Question

Test whether the proportion of iphone owners is more than the proportion of android owners. Take two samples of at least 20 each. In the first sample ask "Do you own an iphone?" in the second sample ask "do you own an android?" Use a significance level of 0.01.

a. survey results: iphone -> 13 yes, 7 no android -> 11 yes, 9 no

b. state your claim

c. null hypothesis and alternative hypothesis

d. which type of test are you running? show calculator input.

e. P-value

f. Decision: reject null or fail to reject null

g. conclusion

h. calculate a 98% confidence interval to estimate the difference in proportions of iphone and android users. write it in sentence format.

i. does the confidence interval agree or contradict your hypothesis test conclusion? why?

Answer 1

b)

claim: proportion of iphone owners is more than the proportion of android owners or p1 > p2

c)

Ho:   p1 - p2 =   0
Ha:   p1 - p2 >   0

d)

Hypothesis Test: Difference of two Proportions              

e)

sample #1   ----->   iphone          
first sample size,     n1=   20          
number of successes, sample 1 =     x1=   13          
proportion success of sample 1 , p̂1=   x1/n1=   0.6500          
                  
sample #2   ----->   android          
second sample size,     n2 =    20          
number of successes, sample 2 =     x2 =    11          
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.550          
                  
difference in sample proportions, p̂1 - p̂2 =     0.6500   -   0.5500   =   0.1000
                  
pooled proportion , p =   (x1+x2)/(n1+n2)=   0.6000          
                  
std error ,SE =    =SQRT(p*(1-p)*(1/n1+ 1/n2)=   0.1549          
Z-statistic = (p̂1 - p̂2)/SE = (   0.100   /   0.1549   ) =   0.6455
                  
z-critical value , Z* =        2.3263   [excel function =NORMSINV(α)]      
p-value =        0.2593 [excel function =NORMSDIST(-z)]      

f)

decision :    p-value>α,Don't reject null hypothesis   

g)Conclusion:   There is not enough evidence to conclude that      the proportion of iphone owners is more than the proportion of android owner

h)

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) =     0.1541          
margin of error , E = Z*SE =    2.326   *   0.1541   =   0.3585
                  
confidence interval is                   
lower limit = (p̂1 - p̂2) - E =    0.100   -   0.3585   =   -0.2585
upper limit = (p̂1 - p̂2) + E =    0.100   +   0.3585   =   0.4585
                  
so, confidence interval is (   -0.2585   < p1 - p2 <   0.4585   )  

we are 95% confident that the difference in proportions of iphone and android users lies within confidence interval

i)

since, confidence interval contains zero, null hypothesis is not  rejected.so,  confidence interval agree your hypothesis test conclusion