Question

1) The U.S. Department of Justice Statistics collects statistics about the provision of victim services following crimes. Suppose that 12% of victims received victim services in Seattle as compared to 5% of victims in Birmingham. Suppose that there are 1,000 victims in Seattle and 1,200 victims in Birmingham.

a) Find the confidence interval for the difference in the proportions receiving victim services for those in Seattle compared to those in Birmingham. Use a 98% level of confidence.

b) Provide an interpretation of the interval.

2) Suppose we want to determine whether the servers at the three Italian restaurants are getting the same average amount in tips. The data is shown below:

Ciao!	97	114	105	85	120
Spiro's	67	85	92	78	90
Marzano's	93	102	98	80	91

Would a student seeking a servers job at one of these three Italian restaurants conclude that the mean tips are the same? Use a 5% level of significance.

a) State the null and alternative hypothesis.

b) Calculate the test statistic, given that SSB=810.5 and MSW=121.94

b) Draw a diagram with the critical value for a 5% level of significance.

c) State the conclusion as to if and why you reject the null hypothesis.

Answer 1

1)

a)

sample #1   ----->  
first sample size,     n1=   1000

proportion success of sample 1 , p̂1=   x1/n1=   0.1200000
      
sample #2   ----->  
second sample size,     n2 =    1200

proportion success of sample 1 , p̂ 2=   x2/n2 =    0.050000

level of significance, α =   0.02              
Z critical value =   Z α/2 =    2.326   [excel function: =normsinv(α/2)      
                  
Std error , SE =    SQRT(p̂1 * (1 - p̂1)/n1 + p̂2 * (1-p̂2)/n2) =     0.01205          
margin of error , E = Z*SE =    2.326   *   0.0120   =   0.02803
                  
confidence interval is                   
lower limit = (p̂1 - p̂2) - E =    0.070   -   0.0280   =   0.0420
upper limit = (p̂1 - p̂2) + E =    0.070   +   0.0280   =   0.0980
                  
so, confidence interval is (   0.0420   < p1 - p2 <   0.0980   )  

b)

we are 98 confident that true difference of proportions of receiving victim services for those in Seattle compared to those in Birmingham lies within the confidence interval.