Question 1: Easy to Chair Disclosure Study
Problem Description:
The study aimed to investigate whether individuals in an easy-to-chair position disclose more information than those lying on a couch. Two populations, denoted as μ₁ and μ₂, represent the average disclosure for sitting positions. The null hypothesis (H₀) assumes no difference between the two populations, while the alternative hypothesis (H₁) suggests that μ₁ is greater than μ₂.
Solution:
Hypotheses:
H₀: μ₁ = μ₂
H₁: μ₁ > μ₂
Comparison Distribution:
T-distribution for independent samples with equal variances.
Cut-off Value:
Cut-off = t_(∝,df) where α = 0.05 and df = 10+10-2 = 18.
t_0.05,18 = 1.7341
Pooled Standard Deviation:= √(((n1-1) 〖S1〗^2+(n2-1)〖S2〗^2)/(n1+n2-2))
= √((9*〖1.9〗^2+9*〖2.2〗^2)/18)
= 2.055
Standard Error: SP * √(1/n1)+1/n2
Where SP = 2.055
Standard error = 2.055 * √(1/10)+1/10
= 0.9192
T-Value:
t = (M_1-M_2)/(S.E of mean difference)
= (18.2-14.3)/0.9192
= 4.2426
Conclusion:
Since >t> cut-off, reject H₀. Conclude that there is sufficient evidence that individuals in an easy-to-chair position disclose more at the 0.05 significance level.
Question 2: Therapies and Mental Health
Problem Description:
The study examines the effects of different therapies on mental health. The parameter of interest is the presence of anxiety disorder, with the null hypothesis (H₀) stating no difference in therapy effects and the alternative hypothesis (H₁) suggesting different effects.
Solution:
Hypotheses:
H₀: No difference in therapy effects
H₁: Different effects of therapies
Comparison Distribution:
F-distribution
Critical Value:
Degree of freedom between group=K-1 =2
critical(2,9)=4.26Fcritical(2,9)=4.26
Variance Calculations:
〖S^2〗_within = 〖S^2〗_Total - 〖S^2〗_between
〖S^2〗_(Total ) = 72136 – 71765.333
= 370.666
〖S^2〗_between = (〖315〗^2/4+ (〖302〗^(2 )+)/4 〖311〗^2/4) – 71765.333
= 22.1667
Hence 〖S^2〗_within= 370.666 - 22.1667
= 348.5
〖S^2〗_between=22.16667
F-Ratio:
F-ratio = MStreatments/Mserror
= 11.083338/38.7222
= 0.2862
Conclusion:
As the F-ratio is less than the critical F, do not reject H₀. Conclude that, at the 5% significance level, different therapies have no significantly different effects on mental health.
Question 3: Ethnicity and Data Support Claim
Problem Description:
The study investigates whether observed percentages of ethnicity differ from expected values. The parameter of interest is the relationship between observed Oi and expected Ei values for various ethnicities.
Solution:
Hypotheses:
H₀: O_i = E_i
H₁: O_i ≠ E_i
Comparison Distribution:
Chi-square distribution
Cut-off Value:
The cut-off is χ_(3,0.05 )= 9.348
Chi-square Calculation:
| Caucasian | European | Asian | Others | |
|---|---|---|---|---|
| Observed | * 100 = 64.62% | * 100 = 6.15% | * 100 = 23.58% | * 100 = 6.15% |
| Expected | 47% | 28% | 15% | 10% |
Table 1: The expected (Ei) vs. observed (Oi)values on various ethnicities
χ^2 = ∑▒〖(oi-ei)〗^2/ei
= 〖(64.62-47)〗^2/47 + 〖(6.15-28)〗^2/28 + 〖(23.58-15)〗^2/15 + 〖(6.15-10)〗^2/10
= 29.491
Conclusion:
Since 2>χ2> cut-off, reject H₀. Conclude that the data does not support the claim about percentages.
Question 4: Victim of Crime and Attitude to Sentencing
Problem Description: The study examines the association between being a victim of crime and attitudes toward sentencing. The null hypothesis (H₀) assumes no association, while the alternative hypothesis (H₁) suggests an association.
Solution:
Hypotheses:
H₀: No association between victimization and attitude to sentencing
H₁: Association between victimization and attitude to sentencing
Comparison Distribution:
Chi-square distribution
Cut-off Value:
χ_((r-1)(c-1),0.05) = χ_(2,0.05 )= 7.378
Expected Frequencies:
Calculated based on observed frequencies
Chi-square Calculation:
2=9.37χ2=9.37
Conclusion:
Since 2>χ2> cut-off, reject H₀. Conclude that there is enough evidence to say there is a significant association between being a victim of crime and attitude to sentencing.