-way ANOVA on the same data, what would be your Fobt value?

12.25 tsquared = f so (3.50)^2 = 12.25

Has two sources of total variability (SS total) =

SS between and SS within

Post hoc comparisons are comparisons NOT planned in advance of the experiment. T or F:

True

Chi-square:

-uses nominal data -uses discrete, mutually exclusive categories -counts frequencies in each category

Two Types of Multiple Comparisons:

(1) A priori (planned) (2) A posteriori (post hoc)

Rounding Rules

- 5 or *LARGER*, round *UP* (decimals) - 5 or *SMALLER*, Round *DOWN* (decimals) - if last number is .55 (or .5 and an odd number round up so = .56) - if last number is .54 (or .5 and an even number keep same so = .54)

When is t test appropriate?

- experiment has ONE sample. - known population mean - standard deviation is UNKNOWN. - sampling distribution of X is normal. - N > or = 30 or population of raw scores is normal.

Student's T Test for Independent Groups:

- subjects randomly selected from subject population. - randomly assigned to a condition. - subjects NOT paired, each subject tested once. - Exp Condition: receive one level of IV - Control Condition: receive a diff level of IV

Sampling F Distribution:

-Gives ALL possible F values along with p(f) for each value, assuming sampling is random from the population. -F varies with degrees of freedom. -F has two values for df: (1) df for numerator = df between (2) df for denominator = df within - F is ALWAYS positive - F is positively skewed. - When ns are = median of f = 1 - F distributions is a family of curves for each combo of df.

About Chi Square

-Sampling distribution is a family of curves that varies with df = similar to t dist. -At lower df, curves are positively skewed. -For an experiment with ONE variable: df = k -1. (k = number of groups/categories) -The larger the discrepancy between the observed and expected results, the larger the value of Chi Square obt. (more unreasonable; or more likely to reject H0)

ANOVA test used:

independent group designs OR repeated measures design

If H0 is true F obt is expected to equal

Correlation coefficient =

-1.0 to +1.0

Size of effect chart:

0.01 - 0.05 = small effect 0.06 - 0.13 = medium effect > or = to 0.14 = large effect

The value of fe for the cell in row A, column Y is _________. YZ Y Z A 53 22 B 37 38 a. 45 b. 30 c. 53 d. 150

1. Calc row/column marginals. Column 1 = 53+37=90 Column 2 = 22+38=60 Row 1 = 53+22 = 75 Row 2 = 37+38 = 75 Marginal total = 150 2. Multiply Column 1 * Row 1 / total marginal = 90 * 75 / 150 = 45. A

Steps to doing chi-square problem:

1. Calculate row and column marginals. 2. Calculate fe for each cell. fe = r marginal x c marginal / total marginal. 3. Calculate X2 obt = sum of (fo-fe)square / total marginal.

How to find f e =

1. Calculate the row and column marginals. 2. For each cell, multiply the the row and column marginal and divide by the total marginal. Ex) Column Marginal = 45 45 Row Marginal = 65 65 Cell 1: (45)(65) / 220

3 analyses in 2 X 2 design

1. Does factor A have a significant effect? = Main Effect. 2. Does factor B have a significant effect? = Main effect. 3. Is there an interaction between factor A & B? = Interaction effect. - Occurs when the effect of one factor is not the same at all levels of other factor.

Assumptions of Chi-Square

1. Independence exists between each observation in contingency table. 2. Sample size is large enough so that Fe in each cell is at least 5 for tables where r or c are greater than 2. 3. If table 1 by 2 or 2 by 2 than each Fe should be at least 10. 4. Chi-Squared can be used with nominal, ordinal, interval, or ratio data.

Assumptions of Two Way ANOVA:

1. Population from which the samples were taken are normally distributed. 2. Homogeneity of variance. 3. Robust with regards to violations of assumptions, provided samples are of equal size.

Assumptions of ANOVA:

1. Populations from which samples are taken are normally distributed. 2. Samples are drawn from populations of equal variances (homogeneity of variance assumption) 3. ANOVA is a robust test. - F is robust is sample sizes are equal. - F is minimally affected by violations of population normality. F = t squared.

A correlated groups t test is conducted in which the same 9 subjects are tested during a baseline period and after a specific training. Using ɑ = 0.052 tail, what is the value of tcrit?

2.306

An independent groups t test is conducted with 10 subjects in Group 1 and 7 subjects in Group 2. Using ɑ = 0.011 tail, what is the value of tcrit?

2.602

A one-way ANOVA is conducted with a total of 18 subjects. Each subject is randomly assigned to one of three groups. Using ɑ = 0.05, what is the value of Fcrit?

3.68

Relationship between Fobt and the IV

Since F obt = MS between / MS within As IV effect increases, Fobt also increases.

Relationship between MS within and the IV

Since IV effects only the mean and NOT variance, MS within does NOT change.

Power of ANOVA:

:probability rejecting the null hypothesis when the null hypothesis is false. - affected by sample variables and in the same manner as t test for independent groups. (1) sample size: increase sample size = increased power. (2) real effect of IV: increased real effect = increased power (3) sample variability: DECREASED sample variability = increased power to detect a real effect.

"Number of females in your class" is an example of what type of variable? a. discrete variable b. continuous variable

Classifying subjects on the basis of sex is an example of using what kind of scale? a. nominal b. ordinal c. interval d. ratio

If tobt = 3.50 and df = 8, what would you conclude using α = 0.052tail? a. reject H0 b. retain H0

A *Find df & use table to find tcrit.

If dfbetween = 3 and dfwithin = 22 and Fobt = 3.25, what would you conclude using α = 0.05? a. reject H0 b. retain H0

A *Find df for numerator and denominator & use table to find fcrit and evaluate.

As the differences between the group mean increases, ___________. a. MSbetween increases. b. Fobt decreases. c. tobt decreases. d. all of these. e. none of the above.

A. MSbetween INCREASES.

"Time of day" is an example of what type of variable? a. discrete variable b. continuous variable

A main effect for variable A means that _________. a. the effect of variable A is the same over all levels of variable B b. the effect of variable A is significant when averaged over all levels of variable B c. the effect of variable A is not the same over all levels of variable B d. variable A has a greater effect than variable B

A therapist measures the difference between two clients. If the therapist can say that Rebecca's score is higher than Sarah's, but can't specify how much higher, the measuring scale used must have been a(an)_______ scale. a. nominal b. ordinal c. interval d. ratio

Chi-square should not be used if _________. a. df = 1 b. fe is below 5 c. fo is below 5 d. fe = fo

The degrees of freedom for a χ2 contingency table equal _________. a. rc-1 b. (r-1)(c-1) c. (r - 1)(c) d. (c - 1)(r) e. N-1

When analyzing data from experiments that involve more than two groups: a. doing t tests on all possible pairs of means decreases the probability of making Type I errors. b. doing t test on all possible pairs of means increases the probability of making Type I errors. c. it is generally permissible to do t tests between all possible pairs of means and use Student's t distribution. d. doing t tests on all possible pairs of means increases the probability of making Type II errors.

B. doing t test on all possible pairs of means INCREASES the probability of making TYPE I ERROR.

Why do we need the F test?

Because of type I error. - If you do multiple t tests, the probability of making type 1 error increases for EACH comparison.

How to do ANOVA problem:

Before doing problem: find sum of x1, x2,x3.... find sum of x1 squared, x2 squared, x3 squared... find n1,n2,n3... find N find all scores EX, all scores EX2 Step 1: Calc SS between. Step 2: Calc SS within. Step 3: Calc SS total. Step 4: Calc (3) degrees of freedom (dfbetween, dfwithin, and dftotal). Step 5: Calc MS between. Step 6: Calc MS within. Step 7: Calc f obt. Step 8: Evaluate f obt finding f crit.

If MSbetween = 27.9 and MSwithin = 54.2, what is the value of Fobt? a. impossible result, there must be an error b. 1.94 c. 0.51 d. 2.76

The number 83.476499 rounded to three decimal places is _________. a. 83.477 b. 83.480 c. 83.476 d. 83.470

When rounded to two decimal places, the number 3.175000 becomes _________. a. 3.17 b. 3.20 c. 3.18 d. 3.10

When rounded to two decimal places, the number 3.9998 becomes _________. a. 3.99 b. 4 c. 4.00 d. 3.98

Computational formula for chi square obt:

Chi square obt = E [(f o - f e ) ^2] / f e f o = observed frequency in the cell. f e = expected frequency in the cell. E = summation over all cells.

Powerfulness of Tests

Scheffe test is LESS powerful than Tukey hsd BUT both are LESS powerful than planned comparisons.

22.Which of the following is(are) illegal values for a correlation coefficient? a. -3.00 b. 0.50 c. 1.00 d. a and c

A personal trainer takes the weight of a group of dieters. The variable "weight" is measured on what type of scale? a. nominal b. ordinal c. interval d. ratio

In a two-way ANOVA, if there is a significant interaction between Factor A and Factor B, which of the following may be true? a. the effect of Factor A is not the same at all levels of Factor B b. The effect of Factor B is not the same at all levels of Factor A c. the effects of the two Factors do not differ across levels d. the effect of Factor A is not the same at all levels of Factor B and/or The effect of Factor B is not the same at all levels of Factor A

Which of the following are true? a. fo is the symbol for the observed frequency b. fe is the symbol for the expected frequency c. χ2 is the symbol for chi-square d. all of these

Which of the following is(are) true? a. df for MSwithin = N - k b. df for MSbetween = k - 1 c. dftotal = N - 1 d. all of these

CORRELATION DOES OR DOES NOT EQUAL CAUSATION.

DOES NOT.

Compute chi square obt:

Determine the expected frequency if sampling is random from the null hypothesis population. - Called expected frequencies (f e)

Two Way Anova

Evaluate two IV and interaction between them.

Evaluate F obt

F obt > or = F crit REJECT H0 F obt < F crit RETAIN H0

If Fobt < Fcrit, we reject H0? T or F:

False

When doing a one-way ANOVA, increasing the sample variability will increase the power. T or F:

False Sample variability and power have an INVERSE relationship, so... Increase in sample variability = decrease in power OR Decrease in sample variability = increase in power

Chi-Square experiment with 2 categorical variables:

Find row marginal (sum) and find column marginals (sums).

Benefits of student's t test for correlated groups:

Groups treated same except for IV Groups run at the same time Same people collecting data.

Hypothesis Testing:

H1 is ALWAYS NONdirectional. H0 states that different conditions are EQUALLY effective

Evaluating Chi Square Obt

If Chi-Square obt > or = Chi-Square crit = reject H0. If Chi-Square obt < Chi-Square crit = retain H0

Within Groups Variance Estimate:

MS Within - tells us how large the differences are between the scores WITHIN each group and the group mean. - same as sw2 used for t test.

Relationship between MS between and the IV

MS between increases with size of effect of IV

Computation of Chi-Square Obt for: Chi-Square experiment with 2 categorical variables

Need to find: estimated H0 population proportions for each sample. Find it for each group = the number of subjects in that particular category/total number of subjects.

Size of effect using omega squared and eta squared:

Omega -- provides an estimate of the proportion of the total variability of Y that is accounted for by X. (use on quizzes and test) Eta -- estimate is usually larger than true size of effect but very easy to calculate and still commonly used.

One Way Anova

Only one IV, two or more levels (conditions) of IV, subjects randomly assigned to a condition.

You conduct an experiment to determine if the cingulate cortex is involved in learning tasks involving choice behavior. Twenty rats with lesions of the cingulate cortex are tested in a two choice Y-maze with the correct arm of the maze being randomly determined from trial-to-trial. Previous research with a large number of rats on this task has shown that the mean number of trials learn the task is 20 trials. The results of the experiment show a mean of 22 trials with a standard deviation of 5.4 for the lesioned rats to learn the task. What is the nondirectional alternative hypothesis?

The cingulate cortex is involved in learning tasks involving choice behavior

The cingulate cortex is not involved in learning tasks involving choice behavior

An automotive engineer believes that the engine she designed will be a gas saver. A large number of tests on engines of the old design yielded a mean gasoline consumption of 27.5 miles per gallon, with a standard deviation of 5.2. Fifteen new engines are tested. The mean gasoline consumption is 29.6 miles per gallon. What is the null hypothesis?

The new engine either uses more gas (i.e., does not save gas) or there is no difference in gas consumption between the old engine design and the new engine

The new engine saves gas.

A pharmacologist is interested in determining whether or not three different psychoactive drugs differ in how long they remain in the body before they are excreted or metabolized. The pharmacologist randomly assigned 18 subjects to three different groups. Each group is given a different type of psychoactive drug and the number of days the drugs remained at measurable levels in each subject is recorded. What is the dependent variable?

The number of days the drug remained at measurable levels.

The type of psychoactive drug

When doing a one-way ANOVA, increasing the sample size will increase power. T or F:

True

Scheffe test

Type of Post hoc: maintains type I error rate @ alpha level when controlling for ALL possible comparisons.

Tukey HSD (honestly significant difference) Test

Type of Post hoc: maintains type I error rate @ alpha level when controlling for all possible comparisons between PAIRS of means.

General rule between parametric/nonparametric tests:

Use parametric tests when possible.

How to find Chi Square crit?

Use table H. Use df and alpha level. df for one variable: df = k -1. df for two variables: df = ????? alpha levels = .05 and .01 which are "p" in the table.

Summary Points:

Use: 1. To analyze data from experiments that employ more than 2 groups or conditions. 2. Instead of many pairwise t tests in order to hold the probability of making a type I error. F test allows us to make ONE overall comparison that tells us if there is significant difference between means of the group.

Fobt value for Two Way ANOVA:

Variable A if Fobt > or = F crit there is a significant main effect for variable a. Variable B if Fobt > or = F crit there is a significant main effect for variable b. Interaction between A/B if F obt > or = to F crit there is a significant interaction effect.

F Test and Analysis of Variance

Want to test different levels of IV. Experimental design with more than two groups.

Parametric Tests:

Z test, t test, F test (ANOVA) - Are robust. - Are MORE powerful than non parametric tests and MORE versatile.

#44 practice exam Given the following frequency data: a. What is the value of χ2obt? b. What are the degrees of freedom? c. What is the value of χ2crit, with α = 0.01? d. What do you conclude? Use α = 0.01.

a) 18.00; b) df = 2; c) 9.210; d) Reject H0

#44 practice exam Given the following frequency data: a. What is the value of χ2obt? b. What are the degrees of freedom? c. What is the value of χ2crit, with α = 0.05? d. What do you conclude? Use α = 0.05.

a) 21.38; b) df = 4; d) 9.488; e) Reject H0

ANOVA Summary Table **On practice exam**

a) 221.61; b) 116.64; c) 338.25; d) 3; e) 22.80

Given the following data: ∑X1 =81 ∑X12 =1123 ∑X2 =84 ∑X22 =1214 ∑X3 =82 ∑X32 =1156 ∑X4 =53 ∑X42 =499 n1, n2, n3, n4 = 6 each N = 24 a. Using a one-way ANOVA, what is the value of Fobt? _______________ b. What is the value of Fcrit? Use α = 0.05 _______________ c. What do you conclude? Use α = 0.05 _______________ d. What type of error could have been made? _______________

a) Fobt = 5.40; b) Fcrit = 3.10; c) Reject H0; d) Type I error

Given the following data from an experiment with an independent groups design: ∑X1 =161 ∑X12 =3377 n1 = 8 ∑X2 =126 ∑X22 =2156 n2 = 8 a. What is the value of tobt? _______________ b. What is the degrees of freedom? _______________ c. What is the value of tcrit? Use α = 0.051 tail. _______________ d. What do you conclude? Use α = 0.051 tail. _______________

a) tobt = 1.86; b) df = 14; c) tcrit = 1.761; d) Reject H0

Before After 5 17 8 25 5 19 7 32 6 26 9 8 10 9 a. What is the value of tobt? b. What is the appropriate degrees of freedom? c. Using ɑ = 0.052 tail, what would you conclude about the null hypothesis using the t test? d. What type of error could have been made?

a) tobt = 3.25; b) df = 6; c) Reject H0 (tcrit = 2.447); d) Type I error

The alternative hypothesis appropriate in the one-way ANOVA is:

always NONDIRECTIONAL

Nominal Scale:

categories for units - used with variables that are qualitative in nature rather than quantitative. Ex) shoes, kinds of fruit, types of music, nationality

Post Hoc Comparison

comparisons NOT planned in advance of each experiment, not based on specific predictions nor prior research. -Conducted AFTER researcher looks at the data and decides which groups to compare (or all groups to compare) -Correct for inflated probability values that occur when doing multiple comparisons. -Maintains Type I error rate @ alpha level.

A researcher believes that if muscle tension is reduced, tension headaches would decrease or disappear. He designs an experiment using nine subjects that experience tension headaches. Subjects keep daily logs of the number of headaches during a two-week baseline period. Subjects are then trained to lower their muscle tension using a biofeedback device. Once subjects learn to lower their muscle tension, they keep a two-week log of the number of What statistical test is appropriate to analyze the results from this experiment?

correlated groups t test

Type I Error:

decision to reject the null hypothesis when the null hypothesis is actually TRUE.

Type II Error:

decision to retain the null hypothesis when the null hypothesis is actually FALSE.

Degrees freedom numerator ______________________________________________ Degrees freedom denominator

degrees freedom between ________________________________________ degrees freedom within

What are the degrees of freedom with contingency table with 3 rows and 3 columns? a. k-1 b. 1 c. 2 d. 4

df = (r-1)(c-1) df = (3-1)(3-1) df = 2*2 = 4. D

Degrees of freedom for contingency table:

df = (r-1)(c-1) r = number of rows c = number of columns

Single Sample T Test Degrees Freedom:

df = N - 1

Degrees Freedom for t test for Independent Groups

df = N - 2

Degrees Freedom for t test for Correlated Groups

df = N -1 (same as single sample t test)

ANOVA Degrees of Freedom

df between = k - 1 df within = N - k df total = N - 1

MS between increases with...

magnitude of IVs effect. MS within is UNAFFECTED.

Within group decrease variability Between group increase variability =

more likely to REJECT H0!

k =

number of groups in the experiment

Ordinal Scale:

numbers on scale represent rank orderings, rather than raw score magnitudes. - rank objects being measured according to whether they possess more, less, or same amount of variable being measured. Ex) rank ordering of top 5 contestants in a speech contest according to speaking ability. * uses words like more, less, or same, but does NOT tell us about the magnitude of the difference between adjacent units on the scale.

The frequencies obtained in the experiment are called:

observed frequencies (f o)

Discrete Variable:

one in which there are no possible values between adjacent units on scale. ex) number of children in family, number of students in a class -- all whole numbers.

Continuous Variable:

one that theoretically can have an infinite number of values between adjacent units on the scale ex) weight, height, time

one-way anova

Multiple Comparisons:

significant F value indicates that ALL conditions DON'T have the same effect on the DV.

Independent Groups Design:

simple randomized group design or one way analysis of variance

Difference between single sample t test and t test for correlated groups?

single sample t test analyzes RAW scores. correlated groups t test analyzes DIFFERENCE scores.

single sample t test; population parameters are unknown

Planned Comparision

specific comparisons that are planned in advance of each experiment based on specific predictions or prior research. -don't correct for higher probability of type I error. - higher power than post hoc comparisons.

ANOVA assumes...

that IV affects the means of scores, but not variance (same as t test)

Interval Scale:

the units represent raw score magnitudes there are equal intervals between adjacent units on the scale and NO absolute zero point. - distance between variables is meaningful. - "equal intervals between adjacent units" = equal amounts of the variable being measured between adjacent units on the scale. BUT NO ABSOLUTE 0 POINT Ex) Celcius scale -- 25 degrees celcius is 5 degrees warmer than 20 degrees celcius.

Ratio Scale

the units represent raw score magnitudes, there are equal intervals between adjacent units on the scale and there IS AN ABSOLUTE 0 POINT. Ex) centimeter scale 11cm is 1cm taller than 10 cm 0 point = 0cm = no height or length.

An experiment was conducted to assess the effects of a minor tranquilizer on a performance task at different levels of stress. The levels of stress examined were moderate and high and the levels of tranquilizer were none and moderate. What statistical test would be appropriate to analyze this experiment?

two-way ANOVA

z test; population parameters are known

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