Homework answers / question archive / Formulae: Independent samples t-test: 1) Pooled Variance: SP2 = (n1 – 1)S12 + (n2 – 1)S22/n1+n2-2 2) Standard Error: Sx1-x2 = Ösp2/n1 + Ösp2/n2 3) T-Statistic: t = (X1 – X2)/Sx1-x2 Correlation: 1) Do the 7-column table to find SSX, SSY, and SP

Formulae: Independent samples t-test: 1) Pooled Variance: SP2 = (n1 – 1)S12 + (n2 – 1)S22/n1+n2-2 2) Standard Error: Sx1-x2 = Ösp2/n1 + Ösp2/n2 3) T-Statistic: t = (X1 – X2)/Sx1-x2 Correlation: 1) Do the 7-column table to find SSX, SSY, and SP

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Formulae: Independent samples t-test:

1) Pooled Variance: S_P² = (n₁ – 1)S₁² + (n₂ – 1)S₂²/n₁+n₂-2

2) Standard Error: S_x1-x2 = Ös_p²/n₁ + Ös_p²/n₂

3) T-Statistic: t = (X₁ – X₂)/S_x1-x2

Correlation:

1) Do the 7-column table to find SSX, SSY, and SP.

2) Pearson’s r: r = SP/ÖSSX*SSY

Chi-Squared:

1) E = N/C (one variable) or E = R*C/N (two variables)

2) Chi-Square: X² = S(0-E)²/E

Note: For critical values, you can find all of them here:

https://www.socscistatistics.com/tests/criticalvalues/default.aspx

Hypothesis Test Examples:

For each hypothesis test listed, explain who that test would be used on and describe a unique study you could run that would use that test. Be sure to mention the groups or variables being compared and what the dependent variable would be. Be specific to avoid accidently providing an example for a different type of test!

For example, if the prompt said “Z-Test” then I would provide an example, such as “Z-Tests compare one sample mean to the population mean. For example, I could compare a sample of 10 local basketball players to the population of all NBA league players on their 3-point shot ratio.

This would let me see if the team of basketball players are better, or worse, than NBA players.”

1) One Sample T-Test:

One sample t-tests compare data from a sample to a population with the known value.

2) A One-Way ANOVA with 4 Groups:

3) Chi-Squared with 2 Independent Variables:

Smoking Program:

1. You oversee an experimental anti-smoking program. You are worried that participant attrition (1.e., drop-outing out a study) 1s causing your treatment group to bias against heavy-smokers. You run a chi-squared to check the participant distribution.

	Control	Treatment	Total
Light Smokers	100	80	180
Heavy Smokers	90	50	140
Total	190	130	320

 

1. What is the expected value for each cell?

Control-Light:

Control-Heavy:

Treatment-Light:

Treatment-Heavy:

b. Calculate the chi squared. Using a critical value of 3.84, 1s it significant? What does this mean for your treatment?

c. In this study, why might attrition be cause for concern? Explain your reasoning by tying this question back to concepts of internal validity.

Study-Time Focus

2. You are studying the impact of three popular Twitch influencers on brand recognition. Your survey gives you a score based on how much the fan follows the influencer. Higher scores means more positive opinion of the brand.

Use this data to answer the following questions.  Be sure to show your work and round all numbers to two decimals (i.e. x.xx) when applicable.

	n	X	S
Influencer K.K.	20	9	2
Influencer P.C.	25	4	4
Influencer D.J.	15	7	1

 

a. What hypothesis test would we use to compare all three influencer audiences at the same time? How would you describe the null hypothesis in this case?

For b, c, and d, be sure to show your work (e.g. at least the pooled variance and standard error)

b. Find the t-statistic for ‘K.K.’ vs. ‘P.C.’ and the critical value for this comparison

c. Find the t-statistic for °‘K.K.’ vs. “D.J.’ and the critical value for this comparison:

d. Find the t-statistic for ‘P.C.’ vs. ‘D.J.’ and the critical value for this comparison:

e. Compare the t-statistics to each critical value. In your own words, what do your findings say about these influencers and brand they represent?

f. What would you need to add to this study to turn it into a factorial ANOVA? Give me a unique example.

Age and Income

3. You are looking for a correlation between age and income (in thousands):

Use this data to answer the following questions. Be sure to show your work and round final answers to two decimals (i.e. x.xx)

Name	Age	Income
Maria	20	10
Marco	23	20
Melody	23	10
Markus	27	40
Maxwell	33	70
Martin	40	50
Minnie	44	80
Mary	68	100

 

a. State in words and formulas the null and alternative hypotheses

H₀:

H_a:

b. Use the 7-column method to find SSx, SSy and SP (4pt):

X			Y

 

SSx =

SSy =

SP =

c. Calculate r using your numbers above

r =

d. How would you describe this correlation’s strength and direction?

e. Given a critical value of 0.582, is there a significant relationship between age and income? Describe this relationship in your own words.

f. Does this correlation alone provide evidence to show that one variable cause the other? Explain your answer.

ANOVA Interpretation

4. Your pharmacy is testing the impact of an anti-anxiety medication. So far experiments have failed, but you decide to run a 2x2 ANOVA using gender as an additional variable. SPSS provides you with the following table of means (higher is better) and F-test results:

SUMMARY	Control	Experiment	Total
Men	40.0	35.0	37.5
Women	60.0	40.0	50.0
Total	50.0	37.5

 

ANOVA

Source of Variation	Sum of Squares	df	Mean Squares	F	Sig
Gender	562.5	1	562.5	3.551	0.030
Treatment	622.5	1	622.5	4.502	0.025
Gender * Treatment	3062.5	1	3062.5	24.775	0.000
Within	4450	36	123.611
Total	8137.5	39

 

a. Describe the tables above. What does each column indicate about the row?

b. Use the sig to determine significance at a 0.05 and 0.01 confidence level. What do you conclude about the three different ANOVA hypotheses? Be extra careful not to confuse the main effect of a factor and the interaction effect of those factors.

i. The main effect of treatment 

ii. The main effect of gender

iii. The interaction

c. Name and describe the tests you need to run when following up on a significant ANOVA.