Question

6. Suppose that the sample unemployment rate for those aged 25–60 is 8% based on a survey of 150 people, while the unemployment rate for those aged 16–24 is 12% based on a survey of 100 people.

(a) Form a 90% confidence interval for the difference between the two population unemployment rates.

(b) Construct a test statistic to test the null hypothesis that the two population rates are the same against the alternative hypothesis that the rate is higher in for those aged 16–24, and report the associated P-value.

Answer 1

a)

level of significance, α =   0.10              
Z critical value =   Z α/2 =    1.645   [excel function: =normsinv(α/2)      
                  
Std error , SE =    SQRT(p̂1 * (1 - p̂1)/n1 + p̂2 * (1-p̂2)/n2) =     0.0393          
margin of error , E = Z*SE =    1.645   *   0.0393   =   0.0647
                  
confidence interval is                   
lower limit = (p̂1 - p̂2) - E =    -0.040   -   0.0647   =   -0.1047
upper limit = (p̂1 - p̂2) + E =    -0.040   +   0.0647   =   0.0247
                  
so, confidence interval is (   -0.1047   < p1 - p2 <   0.0247   )  

b)

Ho:   p1 - p2 =   0          
Ha:   p1 - p2 <   0          
                  
sample #1   ----->   25-60
first sample size,     n1=   150          
number of successes, sample 1 =     x1=   12          
proportion success of sample 1 , p̂1=   x1/n1=   0.0800          
                  
sample #2   ----->   16-24   
second sample size,     n2 =    100          
number of successes, sample 2 =     x2 =    12          
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.120          
                  
difference in sample proportions, p̂1 - p̂2 =     0.0800   -   0.1200   =   -0.0400
                  
pooled proportion , p =   (x1+x2)/(n1+n2)=   0.0960          
                  
std error ,SE =    =SQRT(p*(1-p)*(1/n1+ 1/n2)=   0.0380          
Z-statistic = (p̂1 - p̂2)/SE = (   -0.040   /   0.0380   ) =   -1.0518
                  

p-value =        0.1465   [Excel function =NORMSDIST(z)      
decision :    p-value>α,Don't reject null hypothesis               
                  
Conclusion:   There is not enough evidence that the rate is higher in for those aged 16–24    

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