## 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.