Announcements

• Wednesday (10/2)
• Lab: Article Worksheet #2 due by 3:59 pm
• Lab: APA style guide

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• Next Mon (10/7, 11:59 pm)
• Problem Set #3 due (pls go through and let this monitor your study progress)

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• Next Wed (10/9)
• No Lab

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Statistics:�Independent-groups t-test�Hypothesis testing with one-tailed and two-tailed testing

(**Calculator needed from this point on)

Learning Objectives

• Go through the steps of hypothesis testing (will be in exam)

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• Know the difference between the null and alternative hypothesis

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• Know the difference between one and two-tailed tests and when to use each

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• Know how to look up critical values (a form, calculate df)

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• Calculate effect size

Statistics:�Independent-groups t-test�Hypothesis testing with one-tailed and two-tailed testing

Independent-groups Design/ t-test

Research Q: Evaluate the mean difference between TWO populations (or experiment conditions)

Examine with TWO samples

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• Group A drinks coffee first thing in the morning, while Group B drinks it at 3 PM (when life really starts to feel like a struggle). You want to know if the timing of coffee affects productivity levels.
• Does watching 2 hours of Netflix or 2 hours of TikTok make you happier?
• Do power naps make you more productive than powering through on a Celsius?

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Example Research Questions:

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Scrolling through TikToks makes you happier than scrolling through Instagram reels

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We almost never have any information about the

Entire Population (Mean, Variability)

Therefore, we use

TWO samples to make inferences about these populations.

TikTok Group

IG Reels Group

n=21

n=21

Four Steps of Hypothesis Testing �

1. State the hypothesis/ hypotheses (null vs. alternative)
2. Set the criteria for a decision
3. Collect data and compute sample statistics
4. Make a decision

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What is IV? DV?

Hypothesis: People who scroll TikTok will score higher on a happiness score than people who scroll IG Reels.

What is IV? DV?

Hypothesis: People who scroll TikTok will score different happiness score than people who scroll IG Reels.

Pre-Step: Prepare Sample Data

Collect & Calculate the following information:

Sample size= n

Sum of scores= ∑X

Sample mean= M= ∑X / n

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Deviation Score= X-M

Sum of Squares (SS)= Sum of Squared Deviations= ∑ (X-M)²

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It will be easy, we will go through them later!

Step 1: State the Hypothesis

H₀: Null hypothesis

Null hypothesis suggests that in the general population, there is no change, no difference, no relationship

🡪 Says that the IV has NO effect in DV

H₁ : Alternative hypothesis

Alternative hypothesis suggests that in the general population, there is a change, a difference, a relationship

🡪 Says that the IV HAS an effect in DV

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Step 1: State the Hypothesis

H₀: Null hypothesis

There is no difference in mean happiness scores between TikTok users and IG Reels users.

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H₁ : Alternative hypothesis

There is a difference in mean happiness scores between TikTok users and IG Reels users

Things to keep in mind about Step 1

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• H₀ and H₁ are mutually exclusive. Cannot both be true and one of them must be true.

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• One-tailed vs. Two-tailed

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Hypothesis: TikTok users are more likely to score higher on happiness scores than IG Reels users.�

One-tailed, Directional Hypothesis

Hypothesis: TikTok users differ from IG Reels users in their overall happiness scores.�

Two-tailed Hypothesis

One-tailed vs. Two-tailed

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Two tail

H₀: μ₁ = μ₂

Null Hyp: Population mean of Group 1 equals Population mean of Group 2.

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H₁: μ₁ ≠ μ₂

Alternative Hyp: Population mean of Group 1 differs from Population mean of Group 2.

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One Tail

H₀: μ₁ ≤ μ₂

Null Hyp: Population mean of Group 1 will be less than or equal to the Population mean of Group 2.

H₁: μ₁ > μ₂

Alternative Hyp: Population mean of Group 1 will be a higher than the population mean of Group 2.

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Step 1:

Step 2: Set the Criteria for a Decision

Set the decision criteria (Using the t-table)

• Sketch out the distribution of sample means and indicate the Critical Region

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2 tail

α= .05 (5%)

α= .01 (1%)

α= .001 (0.1%)

Degrees of freedom

Degrees of Freedom (df)

From your formula sheet 🡪

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Degrees of Freedom (df)

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Determined by the df of TWO separate samples

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df = (5 – 1) + (5 – 1)

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Sample 5 students from Tiktok

Sample 5 students from IG Reels

= 8

Critical t for One-tailed &

Two-tailed Hypothesis

Two-tailed hypotheses:

• Add ± to the value = ± 2.306

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Critical t for One-tailed &

Two-tailed Hypothesis

Two-tailed hypotheses:

• Add ± to the value = ±2.101

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One-tailed hypothesis:

• If the alternative hypothesis states “greater than”: Positive value= +1.860
• If the alternative hypothesis states “less than”: Negative value= −1.860

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Set the decision criteria

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T_critical = ± 2.306

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BOX IT STAR IT REMEMBER IT!!

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Step 2:

Step 1:

Step 2:

Step 3:

Step 4:

Set the null and alternative hypotheses - H₀ & H₁

Set the decision criteria

Use the t-table to find your critical value(s).

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“Collect data” & Compute Sample Statistics

Make a decision + Report in perfect APA style

Remember these steps!

Step 3: Calculate Statistics (t-value)

Independent-groups t statistic (compare the means of two groups)

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Sample Mean difference (between two groups)

Population Mean difference

from Null Hypothesis (usually they are 0 bc null)

Estimated Standard Error

(standard deviation of a sample population)

The reason why we need to calculate this is because data are randomly collected, participants perform differently, and there is individual difference

Sample Mean difference (between two groups)

The reason why we need to calculate this is because data are randomly collected, participants perform differently, and there is individual difference

M₁ = 4

M₂ = 2

= 2

Independent-groups t statistic (compare the means of two groups)

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Population Mean difference

from Null Hypothesis (usually they are 0 bc null)

Estimated Standard Error

(standard deviation of a sample population)

The reason why we need to calculate this is because data are randomly collected, participants perform differently, and there is individual difference

2

Independent-groups t statistic (compare the means of two groups)

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Estimated Standard Error

(standard deviation of a sample population)

The reason why we need to calculate this is because data are randomly collected, participants perform differently, and there is individual difference

2

0

Estimated Standard Error

(standard deviation of a sample population)

The reason why we need to calculate this is because data are randomly collected, participants perform differently, and there is individual difference

Estimated Standard Error

(standard deviation of a sample population)

The reason why we need to calculate this is because data are randomly collected, participants perform differently, and there is individual difference

Estimated standard error formula for Unequal number of groups

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Total amount of error involved in using TWO sample means to approximate TWO population means

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Pooled Variance (pool here means group)

Sample size for Sample 1

Calculate SS₁ and SS₂

M₁ = 4

M₂ = 2

- 4 = 0 🡪 Square the value = 0

- 4 = 0 🡪 Square the value = 0

- 4 = 1 🡪 Square the value = 1

- 4 = -1 🡪 Square the value = 1

- 4 = 0 🡪 Square the value = 0

SUM them = 2

- 2 = -1 🡪 Square the value = 1

- 2 = -1 🡪 Square the value = 1

- 2 = 1 🡪 Square the value = 1

- 2 = 1 🡪 Square the value = 1

- 2 = 0 🡪 Square the value = 0

SUM them = 4

SS₁

SS₂

Estimated standard error formula for Unequal number of groups

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Total amount of error involved in using TWO sample means to approximate TWO population means

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2 + 4

4 + 4

0.75

Estimated standard error formula for Unequal number of groups

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Total amount of error involved in using TWO sample means to approximate TWO population means

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0.75

0.75

0.75

5

5

= 0.54

Independent-groups t statistic (compare the means of two groups)

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The reason why we need to calculate this is because data are randomly collected, participants perform differently, and there is individual difference

2

0

0.54

Set the decision criteria

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T_critical = ± 2.306

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BOX IT STAR IT REMEMBER IT!!

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T = 3.704

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Step 2:

+2.306

-2.306

Set the decision criteria

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T_critical = ± 2.306

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BOX IT STAR IT REMEMBER IT!!

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T = 3.704

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Step 2:

+2.306

-2.306

3.704

Step 4: Make a decision

Use the Sample t and decision criteria (Critical t) to make a decision about the Null (H₀)

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Two options:

If sample data are WITHIN the critical region

🡪 Reject H_0, accept H₁; IV has an effect

If the sample data are NOT in the critical region

🡪 Fail to Reject H_0, accept H₀; IV has NO effect

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Step 4: Make a decision

+2.306

-2.306

3.704

Perfect APA style report of t-test

We report our sample t statistic and decision, and also the criteria we used to make the decision (from Step 2).

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We Reject the null hypothesis. TikTok happiness scores are significantly different than IG Reels scores�t(8)= 3.704, p < 0.05

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1. State the hypothesis/ hypotheses
• Null hypothesis
• Alternative hypothesis
2. Set the criteria for a decision
• Decide α
• Calculate df
• Locate critical region using α and df
• Draw and label distribution
3. Compute sample statistics
• Calculate s²_p
• Calculate S_(M1-M2)
• Calculate t statistic
4. Make a decision + Report in Perfect APA Style
• Reject Null, report statistics, explain statistic, and conclude that….
• Fail to reject Null, report statistics, explain statistic, and conclude that…

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Practice Q: A researcher wants to examine the relationship between watching educational TV as a child and academic performance as a high school student. Researchers predict that high school students who watched Sesame Street regularly as a 5-year old will have higher grades than students who did not watch Sesame Street regularly. Can they make this conclusion? Use α= .05 and calculate & interpret an effect size if it makes sense to do so.

Watched Sesame Street

Did NOT Watch Sesame Street

Practice Q: A researcher wants to examine the relationship between watching educational TV as a child and academic performance as a high school student. Researchers predict that high school students who watched Sesame Street regularly as a 5-year old will have higher grades than students who did not watch Sesame Street regularly. Can they make this conclusion? Use α= .05 and calculate & interpret an effect size if it makes sense to do so.

Statistical Significance vs. Effect Size

Statistical significance

There IS a significant effect/ increase/ decrease

p < .05: There is less than 5% chance of getting this sample mean if the H₀ is true.

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Effect size

What is the size of this difference?

Provides a measurement about the absolute magnitude of a treatment effect.

We ONLY calculate Effect size when we have a significant effect

(Reject the Null hypothesis)

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Calculating Effect Size (estimated d)

Measures the size of treatment effects in terms of standard deviations- always positive!

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Small: .2

Medium: .5

Large: .8

Calculating Effect Size (r²)

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The percentage of the variability in DV explained by the IV

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Interpretation: __ % of the variability in the DV is explained by the IV

Calculating Effect Size (r²)

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The percentage of the variability in DV explained by the IV

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Interpretation: 47 % of the variability in the DV is explained by the IV

What do you do when the df in your test is NOT on the t table?

Two-tailed

α= .05

df= 81

df= 60 🡪 t_crit 2.00

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df= 120 🡪 t_crit 1.98

df= 81

USE THE MORE STRINGENT CRITERIA!

Step 1: Find relative location of Study df

Study df – Smaller df

Larger df – Smaller df

Step 2: Find the difference between the critical t values

Larger t_crit – Smaller t_crit

Step 3: Multiply values from Step1 and Step 2

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Step 4: Subtract value from Step 3 from larger t_crit

df= 60 🡪 t_crit 2.00

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df= 120 🡪 t_crit 1.98

df= 81

An important assumption of the�Independent-groups t test

Assumption of Equality of Variance

The two populations from which the two samples are selected from must have equal variances

Requires a separate Hypothesis Test (F-max Test).

🡪 If two population variances are equal, the two sample variances should be very similar as well.

Technically, you are supposed to run through another hypothesis test to test this assumption.�😱 😫��We’ll look at this with an SPSS output instead!

Hypothesis testing with F-max test

Test of Equality of Variance

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Step 1: State the hypotheses

H₀: σ¹= σ² 🡪 The TWO populations have equal variance

H₁: σ¹≠ σ² 🡪 The TWO populations DO NOT have equal variance

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* Important Note: With the F-max test, we WANT the populations to have equal variance, so the goal is to Fail to Reject the Null hypothesis.

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✨🌟 SPSS Outputs of t-tests 🌟✨

So far, we have gone through running the t-test by hand.

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In psychology, we commonly use statistical programs to conduct t-tests, and the most common one is SPSS!

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We will not be using the program in our course, but you will need to know how to interpret SPSS Outputs of t-tests.

Independent t-test Output Example #1

RQ: Do dog and cat-people differ in their energy levels?

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Independent t-test Output Example #1

Sample

Size

Mean for

each Group

Standard Deviation (s)= Square root of Variance (s²)

Sig = p

In the test F-test, the null hypothesis states that variances ARE equal.

p= .355 🡪 p > .05 🡪 Fail to Reject the Null

🡪 Conclude that the two variances are equal.

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Fmax-test: Testing the assumption of Ind-t-tests: Equal Variances

Independent t-test Output Example #1 continued

RQ: Do dog and cat-people differ in their energy levels?

Fmax-test: Testing the assumption of Ind-t-tests: Equal Variances

Sig = p

Read the TOP row of results for the t-test because we CAN ASSUME equal variances based on the results of the F-test.

t-test Results

Sig = p

M₂-M₁

S_M1-M2

Report the appropriate t-test and interpret!

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Fail to Reject the Null.

t(29)= .45, p > .05 (two-tailed)

Dog and cat-people do not differ in their level of energy

Independent t-test Output Example #2: Try it on your own!

RQ: Does Instruction A differ from Instruction B on assembly time?

(Note: This is the same question as Problem Set #2 Q2. However, SPSS always runs a two-tailed test, so results can differ!)