Chapter 3. Hypothesis. 2018.ppt

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What is a Hypothesis?



Dr. Rosa Padilla de Casamayor

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What is a Hypothesis?

Population mean

(







Population proportion (



)

Example: The mean monthly cell phone bill the student of AUCA is  = 10000 Rwf

Example: The proportion of AUCA students who have a Smartphone is p = .80

A tentative, reasonable, testable assertion regarding the occurrence of certain behaviors, phenomena, or events. A prediction of study outcomes.

Research hypothesis

The Null hypothesis

Stressed children do not differ from normal children in terms of behavior problems

Other example:

Are sexually active teenagers any better informed about IDS and other potential health problem related to sex than teenagers who are sexually inactive? A 15-item test of general knowledge about sex and health was administered to random samples of teens who are sexually inactive, teens who are sexually active but with only a single partner, and teens who are sexually active with more than one partner. Is there any significant difference in the test scores?

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Purpose of hypothesis testing

• The purpose of hypothesis testing is to determine whether there is

enough statistical evidence in favor of a certain belief about a

parameter.

• A hypothesis may be precisely defined as a tentative proposition

suggested as a solution to a problem or as an explanation of some

phenomenon. (Ary, Jacobs and Razavieh, 1984)

Example

:

“There is no significant

difference in the anxiety level of

children of High IQ and those of

low IQ.”

Achievement and Enrollment Status of Suspended Students

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The critical concepts are these:

1. There are two hypotheses: the null and the alternative hypotheses.

2. The procedure begins with the assumption that the null hypothesis is

true.

3. The goal is to determine whether there is enough evidence to infer

that the alternative hypothesis is true, or the null is not likely to be

true.

Statistical Hypothesis Testing

4. There are two possible decisions:

Reject the null: To conclude that

there is enough evidence to infer

that the alternative hypothesis is

true.

Fail to reject the null: To

conclude that there is insufficient

evidence to support the

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• The Null Hypothesis is a statement about the population value that will be tested. This hypothesis will be rejected only if the sample data provide substantial contradictory evidence.

Research Hypothesis

“The mean monthly cell phone bill the student of AUCA is less than 10000 Rwf”

Example of the Null Hypothesis:

The mean monthly cell phone bill the student of AUCA is at least 10000 Rwf.

The Null Hypothesis (Ho) and Alternative Hypothesis (Ha)

H₀:  10000

•Always contains “=” , “≤” or “” •May or may not be rejected

The Alternative Hypothesis is the opposite

of the null hypothesis

Example

The mean monthly cell phone bill the

student of AUCA is less than 10000

Rwf.

H_a:  < 10000

•Never contains the “=” , “≤” or “” sign

•May or may not be accepted •Is generally the hypothesis that is believed (or needs to be supported) by the researcher.

This is

what

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An alternative hypothesis is a statement that suggests a potential

outcome that the researcher may expect. (H1 or Ha)

• Comes from prior literature or studies.

• It is established only when a null hypothesis is rejected.

• Often an alternative Hypothesis is the desired conclusion of the

investigator.

• The two types of alternative hypothesis are: Directional Hypothesis

Non-directional Hypothesis.

Alternative Hypothesis (Ha)

Directional: Is a type of alternative hypothesis that specifies the direction of expected findings. Sometimes directional hypothesis are created to examine the relationship among variables rather than to compare groups. Directional hypothesis may read,”…is more than..”, “…will be lesser..”

Example: “Children with high IQ will exhibit more anxiety than children with low IQ”

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.

Statement:

the mean life

expectancy in Rwanda,

2018 is 60.93 years

(Geoba.se)

is 60: x = 60 years

If X=60 likely if Ho:



≤ 60.93

REJECT

Null Hypothesis

If not likely,

Hypothesis Testing Process

Suppose the

sample mean of the Life expectancy H0:  = 60.93

Ha:  ≠ 60.93

Sample

Population

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To make a decision we need to interpret Sig. or P_value

The smaller the p-value, the more statistical evidence exists to support the

alternative hypothesis.

•If Sig.

is less than 1%, there is

overwhelming evidence

that supports the

•If Sig.

is between 1% and 5%, there is a

strong evidence

that supports the

•If Sig.

is between 5% and 10% there is a

weak evidence

that supports the

•If Sig.

exceeds 10%, there is

no evidence

that supports the alternative

hypothesis.

Overwhelming

Evidence

(Highly

Significant)

Strong Evidence

(Significant)

Weak Evidence

(Not Significant)

No Evidence

(Not Significant)



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Actual

situation

Our

decision

Null (Ho)

hypothesis is

false

Null (Ho)

hypothesis is

true

Reject the null

(Ho) hypothesis

Correct

decision

Type I

error (α)

Called Level of Significance

Do not reject the

null (Ho)

hypothesis

Type II

error (β)

Correct

decision

(1-β)

Types of Errors

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Steps in Hypothesis Testing

1. Specify the population value of interest.

2. Making assumptions and meeting test requirements

:

Random sampling, quantitative variable, normal

population distribution (robustness?)

3. Formulate the appropriate null and alternative

hypotheses.

4. Specify the desired level of significance.

5. Determine the rejection region or p_value (Sig.).

6. Obtain sample evidence and compute the test statistic.

7. Making a decision and interpreting the results of the

test.

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Example 1: Upper Tail t Test for Mean

The economy minister announced that the salaries of primary school teachers are better compared to the last 5 years. To prove this, we take a sample of the monthly salaries of teachers in Rwanda. The following data shows average salaries in ten local school districts.

a. Find the 95% confidence interval of the average teacher salary in Rwanda. b. According to "The New Times", it is said that the minimum monthly salary is Rwf 80,000 for primary teachers. Test if salaries are really improving significantly to 95% confidence.

1. Specify the population value of interest

Mean monthly salary of primary teachers in Rwanda

2. Formulate the appropriate null and alternative hypotheses

 Ho: μ _≤ 80,000

 Ha: μ > 80,000 (This is a lower tail test)

3. Specify the desired level of significance

  = .05 is chosen for this test

4. Statistic

5. Make decision and interpret result

Note: Then pass the data in SPSS software

Monthly wage

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Step in SPSS

First, fill the data in SPSS, them to do this, click on Analyze, and then

Compare means followed by One-Sample T test and then continue the followed steps that shown in the figure bellow:

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Output from SPSS

Note:The value of p_value or Sig. gives us the SPSS default is bilateral, unilateral if we value: Sig /2 (.000/2 = .000)

Making decision: Reject null hypothesis because Sig. = .000 <  = .05

Interpretation: The difference found is highly significant therefore we can conclude at 5% of significance level, there is evidence than supports the alternative

hypothesis, i.e. teacher salaries are increasing significantly if we compare previous years

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• Decision with Sig. or p_value with =0.05

– If Sig. <  , reject H_0, If Sig.   , do not reject H₀

Dr. Rosa Padilla de Casamayor One-Sample Statistics

N Mean

Std.

Deviation Std. Error Mean Salaries of

primary teachers

10 70500.00 29997.22 9485.95

One-Sample Test

Test Value = 80 t df Sig. (2-tailed)

Mean Difference

95% Confidence Interval of the Difference

Lower Upper Salaries of

primary teachers

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Comparison Test according the objective and the type of variable

15 Dr. Rosa Padilla de Casamayor

Comparing:

Dependent

(outcome)

Variable

Independent

(explanatory)

variable

Parametric

test (data is

normally

distributed)

Non-parametric test

(ordinal/ skewed data)

or 'assumption not

assumed'

The average score of

two independent groups

Scale

Nominal

(binary)

Independent

t-test

Mann-Whitney

test/wilcoxon rank

sum

The average of 3+

independent groups

Scale

Nominal

One-way

ANOVA

Kruskal-Wallis test

The average difference

between paired

(matched) samples

'before and after'

Scale

Time/

condition

variable

Paired t-test

Wilcoxon signed rank

test

The 3+ measurements

on the same subject

Scale

Time/

condition

variable

Repeated

measures

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Association Test according the objective and the type of variable

Test of association

Dependent (outcome) Variable Independent (explanatory) variable Parametric test (data is

normally distributed)

Non-parametric test (ordinal/ skewed

data

Relationship between 2

continuos variables Scale Scale

Simple Pearson's Correlation Coefficient Spearman's Correlation Coefficient Predicting the value of one

variable from the value of a predictor variable or looking

for significant relationships Scale Any

Simple Linear or non-linear

Regression Transform the data

Predicting the value of one variable from the value of a

predictors variable or looking for significant

relationships Scale Any (more than one variable) Simple Linear or non-linear Regression Nominal (Binary) Any Logistic regression Assessing the relationship

between two or more

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t

-Test for Comparing Two Means

Example

: Researchers are

interested in test anxiety. They

administered an inventory of

anxiety to the students just

before the final exam in a

Sociology class. They also

administered it before the

final exam in a business class.

To compare the two sets of

scores, they use

t-test for independent samples

Example

: Researchers are

interested in exam anxiety.

They administer an anxiety

inventory on the second day of

class. Then they give it again on

the day of the midterm. To

compare the two sets of

scores,

they use

t-test for paired samples

Anxiety test score

Student

Second day of class

Midterm

1 67 89

2 45 55

3 75 70

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Determining when an independent samples t-test is appropriate:

Is the dependent variable interval or ratio?

Can the dependent variable scores be grouped based upon some categorical variable?

Does the grouping result in scores drawn from independent samples?

Are two groups involved in the research question?

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t

-Test for Comparing Two Means (independent) Cont’d

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Assumption when we use t test (parametric test)

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Remember that for proper use of the distribution "t" or normal distribution "Z", the data must satisfy the following assumptions:

Assume that the random samples are independent

Level of measurement of dependent variable is interval-ratio

Randomness: samples were selected using a probabilistic method. Otherwise inference is not applied.

Normality: The variables of analysis, in both populations are normally

distributed. (Boxplot, histogram with normal curve, Normal Q-Q plot, Shapiro-Wilk, KS, etc.). If not satisfy these conditions do using a nonparametric test or you transform your variable .

Homogeneity of variances: The population variances are not different. That is: (Levene test, F, etc.). If not corrected the number of degrees of freedom and

used the t test cuff applies a nonparametric test. When samples are very unequal are more likely to violate this assumption.

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Example for hypothesis testing between two independent groups

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a. Do the results indicate that there is a significant difference in the average score with respect to the different teaching approaches?

b. If the assumption of homogeneity of variance met? (Levene's test for equality of variances).

c. What is the value of the t test?

d. How many degrees of freedom are there? e. What is the obtained p value or Sig.?

f. What is the dependent and independent variable?

Solution

a. Formulate the appropriate hypothesis: B A B A

Ha

Ho







:

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Step in SPSS for hypothesis testing between two independent groups

Step using SPSS (Create data file)

Enter this data in SPSS. (TIP: to do this, you must enter two columns of data: one column for the respective test scores and the other for the class (the first 12 rows will have a 1 indicator to indicate reading and the second 12 rows will have a 2 indicator for indicate computer.) See the figure below:

Lecture Based

Approach 10 23 11 17 7 4 18 11 11 14 10 19 Computer Based

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Step in SPSS for hypothesis testing between two independent groups

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To do this, click on Analyze, and then Compare means followed by

independent samples T test and then continue the followed steps as shown in the figure below < continue < OK

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Interpretation: the report shows the descriptive statistics, the average score of Computer Based Approach is greater than the average of lecture approach, and standard deviation for both are similar; but do not know whether this difference observed is significant.

So we ask the t test for independent samples, which we gives t = -4.422, looking at the next Sig. (2-tailed) the value is .000, lower than .05.

Decision rule: given that Sig. < 0.05 was rejected Null hypothesis, and we conclude that the computer approach produced greater score than the Lecture approach, with Sig. p = 0.000

Output from SPSS

Independent Samples Test

Levene's Test for Equality

of Variances t-test for Equality of Means

F Sig. t df Sig. (2-tailed)

Statistics exam score

Equal variances

assumed .289 .596 -4.422 22 .000 Equal variances

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Assumption

24 Basic assumptions:

To check if the variable is normally distributed, the following steps in the SPSS

Analyze <descriptive statistics <explore <follow the steps as shown in the figure on the side <accept

Box plot (to check if there are no outlier values and if the boxes behave symmetrically

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Assumption

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Making a Decision and Interpreting the Result of the Test

We observed Shapiro-Wilk statistic given the p_values or (Sig.) Lecture: Sig= .778

Computer: Sig= . 640

Decision: From Shapiro-Wilk test of normality are both Sig. greater than 0.05, therefore we don’t reject Null hypothesis, which imply that it is acceptable to assume that the average of score for both groups follow normal distribution (or bell-shaped).

Hypothesis testing to determine the normality

Ho: The variables follow a normal distribution

Ha: Variables do not follow a normal distribution

Tests of Normality

Approach

Kolmogorov-Smirnova _Shapiro-Wilk

Statistic df Sig. Statistic df Sig. Statistics

exam score

Lecture Based Approach

.221 12 .109 .960 12 .778

Computer Based Approach

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Assumption

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Assumption of Homogeneity

Through the Levene test can see if this assumption very important to compare groups met.

The report of SPSS gives without asking

Ho: the variances are equal

Ha: not assume equal variances

Decision: The p-value or Sig.=596, provides the Levene test trough Sig. is greater than 5%, and then we can not reject Ho and conclude that equal variances assumed.

Independent Samples Test

Levene's Test for Equality

of Variances t-test for Equality of Means

F Sig. t df Sig. (2-tailed)

Statistics exam score

Equal variances

assumed .289 .596 -4.422 22 .000 Equal variances

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Hypothesis test for Dependent sampling – matched pairs

t-test (paired or related samples)

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One of the most common experimental designs is the "pre-post" design or “before and after”. A study of this type often consists of two measurements taken on the same subject or one measurement taken on a matched pair of subjects, one before and one after the introduction of a treatment or a stimulus. The basic idea is simple. If the treatment had no effect, the average difference between the measurements is equal to 0 and the null hypothesis holds. On the other hand, if the treatment did have an effect (intended or unintended!), the average difference is not 0 and the null hypothesis is rejected.

The Paired-Samples t test procedure is used to test the hypothesis of no difference between two variables.

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Steps of a Hypothesis test for paired comparisons or related

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Example for paired comparisons or related

Athlete

Weight

before

Weight

after

1

127

135

2

195

200

3

162

160

4

170

175

5

143

147

6

205

200

7

168

172

8

175

186

9

197

194

10

136

130

10 athletes were subjected to a program of intensive physical training

by the coach. Their weights were recorded (in pounds) before and after

the training with the following results:

Does it affect the program the average weight of athletes?

Step for fill the data when is related

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Bivariate Tests of Differences

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Output from SPSS

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Interpretation: the report shows the descriptive statistics, the average weight before shows a slight difference with the weight after implementing the training, but do not know whether this difference observed is significant.

So we ask the t test for related samples, which we gives t =- 1.156. Looking at the next (Sig. (2-tailed) the value is .277, lower than proposed (.05).

Note: If our hypothesis is one-tailed, and the SPSS always does two-tailed; then to convert into a tail you have to divide between two, that is, 0.277 / 2 = 0.139. Therefore, the value of p for this problem is 0.139

Decision rule: Sig> 0.05, at a level of significance of 5% we can not reject Ho, we say that the training was not effective with respect to the increase or decrease of the average weight of the athletes. Dr. Rosa Padilla de Casamayor

Paired Samples Statistics

Mean N Std. Deviation Std. Error Mean

Pair 1 Before 167.800 10 26.578 8.405

After 169.900 10 26.066 8.243

Paired Differences

t df

Sig. (2-tailed) Mean

Std. Deviation

Std. Error Mean

95% Confidence Interval of the

Difference

Lower Upper Pair 1 Before -

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Many analyses involve experiments in which you want to test whether

differences exist in the means of more than two groups. Evaluating

differences between groups is often viewed as a one-factor experiments

(also known as a completely randomized design) in which the variable

that defines the groups is called the factor of interest.

One-way analysis of variance - ANOVA

A particular combination of factor levels, or categories, is called a

treatment.



One-way analysis of variance

involves only one categorical

variable, or a single factor. In one-way analysis of variance, a

treatment is the same as a factor level.



If two or more factors are involved, the analysis is called

n

-way

analysis of variance.

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Relationship Amongst Test, Analysis of Variance,

Analysis of Covariance & Regression

One Independent One or More

Metric Dependent Variable

t Test Variable

One-Way Analysis of Variance One Factor

N-Way Analysis of Variance

More than One Factor Analysis of

Variance Categorical:

Factorial

Analysis of Covariance Categorical and Interval

Regression Interval Independent Variables

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The following assumptions must be true before the ANOVA technique can be applied to a decision-making situation:

The samples are drawn randomly, and each sample is independent of other samples.

The error term is normally distributed, with a zero mean and a constant variance.

The populations have equal variances. The assumption of homogeneity of variance can be tested using tests such as Levene’s test or the Brown-Forsythe Test

• If the Levene’s test is not significant (p >.05), assume equal variances

• If Levene’s test is significant (p <.05), equal variances cannot be assumed (this information makes adjustments to the violation of equal variances).

 The error terms are uncorrelated. If the error terms are correlated (i.e., the observations are not independent), the F ratio can be seriously distorted.

Model Assumptions - ANOVA

It is important to note that ANOVA is not robust to violations to the assumption of independence. This is to say, that even if you violate the assumptions of homogeneity or normality, you can conduct statistical procedures that will still enable you to conduct the ANOVA but you cannot with violations to independence. In general, with violations of homogeneity the study can probably carry on if you have equal sized groups.

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• If the null hypothesis of equal category means is not rejected, then

the independent variable does not have a significant effect on the

dependent variable.

• On the other hand, if the null hypothesis is rejected, then the effect of

the independent variable is significant.

• A comparison of the category mean values will indicate the nature of

the effect of the independent variable.

Conducting One-way Analysis of Variance -

Interpret the Results

Illustrative Applications of One-way ANOVA

Are sexually active adolescents better informed about AIDS and other possible health problems related to sex than sexually inactive adolescents?

A 15-item test of general knowledge about sex and health was administered to random samples of teens who are sexually inactive, teens who are sexually active but with only a single partner, and teens who are sexually active with more than one partner. Is there any significant difference in the test scores?

The data is shown on the next slide:

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Steps in SPSS: knowledge about sex and health (AIDS)

Inactive

Active-One partner

Active-More than

one partner

14 11 8

12 11 12

8 6 10

14 5 4

11 12 3

12 10 5

1

2

To do this, click on

Analyze, and then

Compare means

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Output from SPSS - Descriptives

Descriptives

General knowledge about sex and health (AIDS) in teenagers

N Mean

Std.

Deviation Std. Error

95% Confidence Interval for Mean

Minimu m

Maximu m Lower

Bound

Upper Bound

Inactive 6 11.833 2.229 0.910 9.495 14.172 8.00 14.00

Active-One

partner 6 9.167 2.927 1.195 6.095 12.238 5.00 12.00

Active-More than one partner

6 7.000 3.578 1.461 3.245 10.755 3.00 12.00

Total 18 9.333 3.447 0.812 7.619 11.048 3.00 14.00

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Output from SPSS - ANOVA

ANOVA

Sum of

Squares df Mean Square F Sig. Between Groups _70.333 ₂ _35.167 _4.006 _.040 Within Groups _131.667 ₁₅ _8.778

Total 202.000 17

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Output from SPSS

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Output from SPSS – Post Hoc Analysis

Looking at the data, the researcher ask:

• Are the two groups between the inactive and active-more than one partner really different? (Sig. = 0.032, therefore the difference between the two groups are statistically significant)

Dependent Variable: Tukey HSD

(I) Group

Mean Difference

(I-J) Std. Error Sig.

95% Confidence Interval Lower Bound Upper Bound Inactive Active-One

partner 2.667 1.711 0.293 -1.776 7.110 Active-More

than one

partner 4.833

* _1.711 _0.032 _0.390 _9.276

Active-One partner

Inactive -2.667 1.711 0.293 -7.110 1.776 Active-More

than one

partner 2.167 1.711 0.435 -2.276 6.610 Active-More

than one partner

Inactive _4.833* _1.711 _0.032 _-9.276 _-0.390

Active-One

partner -2.167 1.711 0.435 -6.610 2.276 *. The mean difference is significant at the 0.05 level.

When the Null Hypothesis is rejected in one factor ANOVA, the conclusion is that not all means are the same. This

however leads to an obvious question: Which particular

means are different? Seeking further

information after the results of a test is called post hoc ‐

analysis.

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Output from SPSS – Homogeneous subset

The significance value of the F test in the ANOVA table is .040. Thus, you must reject the hypothesis that average scores are not equal across groups. According to the before table, teenagers who are inactive sexually have more highly score than their counterparts.

General knowledge about sex and health (AIDS) in teenagers

Tukey HSDa

Group N

Subset for alpha = 0.05

1 2

Active-More than one

partner 6 7.0000

Active-One partner 6 9.1667 9.1667

Inactive 6 11.8333

Sig. .435 .293

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• Each group is approximately normal, check this by looking at histograms and boxplot or normal Q-Q plots, or use the test Kolmogorov Smirnov if the sample is big, or use test Shapiro Wilk if the sample is small (< 30).

Assumption: Normality Check

P-values>.05, then do not reject Ho, therefore we conclude that the error term follow a normal distribution or the normality assumption may be assumed valid.

Ho: The errors term follow a normal distribution Ha: The errors term do not follow a normal distribution

Tests of Normality

Group

Statistic df Sig. Statistic df Sig. General

knowledge about sex and health (AIDS) in teenagers

Inactive .196 6 .200* _.890 ₆ _.316

Active-One

partner .279 6 .159 .838 6 .126

Active-More than one partner

.212 6 .200* _.935 ₆ _.619

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This table (Levene’s test) tests the assumption of equal variances for the ANOVA – this is the same assumption found in the t-test but in another table. Look at the sig. or p-value - the value is .207 which is above .05. The result indicates that equal

variances assumption is met.

Therefore the variances are equal.

Box plot (to check if there are no outlier values and if the boxes behave

symmetrically)

ASSUMPTION: homogeneity of variance

Test of Homogeneity of Variances

Levene Statistic df1 df2 Sig.

1.750 2 15 .207

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• Each group is approximately normal

check this by looking at histograms and or normal quantile plots, or use the test Kolmogorov Smirnov if the sample is big, or use test Shapiro Wilk if the sample is small (< 30).

Assumption: Normality Check

P-values>.05, then do not reject Ho, therefore conclude that the

variables follow a normal distribution Ho: The errors terms follow the normal

distributed

Ha: The errors terms were not follow the normal distributed

Tests of Normality

Group

Statistic df Sig. Statistic df Sig. General knowledge

about sex and health (AIDS) in teenagers

Inactive _.196 ₆ _.200* _.890 ₆ _.316

Active-One

partner .279 6 .159 .838 6 .126 Active-More

than one

partner .212 6 .200

* _.935 ₆ _.619

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Steps for the normality plots with test

6

2

3

4

5

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Multivariate Analysis of Variance

•

Multivariate analysis of variance (MANOVA)

is similar to

analysis of variance (ANOVA), except that instead of one

metric dependent variable, we have two or more

, based

on their relationships to categorical and scale predictors.

•

In MANOVA, the null hypothesis is that the vectors of

means on multiple dependent variables are equal across

groups.

•

Multivariate analysis of variance is appropriate when

there are two or more dependent variables that are

correlated.

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1. Everitt, as reported in Hand et al. (1994), presented data on the amount of weight gained by 17 anorexic girls under one of three treatment conditions. One of the conditions were Cognitive Behavior Therapy. The weight gain of each anorexic girls who received cognitive behavioral therapy, what null hypothesis would we be likely to test in this situation? Answer: t=.156, sig. = .878

The researcher considers that the program was successful with respect to gaining weight in the girls if on average it is more than 3 pounds. (Note. Interpret descriptive statistics)

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2. A researcher investigating the effectiveness of different forms of advertising randomly selects ten subjects to take part in an experiment to determine if reaction time to a visible stimulus is different from reaction time to an audible stimulus. The null hypothesis (Ho) in this case is that there is no difference between reaction time to visual and auditory stimuli. The data are set in the following table:

Subjects A B C D E F G H I J Reaction time to

visual stimulus

(m. secs) 259 275 304 285 288 314 291 304 285 246 Reaction time to

auditory stimulus

(m. secs) 201 198 245 287 190 250 285 295 231 201 ID 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Gain 1.7 0.7 -0.1 -0.7 -3.5 14.9 3.5 17.1 -7.6 1.6 11.7 6.1 1.1 -4.0 20.9 -9.1 2.1

Assignment 3

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3. The data are from study involving the emotionality of children from lone-parent and two-parent families. The independent variable is family type which has two levels (the lone-parent type and two-lone-parent type). The dependent variable is emotionality on a standard psychological measure- in score. Is there a significance different between the two groups in terms of their emotionality? Answer: t=2.813, Sig. = 0.011

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4. A company wants to study the effect of the interruption of work on the productivity of its workers. 7 workers were selected and their productivity is measured on an ordinary day, and then the same workers are re-measured but given a break in the work. The

productivity figures measured are the following:

Do these results indicate that rest from work increases productivity?

•Conduct the appropriate statistical test of your hypothesis, using a significance level  = 0.05.

Two-parent family 12 18 14 10 19 8 15 11 10 13 15 16 Lone-parent family 6 9 4 13 14 9 8 12 11 9

5. They have total cholesterol levels of a sample of eight patients before and after participating in a diet-exercise program. Can be concluded that the program had positive impact?

Patient 1 2 3 4 5 Before 201 231 221 260 228 After 200 236 216 233 224

Assignment 3

Answer: t=-2.977, sig. = .025

Without pause 23 35 29 33 43 40 32 With pause 28 38 39 37 42 56 38

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49 6. Do athletes in different sports vary in terms of intelligence? Bellow is reported College Board

scores of random samples of colleges’ basketball and football players. Is there a significant difference?

Conduct the appropriate statistical test of your hypothesis, using a significance level  = 0.05

7. A Social Work Department wishing to validate an empathy scale gives the measure to intending Social workers and to a group of student matched by age and sex whose career choices are other than Social Work. The following table shows the scores of the two groups on the empathy measure. Do the scores indicated that intending Social Workers have higher empathy scores than non-Social Workers? The null hypothesis in this case is that there is no significant difference between the two groups. Answer: t= 5.243, Sig.=.000 (Check assumptions)

Social workers 80 79 78 69 68 78 75 74 73 81

Non-social worker 68 71 58 62 52 67 63 70 59 61

8. A researcher might compare the amount of bulling in two schools, one with a strict and punitive policy and the other with a police of counseling on discipline infringements. A sample of children from each school is interviewed and the number of times they have been bullied in the previous school year obtained. The independent variable is policy on discipline (which has two levels year obtained. The independent variable is policy on discipline (which has two levels – strict versus counseling); and the dependent variable is the number of times a child has been bullied in the previous school year.

Answer: t= -.178, Sig. = .870

Is a difference significantly between the two groups of scores?

Assignment 3

Strict policy 8 5 2 6 7

Counselling 12 1 3 10 4

Answer: t= -.416, Sig. = .688

Basketball players 90 70 50 56 45

(50)

a. Is a difference significantly between the two groups of scores? Answer: F=47.286, sig .000

b. Which differences are significant?

Assignment 3

9. Three groups of students involved in the same curriculum are obliged to study for 15 minutes, 30 minutes, or 90 minutes a night for 8 weeks before taking a mathematics test. Their scores are as follows:

a.Are the differences statistically significant? Answer: F=4.57, sig .033

b.If F is significant, which group(s) is(are) significantly different from which? (15’ compare 90’) c.How much of the difference in mathematics performance can be explained by how long students study?

d.What is the independent variable?

e.What is the data scale of the independent variable? f.What is the dependent variable?

g.What is the data scale of the dependent variable?

10. Here are the number of errors on an analysis of variance problem for each of nine students after drinking grape Kool-Aid, Frankenberry Punch, or water.

H2O GKA FBP

5 8 10

3 7 11

4 7 12

15 minutes 43 39 55 56 73

30 minutes 55 58 66 79 82

(51)

Rosa Padilla Castro de Casamayor

En Excel: insert function(fx)<select category Statistical<select function (NORMSDISTR) Write the value (1.43) =0.9236 . When Probability en excel >, then (1-0.9236=0.0764)

Do your best to present yourself to God as one approved, a worker who

does not need to be ashamed and who correctly handles the word of truth

2 Timothy 2:15

.

t-test for independent samples

t-test for paired samples