Question 1: Easy to Chair Disclosure Study
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.
H₀: μ₁ = μ₂
H₁: μ₁ > μ₂
T-distribution for independent samples with equal variances.
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))
Standard Error: SP * √(1/n1)+1/n2
Where SP = 2.055
Standard error = 2.055 * √(1/10)+1/10
t = (M_1-M_2)/(S.E of mean difference)
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
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.
H₀: No difference in therapy effects
H₁: Different effects of therapies
Degree of freedom between group=K-1 =2
〖S^2〗_within = 〖S^2〗_Total - 〖S^2〗_between
〖S^2〗_(Total ) = 72136 – 71765.333
〖S^2〗_between = (〖315〗^2/4+ (〖302〗^(2 )+)/4 〖311〗^2/4) – 71765.333
Hence 〖S^2〗_within= 370.666 - 22.1667
F-ratio = MStreatments/Mserror
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
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.
H₀: O_i = E_i
H₁: O_i ≠ E_i
The cut-off is χ_(3,0.05 )= 9.348
|Observed||* 100 = 64.62%||* 100 = 6.15%||* 100 = 23.58%||* 100 = 6.15%|
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
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.
H₀: No association between victimization and attitude to sentencing
H₁: Association between victimization and attitude to sentencing
χ_((r-1)(c-1),0.05) = χ_(2,0.05 )= 7.378
Calculated based on observed frequencies
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.