Question

In answering the question(s), make sure to write down the following 7 steps.

Step 1: Establish null and alternate hypotheses State the null and alternative hypothesis (as a sentence and formula).

Step 2: Calculate the degrees of freedom

Step 3: Calculate t _critical using critical t – table

Step 4: Calculate the Sum of Square deviation (SS_D)

Step 5: Calculate t _obtained

Step 6: Specify the critical value and the obtained value on a t-distribution curve

Step 7: Decision and Conclusion Write a clear and concise conclusion.

A researcher is interested in whether gender predicts the type of costume people would wear to an upcoming party. He asks 6 guys and 6 girls: How likely would you be to wear a “Creepy Clown” costume to the upcoming Halloween party? (1 = very unlikely to 7 = very likely). The data are shown below.

Guys: 4 6 4 3 6 7

Girls: 2 5 4 1 2 3

1. Use alpha = .01 to see whether gender impacts willingness to wear a Creepy Clown costume. (Note: You need to write down all the 7 steps.)

2. Use alpha = .01 to see whether guys are more willing than girls to a Creepy Clown costume. (Note: You only need to write down the steps that are different from part (1).)

3. Use alpha = .05 to see whether guys are more willing than girls to a Creepy Clown costume. (Note: You only need to write down the steps that are different from part (1).)

Answer 1

Solution:

Part 1

Step 1: Establish null and alternate hypotheses. State the null and alternative hypothesis (as a sentence and formula).

Null hypothesis: H₀: Gender do not impact significantly on willingness to wear a Creepy Clown costume.

Alternative hypothesis: H_a: Gender significantly impact on willingness to wear a Creepy Clown costume.

H₀: µ₁ = µ₂ v/s H_a: µ₁ ≠ µ₂

This is a two tailed test.

Step 2: Calculate the degrees of freedom

We are given n1 = 6 and n2 = 6

df = n1 + n2 – 2 = 6 + 6 – 2 = 10

Step 3: Calculate t critical using critical t – table

t-critical value = ±3.1693

Step 4: Calculate the Sum of Square deviation (SSD)

SSD = sqrt(∑(X - mean)^2/(n – 1))

No.	Guys X	(X - mean)^2	Girls Y	(Y - mean)^2
1	4	1	2	0.69438889
2	6	1	5	4.69458889
3	4	1	4	1.36118889
4	3	4	1	3.36098889
5	6	1	2	0.69438889
6	7	4	3	0.02778889
Total	30	12	17	10.83333334
Mean	5		2.833333

SSD for guys = sqrt(12/(6 – 1)) = sqrt(12/5) = sqrt(2.4) = 1.549193

SSD for girls = sqrt(10.83333334/(6- 1) = sqrt(10.83333334/5) = 1.471960145

Step 5: Calculate t obtained

Test statistic = t = (X1bar – X2bar) / sqrt[S_p²*((1/n1)+(1/n2))]

Where S_p² is pooled variance

S_p² = [(n1 – 1)*S1^2 + (n2 – 1)*S2^2]/(n1 + n2 – 2)

S_p² = [(6 – 1)* 1.549193^2 + (6 – 1)* 1.47196^2]/(6 + 6 – 2)

S_p² = 2.2833

(X1bar – X2bar) = 5 - 2.833333 = 2.1667

t = 2.1667/sqrt[2.2833*((1/6)+(1/6))]

t = 2.4835

Step 6: Specify the critical value and the obtained value on a t-distribution curve

Critical value = t-critical = ±3.1693

P-value = 0.0323

α = 0.01

P-value > α = 0.01

Step 7: Decision and Conclusion Write a clear and concise conclusion.

P-value = 0.0323

α = 0.01

P-value > α = 0.01

So, we do not reject the null hypothesis H₀

There is insufficient evidence to conclude that Gender significantly impact on willingness to wear a Creepy Clown costume.

Part 2

Step 1: Establish null and alternate hypotheses. State the null and alternative hypothesis (as a sentence and formula).

Null hypothesis: H₀: There is no any significant difference in willing between guys and girls.

Alternative hypothesis: H_a: Guys are more willing than girls to a Creepy Clown costume.

H₀: µ₁ = µ₂ v/s H_a: µ₁ > µ₂

This is an upper tailed test.

Step 2: Calculate the degrees of freedom

We are given n1 = 6 and n2 = 6

df = n1 + n2 – 2 = 6 + 6 – 2 = 10

Step 3: Calculate t critical using critical t – table

t-critical value = ±3.1693

Step 4: Calculate the Sum of Square deviation (SSD)

SSD = sqrt(∑(X - mean)^2/(n – 1))

No.	Guys X	(X - mean)^2	Girls Y	(Y - mean)^2
1	4	1	2	0.69438889
2	6	1	5	4.69458889
3	4	1	4	1.36118889
4	3	4	1	3.36098889
5	6	1	2	0.69438889
6	7	4	3	0.02778889
Total	30	12	17	10.83333334
Mean	5		2.833333

SSD for guys = sqrt(12/(6 – 1)) = sqrt(12/5) = sqrt(2.4) = 1.549193

SSD for girls = sqrt(10.83333334/(6- 1) = sqrt(10.83333334/5) = 1.471960145

Step 5: Calculate t obtained

Test statistic = t = (X1bar – X2bar) / sqrt[S_p²*((1/n1)+(1/n2))]

Where S_p² is pooled variance

S_p² = [(n1 – 1)*S1^2 + (n2 – 1)*S2^2]/(n1 + n2 – 2)

S_p² = [(6 – 1)* 1.549193^2 + (6 – 1)* 1.47196^2]/(6 + 6 – 2)

S_p² = 2.2833

(X1bar – X2bar) = 5 - 2.833333 = 2.1667

t = 2.1667/sqrt[2.2833*((1/6)+(1/6))]

t = 2.4835

Step 6: Specify the critical value and the obtained value on a t-distribution curve

Critical value = t-critical = 2.7638

P-value = 0.0162

α = 0.01

P-value > α = 0.01

Step 7: Decision and Conclusion Write a clear and concise conclusion.

P-value = 0.0162

α = 0.01

P-value > α = 0.01

So, we do not reject the null hypothesis H₀

There is insufficient evidence to conclude that Gender significantly impact on willingness to wear a Creepy Clown costume.

Part 3

Step 1: Establish null and alternate hypotheses. State the null and alternative hypothesis (as a sentence and formula).

Null hypothesis: H₀: There is no any significant difference in willing between guys and girls.

Alternative hypothesis: H_a: Guys are more willing than girls to a Creepy Clown costume.

H₀: µ₁ = µ₂ v/s H_a: µ₁ > µ₂

This is an upper tailed test.

Step 2: Calculate the degrees of freedom

We are given n1 = 6 and n2 = 6

df = n1 + n2 – 2 = 6 + 6 – 2 = 10

Step 3: Calculate t critical using critical t – table

t-critical value = ±3.1693

Step 4: Calculate the Sum of Square deviation (SSD)

SSD = sqrt(∑(X - mean)^2/(n – 1))

No.	Guys X	(X - mean)^2	Girls Y	(Y - mean)^2
1	4	1	2	0.69438889
2	6	1	5	4.69458889
3	4	1	4	1.36118889
4	3	4	1	3.36098889
5	6	1	2	0.69438889
6	7	4	3	0.02778889
Total	30	12	17	10.83333334
Mean	5		2.833333

SSD for guys = sqrt(12/(6 – 1)) = sqrt(12/5) = sqrt(2.4) = 1.549193

SSD for girls = sqrt(10.83333334/(6- 1) = sqrt(10.83333334/5) = 1.471960145

Step 5: Calculate t obtained

Test statistic = t = (X1bar – X2bar) / sqrt[S_p²*((1/n1)+(1/n2))]

Where S_p² is pooled variance

S_p² = [(n1 – 1)*S1^2 + (n2 – 1)*S2^2]/(n1 + n2 – 2)

S_p² = [(6 – 1)* 1.549193^2 + (6 – 1)* 1.47196^2]/(6 + 6 – 2)

S_p² = 2.2833

(X1bar – X2bar) = 5 - 2.833333 = 2.1667

t = 2.1667/sqrt[2.2833*((1/6)+(1/6))]

t = 2.4835

Step 6: Specify the critical value and the obtained value on a t-distribution curve

Critical value = t-critical = 2.7638

P-value = 0.0162

α = 0.05

P-value < α = 0.05

Step 7: Decision and Conclusion Write a clear and concise conclusion.

P-value = 0.0162

α = 0.05

P-value < α = 0.05

So, we reject the null hypothesis H₀

There is sufficient evidence to conclude that Gender significantly impact on willingness to wear a Creepy Clown costume.