Respuesta :
Answer:
The standard error for the difference in proportions of males and females who use public transportation every working day is 0.015.
The conditions are met.
Step-by-step explanation:
The sample 1 (males), of size n1=2570 has a proportion of p1=0.389.
[tex]p_1=X_1/n_1=1000/2570=0.389[/tex]
The sample 2 (females), of size n2=2055 has a proportion of p2=0.745.
[tex]p_2=X_2/n_2=1532/2055=0.745[/tex]
The difference between proportions is (p1-p2)=-0.356.
[tex]p_d=p_1-p_2=0.389-0.745=-0.356[/tex]
The pooled proportion, needed to calculate the standard error, is:
[tex]p=\dfrac{X_1+X_2}{n_1+n_2}=\dfrac{1000+1532}{2570+2055}=\dfrac{2532}{4625}=0.547[/tex]
The estimated standard error of the difference between means is computed using the formula:
[tex]s_{p1-p2}=\sqrt{\dfrac{p(1-p)}{n_1}+\dfrac{p(1-p)}{n_2}}=\sqrt{\dfrac{0.547*0.453}{2570}+\dfrac{0.547*0.453}{2055}}\\\\\\s_{p1-p2}=\sqrt{0.000096+0.000121}=\sqrt{0.000217}=0.015[/tex]
Conditions for a normal distribution approximation:
The expected number of "failures" or "successes", whichever is smaller, has to be larger than 10.
For the males sample, we have p=0.389 and (1-p)=0.611. The sample size is n=2570, so we take the smallest proportion and chek the condition:
[tex]n\cdot p=2570\cdot 0.389=999>10[/tex]
For the females sample, we have p=0.745 and (1-p)=0.255. The sample size is n=2055, so we take the smallest proportion and chek the condition:
[tex]n\cdot (1-p)=2055\cdot 0.255=524>10[/tex]
The conditions are met.
