Question

A Pew Internet Project Data Memo presented data comparing adult gamers with teen gamers with respect to the devices on which they play. The data are from two surveys. The adult survey had 1063 gamers while the teen survey had 1064 gamers. The report presented data for gaming on computers (desktops or laptops). These devices were used by 73% of adult gamers and by 76% of teen gamers.

You are required to test the null hypothesis that the two proportions are equal.

Fill in the blanks. Give the value of Answer 1 to four decimal places and the value of Answer 2 to two decimal places.

The pooled estimate of the proportion, p^, is _Answer 1_.

The test statistic, z=Answer 2___.

Answer 1

p1 = 0.73 73% of adult gamers used the devices     
n1 = 1063 Sample size of adult gamers     
p2 = 0.76 76% of teen gamers used the devices     
n2 = 1064 Sample size of teen gamers     
       
1) the null and alternative hypothesis are       
Ho : p1 = p2      
Ha : p1 ≠ p2       
       
2) Let α = 0.05  Level of significance    
Thus the decision rule is       
Reject the null hypothesis if p-value < α      
       
3) This is a two sided test      
We find the critical value Z-crit such that P(Z < Z-crit) = α/2 = 0.025      
We find Z-crit using Excel function NORM.S.INV      
Z-crit = NORM.S.INV(0.025)      
Z-crit = -1.95996      
Since this is a two sided test, Z-crit is ±1.95996       
Rejection region is       
Reject Ho if Z < -1.96 or Z > 1.96      
       
4) Calculate test statistic Zcalc      
Finding the Standard error      
We find the pooled proportion      

       where pdiff = 0 since null hypothesis states p1 = p2   
       

Answer 1: The pooled estimate of the proportion, p̂ = 0.7450

Answer 2: The test statistic, z = -1.59       
        
5) -1.96 < -1.59 < 1.96       
Thus Z-calc does not lie in the Rejection region       
        
6) Conclusion :       
Since z-calc does not lie in the rejection region, we DO NOT reject Ho            
Thus there is no difference between the two proportions       
There is not enough statistical evidence to prove otherwise