# Quantifying Truths: Explorations Across Diverse Realms in Statistics

Embark on a statistical odyssey with our data analysis assignment as we traverse through four compelling scenarios, unraveling nuanced insights through the lens of data analysis. From decoding the disclosure dynamics in different seating positions to dissecting the impact of diverse therapies on mental health, our journey delves deep into the statistical intricacies that govern these phenomena. As we scrutinize observed versus expected percentages in various ethnic groups and probe the association between being a victim of crime and attitudes toward sentencing, our analytical prowess unveils the richness of statistical inquiry. Join us in this unique exploration where numbers tell tales, hypotheses shape narratives, and statistical insights forge a path toward a deeper understanding of diverse realms.

## 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.