**Statistical Methods in Psychology**

**Homework 8**

**Single Factor Independent Measures ANOVA**

- Why should you use ANOVA instead of several t-tests to evaluate mean differences when an experiment consists of three or more treatment conditions?

**With 3 or more treatment conditions, you need three or more t tests to evaluate all the mean differences. Each test involves a risk of a Type I error (which is equal to the alpha level). The more tests you do, the more risk there is of a Type I error. The ANOVA performs all of the tests simultaneously with a single, fixed alpha level.**

- Explain what the purpose of post-hoc tests is.

**ANOVA tell us that at least one mean difference is significant. It does not tell us which one, or which ones, are. If we reject the null hypothesis, Post Hoc tests are used to determine exactly which treatment conditions are significantly different from each other.**

- When should you use post- hoc tests?

**When you have rejected the null hypothesis, and then have concluded that there is at least one significant mean difference (at least two means are different from each other). And when your IV has 3 or more levels or conditions.**

**Questions 4 to 8**

The following data summarize the results from an independent measures study comparing three treatment conditions.

Use an ANOVA with α = .01 to determine whether there are any significant differences among the three treatment means

**Null Hypothesis**

**There are no significant differences among the three treatment means. **

**H _{0}: μ_{1} = μ_{2} = μ_{3} **

**Alternative Hypothesis**

**H _{1}: There is at least one significant mean difference: At least two means significantly differ from each other**

Critical F-value

N = 15

k = 3

dfbetween = k – 1 = 3 – 1 = 2

dfwithin = N – k = 15 – 3 = 12

In the *F* Distribution Table (Table B.4, pages 705 to707) we find the* F *critical associated with α = .01, and df (2,12): *F *critical = 6.93

If *F*-statistic (that we calculate in step 3 below) is greater than 6.93, we reject the null hypothesis.

**F****-Statistic**

*SS* = 16 *SS* = 12 *SS* = 20

N = 15 G = 60 _{ }ΣX^{2}= 298

- Calculate:
*SS*_{total },*SS*_{between}, and,*SS*_{within}

*SS*_{total} = ΣX^{2}_{ }– G^{2 }

^{ }*N*

*SS*_{total }= 298 – 60^{2 }= 58

15

*SS*_{within} = ∑*SS*_{inside each condition} = 16 + 12 + 20 = 48

*SS*_{between} =* SS*_{total}* – SS*_{within}

*SS*_{between} = 58 – 48 = 10

- Compute the
*df*_{between},*df*_{within}, and,*df*_{total}

*df*_{between} = k – 1 = 3 – 1 = 2

*df*_{within} = N – k = 15 – 3 = 12

- Calculate the mean squares and the F

*Source Sum of squares df Mean squares F*

Between 10 **÷** 2 = 5 = 1.25

Within 48 **÷** 12 = 4

Total 58 14

**F-statistic = 1.25 **

**Your decision is**__: __*F* = 1.25 is not greater than the *F* critical = 6.93. **Fail to reject the null hypothesis and conclude that there are no significant mean differences among the treatments**

**Questions 9 to 13**

A study surveys student to determine the amount of Facebook use during the time they are doing Math homework. Students are classified into three groups: Non- User, Rarely-Use, Regularly-Use and their Math scores are recorded. The following data summarize the results.

Use an ANOVA with α = .01 to determine whether there are significant differences among the means of the three groups.

**Null Hypothesis**

**There are no significant differences in math scores means between the three groups of Facebook users (non-user, rarely-use, regularly use). **

**H _{0}: μ_{1} = μ_{2} = μ_{3} **

**Alternative Hypothesis**

**H _{1}: There is at least one significant mean difference among the three groups: The math score means of at least two of the groups of Facebook users significantly differ from each other**

**Critical F-Value**

N = 22

k = 3

dfbetween = k – 1 = 3 – 1 = 2

dfwithin = N – k = 22 – 3 = 19

With α = .01

*In the F Distribution Table (Table B.4, pages 705 to707) we find the F critical associated with α = .01, and df (2,19): F *

If *F*-statistic (that we will calculate in step 3 below) is greater than 5.93, we reject the null hypothesis.

*F***-Statistic**

*SS* = 30 *SS* = 33 *SS* = 42

N = 22 G = 72 _{ }ΣX^{2}_{tot} = 393

- Calculate:
*SS*_{total },*SS*_{between}, and,*SS*_{within}

*SS*_{total} = ΣX^{2} – G^{2 }

^{ }*N*

*SS*_{total }= 393 – 72^{2 }= 157.36

22

*SS*_{within} = ∑*SS*_{inside each condition} = 30 + 33 + 42 = 105

*SS*_{between} =* SS*_{total}* – Ss*_{within}

*SS*_{between} = 157.36 – 105 = 52.36

- Compute the
*df*_{between},*df*_{within}, and,*df*_{total}

*df*_{between} = k – 1 = 3 – 1 = 2

*df*_{within} = N – k = 22 – 3 = 19

- Calculate the mean squares and the F

*Source Sum of squares df Mean squares F*

Between 52.36 **÷** 2 = 26.18 = 4.73

Within 105 **÷** 19 = 5.53

Total 157.36 21

*F*** = 4.73**

**Your decision is**: *F* = 4.73 is not greater than the *F* critical = 5.93.

**Fail to reject the null hypothesis and conclude that there are no significant mean differences among the three groups regarding Math scores**

**APA Report**

**There are no significant mean differences among the three groups regarding Math scores, F (2,19) = 4.73, p > .01**

**Questions 14 to 19**

The following data were obtained from an independent-measures research study comparing three treatment conditions.

Use an ANOVA with α = .05 to determine whether there are any significant mean differences among the treatments.

**Null Hypothesis**

**There are no significant differences among the three treatment means. **

**H _{0}: μ_{1} = μ_{2} = μ_{3} **

**Alternative Hypothesis**

**H _{1}: There is at least one mean difference among the three treatment means.**

**In Anova, we do not state the alternative hypothesis in symbols**

**Critical F-value**

N = 14

k = 3

dfbetween = k – 1 = 3 – 1 = 2

dfwithin = N – k = 14 – 3 = 11

In the *F *Distribution Table (Table B.4, pages 705 to707) we find the F critical associated with α = .05, and df (2,11): *F *critical = 3.98

If *F*-statistic (that we will calculate in step 3 below) is greater than 3.98, we reject the null hypothesis.

*F*-Statistic

*SS* = 14 *SS* = 9 *SS* = 10

N = 14 G = 42 _{ }ΣX^{2} = 182

- Calculate:
*SS*_{total },*SS*_{between}, and,*SS*_{within}

*SS*_{total} = ΣX^{2} – G^{2 }

^{ }*N*

* *

*SS*_{total }= 182 – 42^{2 }= 56

14

*SS*_{within} = ∑*SS*_{inside each condition} = 14 + 9 + 10 = 33

*SS*_{between} =* SS*_{total}* – Ss*_{within}

*SS*_{between} = 56 – 33 = 23

- Compute the
*df*_{between},*df*_{within}, and,*df*_{total }

*df*_{between} = k – 1 = 3 – 1 = 2

*df*_{within} = N – k = 14 – 3 = 11

- Calculate the mean squares and the F

*Source Sum of squares df Mean squares F*

Between 23 **÷** 2 = 11.5 = 3.83

Within 33 **÷** 11 = 3

Total 56 13

**F = 3.83 **

**Your decision is: ***F* = 3.83 is not greater than the *F* critical = 3.98. **Fail to reject the null hypothesis and conclude that there are not significant differences among the conditions**

Use η^{2}to measure the effect size for this study.

** **η

**η ^{2} = 23/56 = .41 (41%)**

**APA report**

**The ANOVA indicates that there were not significant mean differences among the three groups, F (2,11) = 3.83, p > .05,**

The price is based on these factors:

Academic level

Number of pages

Urgency

Basic features

- Free title page and bibliography
- Unlimited revisions
- Plagiarism-free guarantee
- Money-back guarantee
- 24/7 support

On-demand options

- Writer’s samples
- Part-by-part delivery
- Overnight delivery
- Copies of used sources
- Expert Proofreading

Paper format

- 275 words per page
- 12 pt Arial/Times New Roman
- Double line spacing
- Any citation style (APA, MLA, Chicago/Turabian, Harvard)

Delivering a high-quality product at a reasonable price is not enough anymore.

That’s why we have developed 5 beneficial guarantees that will make your experience with our service enjoyable, easy, and safe.

You have to be 100% sure of the quality of your product to give a money-back guarantee. This describes us perfectly. Make sure that this guarantee is totally transparent.

Read moreEach paper is composed from scratch, according to your instructions. It is then checked by our plagiarism-detection software. There is no gap where plagiarism could squeeze in.

Read moreThanks to our free revisions, there is no way for you to be unsatisfied. We will work on your paper until you are completely happy with the result.

Read moreYour email is safe, as we store it according to international data protection rules. Your bank details are secure, as we use only reliable payment systems.

Read moreBy sending us your money, you buy the service we provide. Check out our terms and conditions if you prefer business talks to be laid out in official language.

Read more