Question

In a study of obesity the following results were obtained from samples of males and females between the ages of 20 and 75:

              n         No. Overweight

Male   150           21

Female 200         48

Can we conclude from these data that in the sampled populations there is a difference in the proportions who are overweight? Let α=0.05

Answer 1

Test For Significance of Different Of Proportion

Step1: Set Up Hypothesis

There Is No Significance between them - Under The Null Hypothesis $H_0: p_1 = p_2$

There Is Significance between them - Under The Alternate Hypothesis $H_1: p_1 \neq p_2$

Step2: Test Statistic

Sample 1

Probability Success $( X_1 )=21$

No. of observaed $(n_1)=150$

$P_1= X_1/n_1=0.14$

Sample 2

Probability Success $(X_2)=48$

No. of Observaed $(n_2)=200$

$P_2= X_2/n_2=0.24$

Finding a $\hat P$ value For Proportion $\hat P=\dfrac{X_1+X_2}{n_1+n_2}=0.1971$

$\hat Q$ Value For Proportion= $1-\hat P=0.8029$

if n>30 So, we use Test Statistic $(Z) = \dfrac{(P_1-P_2)}{(\sqrt{\hat P\cdot \hat Q(\frac{1}{n_1}+\frac{1}{n_2}}))}=\dfrac{0.14-0.24}{\sqrt{0.197\times 0.8029(\frac{1}{150}+\frac{1}{200})}}$

$Z_{ cal}=-2.32711\\ | Z _{cal} | =2.3271$

Step3: Tabulated Value

The Value of $|Z_{ tab}|$ at LOS 0.05% is 1.96

We got $|Z _{cal}| =2.3271$ & $| Z_{ tab} | =1.96$

Step4: Make Decision

Hence Value of $| Z_{ cal} | > | Z _{tab}|$ and Here we Accept Ho

There Is Significance between them, i.e there is a difference in the proportions

 P-Value

Two Tailed ( double the one tail ):

$H_1 : ( P \neq -2.3271 ) = 0.02$