Question

Student Name: 1. Test the claim that the mean age of the prison population in one city is less than 27 years. Assume that a random sample has been selected from the normal population and sample data are summarized as n-50, x 252 years, and s-6.8 years. Use significance level of a 0.05. 2. In a study of factors affecting hypnotism, visual analogy scale (VAS) sensory ratings were obtained for 25 subjects. For these sample ratings the mean x 8.90 and standard deviation s-1.80. At the 0.01 level of significance, test the claim that this sample comes from the population with a mean rating less than 10

Answer 1

Question 1)
Null hypothesis  

$H_{0}:\mu=27$

vs   
                  
Alternative hypothesis      

$H_{1}:\mu<27$           
                  
We have for given sample,                  
Population mean for given sample=27  
Sample mean=25.2          
Sample standard deviation s=6.8      
Sample size =n=   50          
                  
Degree of freedom = n-1 =49      
                  
  
                  
t test statistic formula is
$t=\frac{\bar{x}-\mu }{\frac{s}{\sqrt{n}}}=\frac{25.2-27 }{\frac{6.8}{\sqrt{50}}}=-1.87$

P value is 0.0337.........................by using TDIST(-1.87, 49,1)

P value is 0.0337<0.05

Therefore we reject H0.

Conclusion: We have sufficient evidence at $\alpha =0.05$ to say that, the mean age of prison population in one city is less than 27 years.

Question 2)              
      
Null hypothesis  

$H_{0}:\mu=10$   

vs
                  
Alternative hypothesis

$H_{1}:\mu<10$

              
                  
We have for given sample,                  
Population mean for given sample=10  
Sample mean=8.9          
Sample standard deviation s=1.8      
Sample size =n=25          
                  
Degree of freedom = n-1 =24      
                  

                  
t test statistic formula is
$t=\frac{\bar{x}-\mu }{\frac{s}{\sqrt{n}}}=\frac{8.9-10 }{\frac{1.8}{\sqrt{25}}}=-3.06$

P value is 0.0027......................by using TDIST(-3.06,24,1) in Excel.

P value is 0.0027<0.01

Therefore we reject H0.

Conclusion: We have sufficient evidence at $\alpha =0.01$ to say that , this sample comes from the population with mean rating less than 10.