Question

A survey of 900 adults from a certain region​ asked, "What do you buy from your mobile​ device?" The results indicated that 48​% of the females and 40​% of the males answered clothes. The sample sizes of males and females were not provided. Suppose that of 300 ​females, 144 reported they buy clothing from their mobile​ device, while of 600 ​males, 240 reported they buy clothing from their mobile device. Complete parts​ (a) through​ (d) below.

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 0.1 level of​ significance? State the null and alternative​ hypotheses, where pi 1 is the population proportion of females who said they buy clothing from their mobile device and pi 2 is the population proportion of males who said they buy clothing from their mobile device. Determine the value of the test statistic. Upper Z Subscript STATequals nothing ​(Type an integer or a decimal. Round to two decimal places as​ needed.)

Determining the critical values depends on the level of​ significanceUse technology to find the critical​ values, rounding to two decimal places and state the conclusion.

b. Find the​ p-value in​ (a) and interpret its meaning.Use this information to interpret the meaning of the​ p-value.

c. Construct and interpret a 99​% confidence interval estimate for the difference between the proportion of males and females who said they buy clothing from their mobile device. Use this information and the previous results to interpret the confidence interval.

d. What are your answers to​ (a) through​ (c) if 432 males said they buy clothing from their mobile​ device? Is there evidence of a difference between males and females in the proportion who said they buy clothing from their mobile device at the 0.01 level of​ significance?

Determine the value of the test statistic. Determine the proportion of items of interest in sample​ 2, p 2.determine the value of the test​ statistic, rounding to two decimal places.

Determine the critical​ value(s) for this test of hypothesis.Find the​ p-value in​ (a) and interpret its meaning.Use the previous results to interpret the meaning of this​ p-value.

Construct and interpret a 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

a)

Ho:   p1 - p2 =   0          
Ha:   p1 - p2 ╪   0          
                  
sample #1   ----->              
first sample size,     n1=   300          
number of successes, sample 1 =     x1=   144          
proportion success of sample 1 , p̂1=   x1/n1=   0.4800          
                  
sample #2   ----->              
second sample size,     n2 =    600          
number of successes, sample 2 =     x2 =    240          
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.4000          
                  
difference in sample proportions, p̂1 - p̂2 =     0.4800   -   0.4000   =   0.0800
                  
pooled proportion , p =   (x1+x2)/(n1+n2)=   0.4267          
                  
std error ,SE =    =SQRT(p*(1-p)*(1/n1+ 1/n2)=   0.03497          
Z-statistic = (p̂1 - p̂2)/SE = (   0.080   /   0.0350   ) =   2.29
                  
z-critical value , Z* =        1.64 [excel formula =NORMSINV(α/2)]      

test stat > critical value, so reject Ho

conclusion: there is evidence of a difference between males and females in the proportion who said they buy clothing from their mobile device at the 0.1 level of​ significance

b)

p-value =        0.0222   [excel formula =2*NORMSDIST(z)]  

c)

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) =     0.03510          
margin of error , E = Z*SE =    2.576   *   0.0351   =   0.0904
                  
confidence interval is                   
lower limit = (p̂1 - p̂2) - E =    0.080   -   0.0904   =   -0.0104
upper limit = (p̂1 - p̂2) + E =    0.080   +   0.0904   =   0.1704
                  
so, confidence interval is (   -0.0104   < p1 - p2 <   0.1704   )  

d)

Ho:   p1 - p2 =   0          
Ha:   p1 - p2 ╪   0          
                  
sample #1   ----->              
first sample size,     n1=   300          
number of successes, sample 1 =     x1=   144          
proportion success of sample 1 , p̂1=   x1/n1=   0.4800          
                  
sample #2   ----->              
second sample size,     n2 =    600          
number of successes, sample 2 =     x2 =    432          
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.7200          
                  
difference in sample proportions, p̂1 - p̂2 =     0.4800   -   0.7200   =   -0.2400
                  
pooled proportion , p =   (x1+x2)/(n1+n2)=   0.6400          
                  
std error ,SE =    =SQRT(p*(1-p)*(1/n1+ 1/n2)=   0.03394          
Z-statistic = (p̂1 - p̂2)/SE = (   -0.240   /   0.0339   ) =   -7.07
                  
z-critical value , Z* = ± 2.58 [excel formula =NORMSINV(α/2)]      

Z stat <-2.58, so, reject Ho

there is evidence of a difference between males and females in the proportion who said they buy clothing from their mobile device at the 0.01 level of​ significance

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p-value =        0.0000
decision :    p-value<α,Reject null hypothesis   
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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) =     0.03418          
margin of error , E = Z*SE =    2.576   *   0.0342   =   0.0880
                  
confidence interval is                   
lower limit = (p̂1 - p̂2) - E =    -0.240   -   0.0880   =   -0.3280
upper limit = (p̂1 - p̂2) + E =    -0.240   +   0.0880   =   -0.1520
                  
so, confidence interval is (   -0.3280   < p1 - p2 <   -0.1520   )