Problem 1: Average Grooming Time
Hypothesis Test:
- Null Hypothesis Ho: μ=0.8
- Alternative Hypothesis Ha:μ≠0.8
Test Statistic:
t=(x ̅-μ)/(s/√n) ~ t_((n-1)),
x=1.2408, S=1.002254, μ=0.8 and n=37
t=(1.2408-0.8)/(1.002254/√37)= 2.6753
P-value = 2*pt>tcal for two-tailed test
p=2*pt>2.6753, df=36
p=2*0.005581
Hence, p-value = 0.011162
Conclusion: The p-value (0.0112) is less than the significance level (0.05). Therefore, we reject the null hypothesis, concluding that the average grooming time significantly differs from 0.8 hours.
95% Confidence Interval
95% CI= x ̅±t_((n-1, α/2))*s/√n
95% CI= 1.2408±t_((37-1, 0.05/2))*1.002254/√37,where t_((37-1,0.05/2) )=2.028094
95% CI= 1.2408±2.028094*1.002254/√37
95% CI= 1.2408±0.3341681=[ 0.9066319,1.574968]
Hence, 95% CI=[0.9066,1.57450]
Interpretation of CI: we are 95% confident that the population mean number of hours MAT115 students spend a day grooming lies between 0.90 and 1.57
Problem 2: Weekly Study Hours
Problem Description:The goal is to determine if the average weekly study hours for MAT115 students are significantly less than 17. The sample data includes 37 observations with a sample mean (xˉ) of 10.56757 and a known population standard deviation (σ) of 10.
Hypothesis Test:
- Null Hypothesis (0H0): =17μ=17
- Alternative Hypothesis (Ha): <17μ<17
Test Statistic: Z=(x ̅-μ)/(σ/√n) ~ N(0,1)
x=10.56757, σ=10, μ=17 and n=37
Z=(10.56757-17)/(10/√37)= -3.912694
P-value = p(Z
P-value = pZ
p=pZ<-3.912694
p=0.0000
Hence, p-value = 0.0000
Conclusion: The p-value (0.0000) is less than 0.05. Thus, we reject the null hypothesis, concluding that the weekly study hours are significantly less than 17.
95% Confidence Interval
95% CI= x ̅±1.96*σ/√n
95% CI= 10.56757±1.96*1.002254/√37,
95% CI= 10.56757±1.96*10/√37
95% CI= 10.56757±3.22222=[ 7.34535,13.78979]
Hence, 95% CI=[ 7.34535,13.78979]
Interpretation of CI: we are 95% confident that the population mean number of hours a week a MAT115 student studies lie between 7.35 and 13.79
Problem 3: Dog Ownership Percentage
Problem Description: This analysis aims to determine if the percentage of MAT115 students owning a dog is significantly different from 36.5%. The sample includes 37 students, with 18 owning a dog.
Hypothesis Test:
- Null Hypothesis (0H0): =0.365p=0.365
- Alternative Hypothesis (Ha): ≠0.365p=0.365
Test Statistic: Z=(p ̂-p)/√((p(1-p))/n) ~ N(0,1),
p ̂=0.4865,p=0.365 and n=37
Z=( 0.4865-0.3650)/√((0.365*(1-0.365))/37)= 1.535125
P-value = p(Z>Z_cal ) for a left-tailed test
p=2*p(Z>1.535125)
p=2*0.06237656
p= 0.1247531
Hence, p-value = 0.1248
Conclusion: The p-value (0.1248) is greater than 0.05, leading to the acceptance of the null hypothesis. There is no significant difference in the percentage of students owning a dog.
95% Confidence Interval
95% CI= p ̂±1.96*√((p ̂(1-p ̂))/n)
95% CI= 0.4865±1.96*√(( 0.4865*(1- 0.4865))/37)
95% CI= 0.4865±0.1610523=[ 0.3254477,0.6475523]
Hence, 95% CI=[ 0.3254477,0.6475523]
Interpretation of CI: we are 95% confident that the percentages of MAT115 students who own a dog lie between 32.54% and 64.76%
Problem 4: Sleep Hours Comparison
Problem Description: The aim is to assess if the nightly sleep hours for two groups of students (DA and DB) differ significantly. The samples include 22 observations for DA and 15 for DB.
Hypothesis Test:
- Null Hypothesis (0H0): μDA=μDB
- Alternative Hypothesis (Ha): μDA=μDB
Test Statistic:t=(x ̅_DA-x ̅_DB)/√((S_DA^2)/n_DA +(S_DB^2)/n_DB ) ~ t_((n_DA+n_DB-2)),
x ̅_DA=7.22727,S_DA^2=1.517316017,n_DA=22,x ̅_DB=5.683333,S_DB^2=2.129167 ,n_DB=15
t=(7.22727-5.683333)/√(1.517316017/22+2.129167/15)= 3.361844
P-value = 2*p(t>t_cal ) for two-tailed test
p=2*p(t>3.361844,df=35)
p=2*0.00094
Hence, p-value =0.00188
Conclusion: The p-value (0.00188) is less than 0.05, leading to the rejection of the null hypothesis. The nightly sleep hours for DA students are significantly different from DB students.
95% Confidence Interval: 95% CI=(x ̅_DA-x ̅_DB )±t_((35,0.05/2 ))*√((S_DA^2)/n_DA +(S_DB^2)/n_DB )
95% CI=(7.22727-5.683333)±2.030108*√(1.517316017/22+2.129167/15)
95% CI= 1.543937± 0.9323333
95% CI=[ 0.6116037,2.47627]
Hence, 95% CI=[ 0.6116,2.4763]
Interpretation of CI: we are 95% confident that the difference in the population means of the number of hours a night DA students sleep and the number of hours a night DB students sleep lie between 0.6116 and 2.4763.