The Organization for Economic Cooperation and Development (OECD) conducts studies on unemployment rates by country and publishes its findings in the document Main Economic Indicators. Independent random samples of $100$and$75$ people in the civilian labor forces of Finland and Denmark, respectively, revealed $7$and $3$ unemployed, respectively. Find a $95\%$ confidence interval for the difference between the unemployment rates in Finland and Denmark.

Given in the question that, The Organization for Economic Cooperation and Development (OECD) conducts studies on unemployment rates by country and publishes its findings in the document Main Economic Indicators. Independent random samples of$100$and $75$people in the civilian labor forces of Finland and Denmark, respectively, revealed $7$and $3$unemployed, respectively.

We need to find a $95\%$confidence interval for the difference between the unemployment rates in Finland and Denmark.

The given values are, $x_1=7,n_1=100,x_2=3,n_2=75$, and $95\%$confidence interval.

The formula for ${\tilde{p}}_1$is given by,

${\tilde{p}}_1=\frac{x_1+1}{n_1+2}$

Substitute $x_1=7,n_1=100$

$=\frac{7+1}{100+2}$

$=0.078$

The formula for ${\tilde{p}}_2$is given by,

${\tilde{p}}_2=\frac{x_2+1}{n_2+2}$

Substitute $x_2=3,n_2=75$

$=\frac{3+1}{75+2}$

$=0.051$

As a result, $0.078$and $0.051$are the sample proportions.

When both sample sizes are 5 or higher, you can utilise the two proportions plus-four $z$confidence interval technique.

Foe a confidence level of $(1-\alpha )$the plus-four $z$confidence interval for $P_1-P_2$is

For the difference between the two-population proportion, the needed confidence interval is determined as

$\left({\tilde{p}}_1-{\tilde{p}}_2\right)\pm z_{\alpha /2}·\sqrt{\frac{{\tilde{p}}_1\left(1-{\tilde{p}}_1\right)}{n_1+2}+\frac{{\tilde{p}}_2\left(1-{\tilde{p}}_2\right)}{n_2+2}}=(0.078-0.051)\pm 1.960$

$.\sqrt{\frac{0.078(1-0.078)}{102}+\frac{0.051(1-0.051)}{79}}$

$=0.028\pm 0.036$

$=-0.008$to $0.064$

Therefore, the difference between the percentage of Finland and Denmark is $-0.008$ to$0.064$