Answer to Question #188272 in Statistics and Probability for padma raga. A

Question

1. Before an increase in excise duty on tea, 800 persons out of a sample of 1000 persons were found to be tea drinkers. After an increase in duty 800 people were tea drinkers in the sample of 1200 people. Using standard error of proportions state whether there is a significant decrease in the consumption of tea after the increase in the excise duty. 

Answer 1

Before increasing excise duty-

 $n_1=100, x_1=800$

After excise duty-

$n_2=1200,x_2=800$

Sample proportion of taking coffee before increasing excise duty-

$=p_1=\dfrac{x_1}{n_1}+\dfrac{800}{1000}=0.8$

Sample proportion of taking coffee after increasing excise duty-

$=p_2=\dfrac{x_2}{n_2}=\dfrac{800}{1200}=0.67$

Let 

Null hypothesis:$H_o:p_1=p_2=p$ i.e. there is no significant difference in the consumption of coeefe before and after the increase in excise duty.

Alternate Hypothesis:$H_1=P_1>P_2$ i.e. there is significant difference in the consumption of coeefe before and after the increase in excise duty.

Test statistics is-

$Z=\dfrac{p_1-p_2}{\sqrt{\hat{p}(1-\hat{p})(\dfrac{1}{n_1}+\dfrac{1}{n_2})}}$

Where,$\hat{p}=\dfrac{n_1p_1+n_2p_2}{n_1+n_2}=\dfrac{x_1+x_2}{n_1+n_2}=\dfrac{800+800}{1000+1200}=\dfrac{1600}{2200}=0.73$

so , $Z=\dfrac{0.8-0.67}{\sqrt{0.73\times 0.23(\dfrac{1}{1000}+\dfrac{1}{1200}})}=\dfrac{0.13}{\sqrt{0.0031}}=7.22$

Tabulated value of z at 5% level of significance for right tailed test is $1.645 i.e. Z_{0.05}=1.645$

Conclusion: Since Calculated value of Z is greater than the tabulated value of Z, It is significant and $H_o$ is rejected i.e.There has been a significant decrease in the consumption of coeffe.