Question

A survey of 800 adults from a certain region​ asked, "What do you buy from your mobile​ device?" The results indicated that 57​% of the females and 48​% of the males answered clothes. The sample sizes of males and females were not provided. Suppose that of 500 ​females, 285 reported they buy clothing from their mobile​ device, while of 300 ​males, 144 reported they buy clothing from their mobile device.

A. is there evidence of a difference between males and females in the proportion who said they buy clothing from their mobile device at the 0.05 level of​ significance?

The null and alternative​ hypotheses is:

H0​: π1=π2

H1​: π1≠π2

Determine the value of the test statistic

ZSTAT=

Determine the critical​ value(s) for this test of hypothesis=

State the conclusion:

B. Find the​ p-value in​ (a) and interpret its meaning:

C. Construct and interpret a 90%, 95%, and 99​% confidence interval estimate for the difference between the proportion of males and females who said they buy clothing from their mobile device.

D: What are your answers to​ (a) through​ (c) if 432 males said they buy clothing from their mobile​ device?

Determine the value of the test statistic

Determine the value of X2=

Determine the proportion of items of interest in sample​ 2, p2=

ZSTAT=

The critical values for this test of hypothesis =

​p-value=

Construct and interpret a90%, 95%, and 99​% confidence interval estimate for the difference between the proportion of males and females who said they buy clothing from their mobile device.

Answer 1

Ho:   p1 - p2 =   0          
Ha:   p1 - p2 ╪   0          
                  
sample #1   -----> female
first sample size,     n1=   500          
number of successes, sample 1 =     x1=   285          
proportion success of sample 1 , p̂1=   x1/n1=   0.5700          
                  
sample #2   -----> male
second sample size,     n2 =    300          
number of successes, sample 2 =     x2 =    144          
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.4800          
                  
difference in sample proportions, p̂1 - p̂2 =     0.5700   -   0.4800   =   0.0900
                  
pooled proportion , p =   (x1+x2)/(n1+n2)=   0.53625          
                  
std error ,SE =    =SQRT(p*(1-p)*(1/n1+ 1/n2)=   0.03642          
Z-statistic = (p̂1 - p̂2)/SE = (   0.090   /   0.0364   ) =   2.47
                  
z-critical value , Z* =        1.9600   [excel formula =NORMSINV(α/2)]

test stat > critical value, reject Ho

there evidence of a difference between males and females in the proportion who said they buy clothing from their mobile device at the 0.05 level of​ significance

.............

b)

p-value =        0.0135   [excel formula =2*NORMSDIST(z)]      
if null hypothesis is true then there is probability of 0.0135

.................

c)

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) =     0.03636          
margin of error , E = Z*SE =    1.645   *   0.0364   =   0.05981
                  
confidence interval is                   
lower limit = (p̂1 - p̂2) - E =    0.090   -   0.0598   =   0.0301897
upper limit = (p̂1 - p̂2) + E =    0.090   +   0.0598   =   0.1498103
                  
so, confidence interval is (   0.030   < p1 - p2 <   0.150   )  
................

................

level of significance, α =   0.05              
Z critical value =   Z α/2 =    1.960   [excel function: =normsinv(α/2)      
                  
Std error , SE =    SQRT(p̂1 * (1 - p̂1)/n1 + p̂2 * (1-p̂2)/n2) =     0.03636          
margin of error , E = Z*SE =    1.960   *   0.0364   =   0.07127
                  
confidence interval is                   
lower limit = (p̂1 - p̂2) - E =    0.090   -   0.0713   =   0.0187317
upper limit = (p̂1 - p̂2) + E =    0.090   +   0.0713   =   0.1612683
                  
so, confidence interval is (   0.019   < p1 - p2 <   0.161   )  
.................

level of significance, α =   0.01              
Z critical value =   Z α/2 =    2.576   [excel function: =normsinv(α/2)      
                  
Std error , SE =    SQRT(p̂1 * (1 - p̂1)/n1 + p̂2 * (1-p̂2)/n2) =     0.03636          
margin of error , E = Z*SE =    2.576   *   0.0364   =   0.09366
                  
confidence interval is                   
lower limit = (p̂1 - p̂2) - E =    0.090   -   0.0937   =   -0.0036625
upper limit = (p̂1 - p̂2) + E =    0.090   +   0.0937   =   0.1836625
                  
so, confidence interval is (   -0.004   < p1 - p2 <   0.184   )  

...............